The correct answer is a) number of charge carriers can change with temperature T b) time interval between two successive collisions can depend on T

### Temperature dependence of resistivity ρ(T) of semiconductors, insulators, and metals is significantly based on the following factors:

### A rectangular box lies on a rough inclined surface. The coefficient of friction between the surface and the box is μ. Let the mass of the box be m. a) at what angle of inclination θ of the plane to the horizontal will the box just start to slide down the plane? b) what is the force acting on the box down the plane, if the angle of inclination of the plane is increased to a > θ c) what is the force needed to be applied upwards along the plane to make the box either remain stationary or just move up with uniform speed? d) what is the force needed to be applied upwards along the plane to make the box move up the plane with acceleration a.

a) As the box starts to slide down the plane, $\mu=\tan \theta$ $ \theta=\tan ^{-1}(\mu) $ b) If $a>\theta$, the angle of inclination will be the angle of repose and the net force acting will be...

### There are four forces acting at a point P produced by strings as shown in the figure, which is at rest. Find the forces F1 and F2.

Because the point is at rest with a = 0, the resulting forces on the point are zero. As a result, the X and Y axis net components will be zero. It's difficult to resolve all of the forces along the...

### A cricket bowler releases the ball in two different ways a) giving it only horizontal velocity and b) giving it horizontal velocity and a small downward velocity. The speed vs at the time of release is the same. Both are released at a height H from the ground. Which one will have greater speed when the ball hits the ground? Neglect air resistance.

a) $\frac{1}{2} v_{z}^{2}=g H \Rightarrow v_{z}=\sqrt{2 g H}$ Speed at ground is given as: $\sqrt{v_{s}^{2}+v_{z}^{2}}=\sqrt{v_{s}^{2}+2 g H}$ b)$\frac{1}{2} m v_{s}^{2}+m g H$ is the total energy...

### The displacement vector of a particle of mass m is given bya) show that the trajectory is an ellipse b) show that

a) $ \begin{array}{l} r(t)=\hat{i} A \operatorname{coswt}+\hat{j} B \sin \omega t \\ x=A \cos \omega t \\ y=B \sin \omega t \\ x / A=\cos \omega t \\ y / B=\sin \omega t \end{array} $ Squaring both...

### There are three forces F1, F2, and F3 acting on a body, all acting on a point P on the body. The body is found to move with uniform speed. a) show that the forces are coplanar b) show that the torque acting on the body about any point due to these three forces is zero

a) The body's acceleration is zero because the resultant force of the three forces F1, F2, and F3 on a location on the body is zero. The directions of forces F1 and F2 are in the plane of the paper,...

### A person in an elevator accelerating upwards with an acceleration of 2 m/s2, tosses a coin vertically upwards with a speed of 20 m/s. After how much time will the coin fall back into his hand?

a = 2 m/s2 is the rising acceleration of an elevator. Gravitational acceleration, g = 10 m/s2 As a result, the net effective acceleration, a' = (a + g) = 12 m/s2, is calculated. In light of the...

### Figure shows (x,t) (y,t) diagram of a particle moving in 2-dimension. If the particle has a mass of 500 g, find the force acting on the particle.

For graph a) $ \begin{array}{l} v_{x}=d x / d t=2 / 2=1 \mathrm{~m} / \mathrm{s} \\ a_{x}=d^{2} x / d t^{2}=d v_{x} / d t=0 \end{array} $ For graph b) $ \begin{array}{l} y=t^{2} \\ v_{y}=d y / d t=2...

### A 100 kg gun fires a ball of 1 kg horizontally from a cliff of height 500 m. It falls on the ground at a distance of 400 m from the bottom of the cliff. Find the recoil velocity of the gun.

The ball's horizontal speed is = u m/s. The vertical component is equal to zero. Considering the ball's vertical downward motion $ \begin{array}{l} u=0 \\ s=h=500 \mathrm{~m} \\ g=10 \mathrm{~s} /...

### Block A of weight 100N rests on a frictionless inclined plane of slope angle 30o. A flexible cord attached to A passes over a frictionless pulley and is connected to block B of weight W. Find the weight W for which the system is in equilibrium.

Equilibrium between $A$ or $B$, Then we know that, $\mathrm{mg} \sin 30^{\circ}=\mathrm{F}$ $ \begin{array}{l} 1 / 2 \mathrm{mg}=\mathrm{F} \\ \mathrm{F}=(1 / 2)(100)=50 \end{array} $ Therefore,...

### In the above-given problem if the lower thread is pulled with a jerk, what happens?

The thread CD breaks when the lower thread is jerked. The thread is inertia at rest, and the mass 2 kg above acts on it.

### A mass of 2 kg is suspended with thread AB. Thread CD of the same type is attached to the other end of 2 kg mass. Lower thread is pulled gradually harder and harder in the downward direction so as to apply force on AB. Which of the threads will break and why?

As the mass 2 kg acts downward, the force acting on the thread AB is equal to the force F. As a result, the force exerted on the AB is 2 kg more than on the D, and the thread AB breaks.

### Why are mountain roads generally made winding upwards rather than going straight up?

Because the force of friction is strong, which decreases the risks of sliding, mountain roads are typically built curving uphill rather than straight up. The value of frictional force increases when...

### A woman throws an object of mass 500 g with a speed of 25 m/s. a) what is the impulse imparted to the object? b) if the object hits a wall and rebounds with half the original speed, what is the change in momentum of the object?

a) Mass of the object, $\mathrm{m}=500 \mathrm{~g}=0.5 \mathrm{~kg}$ $ u=0 $ $ \mathrm{v}=25 \mathrm{~m} / \mathrm{s} $ Impulse is given as: $ \underset{F}{\vec{F}} \cdot d t=\frac{d \rightarrow}{d...

### Why are porcelain objects wrapped in paper or straw before packing for transportation?

Because porcelain artefacts are delicate, they are wrapped in paper or straw before being packed for shipment. When the acceleration (v – u)/t drops while carrying these items, the shift in velocity...

### A man wants to reach from A to the opposite comer of the square C. The sides of the square are 100 m. A central square of 50 m × 50 m is filled with sand. Outside this square, he can walk at a speed 1 m/s. In the central square, he walk only at a speed of v m/s. What is smallest value of v for which he can reach faster via a straight path through the sand than any path in the square outside the sand?

Answer: As depicted in the diagram, APQC represents the path taken by the guy through the sand, the time it took him to get from A to C, and the distance travelled by him. $$ \begin{aligned}...

### A river is flowing due east with a speed 3 m/s. A swimmer can swim in still water at a speed of 4 m/s.c) from two different cases as mentioned in a) and b) above, in which case will he reach opposite bank in shorter time?

Answer: c) In case (a), Time taken by the swimmer to cross the river, $t_{1}=\frac{d}{v_{s}}=\frac{d}{4} s$ In case (b), Time taken by the swimmer to cross the river,...

### A river is flowing due east with a speed 3 m/s. A swimmer can swim in still water at a speed of 4 m/s. a) if swimmer starts swimming due north, what will be his resultant velocity? b) if he wants to start from point A on south bank and reach opposite point B on north bank, i) which direction should he swim? ii) what will be his resultant speed?

Answer: Given, The river's velocity, vr, is 3 meters per second. vs = 4 m/s is the speed of the swimmer. a) When the swimmer swims due north, the y-component will have a velocity of 4 meters per...

### A girl riding a bicycle with a speed of 5 m/s towards north direction, observes rain falling vertically down. If she increases her speed to 10 m/s, rain appears to meet her at 45o to the vertical. What is the speed of the rain? In what direction does rain fall as observed by a ground based observer?

Suppose that Vrg is the velocity of the rain drop that appears to the female observer. All of the vectors are drawn with reference to the frame from the ground up, save for one. Let's say the rain...

### A particle falling vertically from a height hits a plane surface inclined to horizontal at an angle θ with speed v0 and rebounds elastically. Find the distance along the plane where it will hit second time.

Answer: When the x and y axes are set in the manner depicted in the diagram, the missile moves from point O to point A. y = 0 uy = vo cos θ ay = -g cos θ t = T Using the kinematics for the y-axis y...

### A particle is projected in air at an angle β to a surface which itself is inclined at an angle α to the horizontal. a) find an expression of range on the plane surface b) time of flight c) β at which range will be maximum

Answer: This problem has two solutions: i)The point P where the particle impacts the plane is the parabola/straight line intersection. In other words, it's not a straight line. ii) It...

### A gun can fire shells with maximum speed and the maximum horizontal range that can be achieved is . If a target farther away by distance has to be hit with the same gun, show that it could be achieved by raising the gun to a height at least

Answer: $R=\frac{v_{0}^{2}}{g}$ is the maximum range. As a result, the projection angle is 45 degrees. The gun is raised to a height of h in order to hit the target. Negative is used to the vertical...

### A hill is 500 m high. Supplies are to be sent across the hill using a canon that can hurl packets at a speed of 125 m/s over the hill. The canon is located at a distance of 800 m from the foot of hill and can be moved on the ground at a speed of 2 m/s so that its distance from the hill can be adjusted. What is the shortest time in which a packet can reach on the ground across the hill? Take g = 10 m/s2.

Answer: The speed of packets is 125 m/s, the height of the hill is 500 m, and the distance between the cannon and the foot of the hill is 800 m, according to the problem. Ideally, the vertical...

### If and , then match the relations in column with the angle between and in column .

Column I Column II a) |A×B|=0 i) θ = 30o b) |A×B|=8 ii) θ = 45o c) |A×B|=4 iii) θ = 90o d) |A×B|=4√2 iv) θ = 0o Answer: a) matches with iv) b) matches with iii) c) matches with i) d) matches with...

### If and , then match the relations in column I with the angle between between and in column II.

Column I Column II a) A.B = 0 i) θ = 0 b) A.B = +8 ii) θ = 90o c) A.B = 4 iii) θ = 180o d) A.B = -8 iv) θ = 0o Answer: a) matches with ii) b) matches with i) c) matches with iv) d) matches...

### Given below in column I are the relations between vectors a, b, and c and in column II are the orientations of a, b, and c in the XY plane. Match the relation in column I to correct orientations in column II.

Answer: a) matches with iv) b) matches with iii) c) matches with i) d) matches with ii)

### a) Earth can be thought of as a sphere of radius 6400 km. Any object is performing circular motion around the axis of earth due to earth’s rotation. What is acceleration of object on the surface of the earth towards its centre? What is it at latitude θ? How does these accelerations compare with g = 9.8 m/s2? b) Earth also moves in circular orbit around sun once every year with an orbital radius of 1.5 × 1011m. What is the acceleration of earth towards the centre of the sun? How does this acceleration compare with g = 9.8 m/s2?

Answer: (a) According to the question, we have been given that, Radius of the earth $(R)=6400 km =6.4 \times 10^{6} m$. Time period of the motion $(T)=1$ day $=24 \times 60 \times 60 s =86400 s$ As...

### A fighter plane is flying horizontally at an altitude of 1.5 km with speed 720 km/h. At what angle of sight when the target is seen, should the pilot drop the bomb in order to attack the target?

Answer: Given, u = 720 km/h = 200 m/s If t is the moment at which the pilot detonates the bomb, then Q is the point vertically above the target T at which the bomb is dropped. The bomb's horizontal...

### In dealing with motion of projectile in air, we ignore effect of air resistance on motion. This gives trajectory as a parabola as you have studied. What would the trajectory look like if air resistance is included? Sketch such a trajectory and explain why you have drawn it that way.

The vertical and horizontal velocity of a projectile reduces due to air resistance. The formula for lowering the height of motion is as follows: R = (u2/g) sin 2θ Hmax = u2 sin2 θ/2g The graphic...

### A boy throws a ball in air at 60o to the horizontal along a road with a speed of 10 m/s. Another boy sitting in a passing by car observes the ball. Sketch the motion of the ball as observed by the bot in the car, if car has a speed of 18 km/h. Give explanation to support your diagram.

Answer: Given, u = 36 km/h = 10 m/s ux = u cos 60o = 5 m/s Speed of the car in the direction of motion of ball = (18)(5/18) = 5 m/s The boy throws a ball when a car goes by. ...

### A boy travelling in an open car moving on a labelled road with constant speed tosses a ball vertically up in the air and catches it back. Sketch the motion of the ball as observed by a boy standing on the footpath. Give explanation to support your diagram.

Answer: Given, v denotes the vertical velocity of the ball that the boy is holding. In this equation, u = horizontal velocity of the ball multiplied by the velocity of the car. The diagram shown...

### A, B, and C are three non-collinear, non co-planar vectors. What can you say about direction of A(B×C)?

Answer: When vector B and C are added together, the direction of the product is perpendicular to the plane that contains vector B and C, which is based on the Right hand grip rule. In this case, the...

### A football is kicked into the air vertically upwards. What is its a) acceleration b) velocity at the highest point

Answer: a) When a football is kicked vertically upwards into the air, as depicted in the diagram below, the situation is considered to be "upwards." It is called acceleration owing to gravity when...

### A ball is thrown from a roof top at an angle of 45o above the horizontal. It hits the ground a few seconds later. At what point during its motion, does the ball have a) greatest speed b) smallest speed c) greatest acceleration? Explain

Answer: a) Assume the ball is thrown $45^o$ from O. The ball's height increases from O to A, lowering its KE. The speed increases as the height lowers from A to B, until it reaches the initial speed...

### A particle is projected in air at some angle to the horizontal, moves along parabola as shown in the figure, where x and y indicate horizontal and vertical directions respectively. Show in the diagram, direction of velocity and acceleration at points A, B, and C.

Answer: The projectile motion is parabolic. The velocity is always tangential to A, B, and C. The trajectory reaches its greatest height at B. So Bvy = 0 and u cos. We know that acceleration follows...

### A cyclist starts from centre O of a circular park of radius 1 km and moves along the path OPRQO as shown in the figure. If he maintains constant speed of 10 m/s, what is his acceleration at point R in magnitude and direction?

Answer: The circular path has a radius of 1 km, with O as the center and R as the distance between the two points. The biker is moving at a speed of 10 meters per second....

### For two vectors A and B, is always true when a)

b)

c) and and are parallel or antiparallel d) when either or is zero

Answer: The correct option is a) & d) Given in question $|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|$, it can be assertive when $| A |=0$ or $| B |=0$ or both are zero. The provided statement can also...

### A body of mass 10 kg is acted upon by two perpendicular forces, 6N and 8N. The resultant acceleration of the body is a) 1 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force b) 0.2 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force c) 1 m/s2 at an angle of tan-1(3/4) w.r.t 8N force d) 0.2 m/s2 at an angle of tan-1(3/4) w.r.t 8N force

The correct answers are a) 1 m/s2 at an angle of tan-1 (4/3) w.r.t 6N force c) 1 m/s2 at an angle of tan-1(3/4) w.r.t 8N force

### For a particle performing uniform circular motion, choose the correct statement from the following: a) magnitude of particle velocity (speed) remains constant b) particle velocity remains directed perpendicular to radius vector c) direction of acceleration keeps changing as particle moves d) angular momentum is constant in magnitude but direction keep changing

Answer: The correct answer is a) magnitude of particle velocity (speed) remains constant, b) particle velocity remains directed perpendicular to radius vector and c) direction of acceleration keeps...

### Following are four different relations about displacement, velocity, and acceleration for the motion of a particle in general. Choose the incorrect one (s): a) = 1/2 [v(t1) + v(t2)] b) = r(t2)-r(t1)/t2-t2 c) r = 1/2 [v(t2)-v(t1)](t2-t1) d) = v(t2)-v(t1)/t2-t1

Answer: \text { The correct answer is a) } v_{ av }=1 / 2\left[ v \left( t _{1}\right)+ v \left( t _{2}\right)\right] \text { and c) } r =1 / 2\left[ v \left( t _{2}\right)- v \left( t...

### A particle slides down a frictionless parabolic track starting from rest at point A. Point B is at the vertex of parabola and point C is at a height less than that of point A. After C, the particle moves freely in air as a projectile. If the particle reaches highest point at P, then a) KE at P = KE at B b) height at P = height at A c) total energy at P = total energy at A d) time of travel from A to B = time of travel from B to P

Answer: The correct answer is c) total energy at P = total energy at A Because energy is always conserved (unless in inelastic collisions), the total energy at A and P will always be equal....

### Two particles are projected in air with speed , at angles θ1 and θ2 to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then tick the right choices a) angle of project: q1 > q2 b) time of flight: T1 > T2 c) horizontal range: R1 > R2 d) total energy: U1 > U2

Answer: The correct answer is a) angle of the project: q1 > q2 and b) time of flight: T1 > T2 Assuming this is true, two particles are pushed into the air at a speed of u and at angles of 1...

### A hockey player is moving northward and suddenly turns westward with the same speed to avoid an opponent. The force that acts on the player is a) frictional force along westward b) muscle force along southward c) frictional force along south-west d) muscle force along south-west

The correct answer is c) frictional force along the south-west

### It is found that . This necessarily implies a) b) A, B are antiparallel c) A, B are perpendicular d) A.B

Answer: The correct answer is a) B = 0 Its mean vector B will have 0 magnitudes, and there will be no direction in which it moves. A→+B→=A→\overrightarrow{ A }+\overrightarrow{...

### Three vectors A, B, and C add up to zero. Find which is false, a) vector is not zero unless vectors are parallel d) vector is not zero unless vectors are parallel c) if vectors define a plane, is in that plane d) such that

Answer: The correct answer is $c$ ) if vectors $A, B, C$ define a plane, $(A \times B) C$ is in that plane and d) $(A \times B) \cdot C=$ $|A \| B||C| \quad$ such that $C ^{2}= A ^{2}+ B ^{2}$ Given...

### In a two dimensional motion, instantaneous speed is a positive constant. Then which of the following are necessarily true? a) the acceleration of the particle is zero b) the acceleration of the particle is bounded c) the acceleration of the particle is necessarily in the plane of motion d) the particle must be undergoing a uniform circular motion

Answer: The correct answer is d) the particle must be undergoing a uniform circular motion In two dimensions, instantaneous speed $v_0$ is positive. The particle's acceleration must be in the plane...

### In a two dimensional motion, instantaneous speed is a positive constant. Then which of the following are necessarily true? a) the average velocity is not zero at any time b) average acceleration must always vanish c) displacements in equal time intervals are equal d) equal path lengths are traversed in equal intervals

Answer: The correct answer is d) equal path lengths are traversed in equal intervals Given that the immediate speed $v_0$ is a positive constant, this motion is two-dimensional. Because acceleration...

### Consider the quantities pressure, power, energy, impulse gravitational potential, electric charge, temperature, area. Out of these, the only vector quantities are a) impulse, pressure, and area b) impulse and area c) area and gravitational potential d) impulse and pressure

Answer: The correct answer is b) impulse and area We know that impulse is defined as $J = F \cdot \Delta t =\Delta p$, where $F$ is force, At is time length, and $\Delta p$ represents momentum...

### A satellite is in an elliptic orbit around the earth with an aphelion of 6R and perihelion of 2R where R = 6400 km is the radius of the earth. Find eccentrically of the orbit. Find the velocity of the satellite at apogee and perigee. What should be done if this satellite has to be transferred to a circular orbit of radius 6R?

Solution: Radius of perigee is given as $r_{p}=2R$ Radius of apogee is given as $r_{a}=6R$ And we know, $r_{p}=a(1-e)=2R$ and, $r_{a}=a(1+e)=6R$ From the above equations, we get $e = 1/2$ From the...

### The horizontal range of a projectile fired at an angle of 15o is 50 m. If it is fired with the same speed at an angle of 45 degrees, its range will be a) 60 m b) 71 m c) 100 m d) 141 m

Answer: The correct answer is c) 100 m Given: The horizontal range of a projectile that is fixed at an angle to the horizontal plane $=15^{\circ}$ Range $=50 m$ In the first case range $=50= u ^{2}...

### The component of a vector r along X-axis will have maximum value if a) r is along positive Y-axis b) r is along positive X-axis c) r makes an angle of 45 degrees with the X-axis d) r is along negative Y-axis

Answer: The correct answer is b) r is along positive X-axis Therefore, the component should have an angle of 0 degrees with respect to the axis specified here, and the component of the vector along...

### Figure shows the orientation of two vectors u and v in the XY plane. If and which of the following is correct? a) and are positive while and are negative b) , and are positive while is negative c) , and are positive while is negative d) , and are all positive

Answer: B) The tail is at the origin, and the x- and y-components are projected on the positive x- and y-axes. So a and b are yes. Now translate v so that its orientation is unaltered and its tail...

### Earth’s orbit is an ellipse with eccentricity 0.0167. Thus, the earth’s distance from the sun and speed as it moves around the sun varies from day to day. This means that the length of the solar day is not constant throughout the year. Assume that earth’s spin axis is normal to its orbital plane and find out the length of the shortest and the longest day. A day should be taken from noon to noon. Does this explain the variation of the length of the day during the year?

Solution: Velocity of the earth at perigee is given as $v_{p}$ Velocity of the earth at apogee is given as $v_{a}$ Angular velocity of the earth at perihelion is given as $\omega_{p}$ Angular...

### Which one of the following statements is true? a) a scalar quantity is the one that is conserved in a process b) a scalar quantity is the one that can never take negative values c) a scalar quantity is the one that does not vary from one point to another in space d) a scalar quantity has the same value for observers with different orientations of the axes

Answer: The correct answer is d) a scalar quantity has the same value for observers with different orientations of the axes (a) Scalars can have both positive and negative values, for example,...

### The angle between and is a) b) c) d)

Answer: The correct answer is b) 90o Given vectors in the question, $\overrightarrow{ A }=\hat{ i }+\hat{ j }$ and $\overrightarrow{ B }=\hat{ i }-\hat{ j }$...

### A satellite is to be placed in equatorial geostationary orbit around the earth for communication

a) calculate height of such a satellite

b) find out the minimum number of satellites that are needed to cover entire earth so that at least one satellites is visible from any point on the equator

a) Mass of the earth is given as $M=6\times 10^{24}kg$ Radius of the earth is given as $R=6.4 \times 10^{3}m$ Time period is given as $24.36 \times 10^{2}s$ $G=6.67 \times 10^{-11}Nm^{2}kg^{-1}$...

### Six-point masses of mass m each are at the vertices of a regular hexagon of side l. Calculate the force on any of the masses.

$AE = AG + EG$ $AG + AG = 2AG$ $= 2l cos 30^{o}$ $AE = AC = \sqrt3l$ $AD = 2l$ As a result, $\frac{Gm^{2}}{l^{2}}$ is the force on A due to B along B to A $\frac{Gm^{2}}{3l^{2}}$ is the force on A...

### A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let r be the distance of the body from the centre of the star and let its linear velocity be v, angular velocity ω, kinetic energy K, gravitational potential energy U, total energy E, and angular momentum l. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease.

When a body moves around a star in equilibrium, the gravitational attraction produces a centripetal force. Consider a body of mass $m$ revolving in a circular path of radius $r$ around the star S of...

### A mass m is placed at P a distance h along the normal through the centre O of a thin circular ring of mass M and radius r. If the mass is removed further away such that OP becomes 2h, by what factor the force of gravitation will decrease, if h = r?

Solution: Let the radius of the ring be r Let the mass of the ring be m When small element dM is considered as the mass, the gravitation force becomes, $dF=\frac{G(dM)m}{x^{2}}$ where...

### An object of mass m is raised from the surface of the earth to a height equal to the radius of the earth, that is, taken from a distance R to 2R from the centre of the earth. What is the gain in its potential energy?

Potential Energy of the body on the earth is given by $-GMm/R$ When the body is closer to the equator, then PE becomes $-GMm/2R$ Gain in PE is $1/2 mgR$

### Shown are several curves. Explain with reason, which ones amongst them can be possible trajectories traced by a projectile

Solution: Amongst the given figures, (c) show the focus of trajectory.

### Show the nature of the following graph for a satellite orbiting the earth.

TE vs orbital radius R

Total energy of the satellite is $-GMm/2R$

### Show the nature of the following graph for a satellite orbiting the earth.

a) KE vs orbital radius R

b) PE vs orbital radius R

a) $\mathrm{K}=1 / 2 \mathrm{mv}^{2}=(1 / 2 \mathrm{~m})(\mathrm{GM} / \mathrm{R})$ b) PE of satellite is $U = -GMm/R = -2K$

### Two identical heavy spheres are separated by a distance 10 times their radius. Will an object placed at the midpoint of the line joining their centres be in stable equilibrium or unstable equilibrium? Give a reason for your answer.

M = mass of the spheres R = radius of the spheres. P = midpoint of A and B. The magnitude of the force is expressed as, $F_{1}=F_{2}=\frac{GMm}{5R^{2}}$ As the resultant force exerted on the object...

### Mean solar day is the time interval between two successive noon when the sun passes through zenith point. The sidereal day is the time interval between two successive transits of a distant star through the zenith point. By drawing the appropriate diagram showing earth’s spin and orbital motion, show that mean solar day is four minutes longer than the sidereal day. In other words, distant stars would rise 4 minutes early every successive day.

The polar axis of the earth and its movement are E and E’ respectively. Translational motion is P’ After every 24 hours, earth's orbit is approximately advanced by $1^{o}$ As a result, time taken...

### What is the angle between the equatorial plane and the orbital plane of

a) polar satellite?

b) geostationary satellite?

a) The equatorial plane and the orbital plane of a polar satellite form a $90^{o}$ angle. b) A geostationary satellite's equatorial plane and orbital plane are at an angle of $0^{o}$.

### Out of aphelion and perihelion, where is the speed of the earth more and why?

According to Kepler’s second law, a real velocity is constant and is given as: $r_{p}\times v_{p}=r_{A}\times v_{A}$ $\frac{r_{A}}{r_{p}}=\frac{v_{p}}{v_{A}}$ $r_{A}>r_{p}$ and...

### The gravitational force between a hollow spherical shell and a point mass is F. Show the nature of F vs r graph where r is the distance of the point from the centre of the hollow spherical shell of uniform density.

R is the spherical shell's radius, while r is the distance between m and M. The point's mass is m, while the hollow spherical shell's mass is M. Now, $F=\frac{GMm}{r^{2}}$ When F = 0,...

### An astronaut inside a small spaceship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity?

The astronaut will experience variation when the size of the space station orbiting the earth is large, and this is due to acceleration due to gravity.

### We can shield a charge from electric fields by putting it side a hollow conductor. Can we shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?

No. As gravitation is independent of the medium, a body can be shielded from the gravitational pull of adjacent matter.

### Is it possible for a body to have inertia but no weight?

Yes. It is possible for a body to have inertia but no weight as inertia is associated with the mass of the body. Satellite revolving around the earth is an example of a body with inertia and has no...

### How is the gravitational force between two point masses affected when they are dipped in the water keeping the separation between them the same?

The gravitational force between point masses is unaffected by the medium and remains constant regardless of their surroundings. As a result, even if the two-point masses are submerged in water, the...

### What is the direction of areal velocity of the earth around the sun?

The direction of the earth's areal velocity around the sun is determined by the product of r and v.

### Give one example each of central force and non-central force.

Example of Central force: Electrostatic force acting on the point charge Example of Non-central force: Nuclear force between the atoms

### Molecules in the air in the atmosphere are attracted by the gravitational force of the earth. Explain why all of them do not fall into the earth just like an apple falling from a tree.

The gravitational force of the earth attracts molecules in the atmosphere, yet they do not descend into the earth since they are in random motion, whereas an apple moves downhill.

### The centre of mass of an extended body on the surface of the earth and its centre of gravity

a) are always at the same point for any size of the body

b) are always at the same point only for spherical bodies

c) can never be at the same point

d) is close to each other for objects, say of sizes less than 100 m

e) both can change if the object is taken deep inside the earth

The correct option is d) is close to each other for objects, say of sizes less than 100 m

### Supposing Newton’s law of gravitation for gravitation forces F1 and F2 between two masses m1 and m2 at positions r1 and r2 read Exemplar Solutions Physics Class 11 Chapter 8 – 21 where Mo is a constant of the dimension of mass, r12 = r1 – r2 and n is a number. In such a case,

a) the acceleration due to gravity on earth will be different for different object

b) none of the three laws of Kepler will be valid

c) only the third law will become invalid

d) for n negative, an object lighter than water will sink in water

The correct options are a) the acceleration due to gravity on earth will be different for different object c) only the third law will become invalid d) for n negative, an object lighter than water...

### There have been suggestions that the value of the gravitational constant G becomes smaller when considered over a very large time period in the future. If that happens for our earth,

a) nothing will change

b) we will become hotter after billions of years

c) we will be going around but not strictly in closed orbits

d) after a sufficiently long time we will leave the solar system

The correct options are c) we will be going around but not strictly in closed orbits d) after a sufficiently long time we will leave the solar system

### If the sun and the planets carried huge amounts of opposite charges,

a) all three of Kepler’s laws would still be valid

b) only the third law will be valid

c) the second law will not change

d) the first law will still be valid

The correct options are a) all three of Kepler’s laws would still be valid c) the second law will not change d) the first law will still be valid

### If the mass of sun were ten times smaller and gravitational constant G were ten times larger in magnitudes

a) walking on ground would become more difficult

b) the acceleration due to gravity on earth will not change

c) raindrops will fall much faster

d) aeroplanes will have to travel much faster

The correct options are a) walking on ground would become more difficult c) raindrops will fall much faster d) aeroplanes will have to travel much faster

### If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube-law

a) planets will not have elliptic orbits

b) circular orbits of planets is not possible

c) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic

d) there will be no gravitational force inside a spherical shell of uniform density

The correct options are a) planets will not have elliptic orbits c) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic Explanation: The planets...

### Which of the following options is correct?

a) acceleration due to gravity decreases with increasing altitude

b) acceleration due to gravity increases with increasing depth

c) acceleration due to gravity increases with increasing latitude

d) acceleration due to gravity is independent of the mass of the earth

The correct options are a) acceleration due to gravity decreases with increasing altitude c) acceleration due to gravity increases with increasing latitude

### Particles of masses 2M, m and M are respectively at points A, B, and C with AB = 1/2 (BC). M is much-much smaller than M and at time t = 0, they are all at rest. At subsequent times before any collision takes place

a) m will remain at rest b) m will move towards M c) m will move towards 2M d) m will have oscillatory motion Solution: The correct option is c) m will move towards 2M

### Choose the wrong option.

a) inertial mass is a measure of the difficulty of accelerating a body by an external force whereas the gravitational mass is relevant in determining the gravitational force on it by an external mass

b) that the gravitational mass and inertial mass are equal is an experimental result

c) that the acceleration due to gravity on earth is the same for all bodies is due to the equality of gravitational mass and inertial mass

d) gravitational mass of a particle-like proton can depend on the presence of neighbouring heavy objects but the inertial mass cannot

The correct option is d) gravitational mass of a particle-like proton can depend on the presence of neighbouring heavy objects but the inertial mass cannot

### In our solar system, the inter-planetary region has chunks of matter called asteroids. They

a) will not move around the sun since they have very small masses compared to the sun

b) will move in an irregular way because of their small masses and will drift away outer space

c) will move around the sun in closed orbits but not obey Kepler’s laws

d) will move in orbits like planets and obey Kepler’s laws

The correct option is d) will move in orbits like planets and obey Kepler’s laws

### Both earth and moon are subject to the gravitational force of the sun. as observed from the sun, the orbit of the moon

a) will be elliptical

will not be strictly elliptical because the total gravitational force on it is not central

c) is not elliptical but will necessarily be a closed curve

d) deviates considerably from being elliptical due to the influence of planets other than earth

The correct option is b) will not be strictly elliptical because the total gravitational force on it is not central

### Satellites orbiting the earth have a finite life and sometimes debris of satellites fall to the earth. This is because

a) the solar cells and batteries in satellites run out

b) the laws of gravitation predict a trajectory spiralling inwards

c) of viscous forces causing the speed of the satellite and hence height to gradually decrease

d) of collisions with other satellites

The correct option is c) of viscous forces causing the speed of the satellite and hence height to gradually decrease

### Different points in the earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the cm causing translation and a net torque at the cm causing translation and a net torque at the cm causing rotation around an axis through the cm. For the earth-sun system

a) the torque is zero

b) the torque causes the earth to spin

c) the rigid body result is not applicable since the earth is not even approximately a rigid body

d) the torque causes the earth to move around the sun

The correct option is a) the torque is zero

### As observed from earth, the sun appears to move in an approximately circular orbit. For the motion of another planet like mercury as observed from earth, this would

a) be similarly true

b) not be true because the force between earth and mercury is not inverse square law

c) not be true because the major gravitational force on mercury is due to sun

d) not be true because mercury is influenced by forces other than gravitational forces

The correct option is c) not be true because the major gravitational force on mercury is due to sun

### The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity

a) will be directed towards the centre but not the same everywhere

b) will have the same value everywhere but not directed towards the centre

c) will be same everywhere in magnitude directed towards the centre

d) cannot be zero at any point

The correct option is d) cannot be zero at any point

### Consider a Carnot’s cycle operating between T1 = 500K and T2 = 300K producing 1 kJ of mechanical work per cycle. Find the heat transferred to the engine by the reservoirs.

Answer: According to the question, the temperature of the source is T1 = 500 K and the temperature of the sink is T2 = 300 K We also have the work done per cycle as W = 1 kJ = 1000 J we have to...

### Figure shows two identical particles 1 and 2, each of mass m, moving in opposite directions with the same speed v along parallel lines. At a particular instant, and are their respective positions vectors drawn from point A which is in the plane of the parallel lines. Choose the correct options:

a) angular momentum of particle 1 is about A is

b) angular momentum of particle 2 about A is

c) total angular momentum of the system about A is

d) total angular momentum of the system about A is

Solution: Correct answers is: d) total angular momentum of the system about A is $l = mv(d_{2}-d_{1})$ Angular momentum of particle 1 about A is given as, $\vec L_1=mvd_1$ Angular momentum of...

### A raindrop falling from a height h above ground, attains a near-terminal velocity when it has fallen through a height (3/4)h. Which of the diagrams show correctly the change in kinetic and potential energy of the drop during its falls up to the ground?

b)

### Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V as shown:If the collision is elastic, which of the following is a possible result after collision?

\b)

### Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass (m) to energy (E) as , where c is the speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in MeV where , the masses are measured in unified equivalent of 1u is 931.5 MeV.

a) Show that the energy equivalent of 1 u is 931.5 MeV.

b) A student writes the relation as 1 u = 931.5 MeV. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.

a) The energy that is comparable to a given mass can be computed using Einstein's mass-energy relation. $1amu=1u=1.67\times 10^{-27}kg$ On Applying $E=mc^{2}$ we get, E = 931.5 MeV b) As $E=mc^{2}$...

### Mars has approximately half of the earth’s diameter. When it is closer to the earth it is at about ½ AU from the earth. Calculate at what size it will disappear when seen through the same telescope.

$D_{mars}/D_{earth}=1/2$ Also, $D_{earth}/D_{sun}=1/100$ So, $D_{mars}/D_{sun}=1/2\times 1/100$ At 1AU, the sun’s diameter is = (1/2) degree Therefore, diameter of mars will be = (1/400) degree At...

### a) How many astronomical units (AU) make 1 parsec?

b) Consider the sun like a star at a distance of 2 parsec. When it is seen through a telescope with 100 magnification, what should be the angular size of the star? Sun appears to be (1/2) degree from the earth. Due to atmospheric fluctuations, eye cannot resolve objects smaller than 1 arc minute.

a) 1 parsec is the distance at which 1 AU long arc subtends an angle of 1s, according to the definition. Using the definition, we can write, 1 parsec = (3600)(180)/π AU = 206265 AU = 2 × 105 AU b)...

### In an experiment to estimate the size of a molecule of oleic acid, 1mL of oleic acid is dissolved in 19mL of alcohol. Then 1mL of this solution is diluted to 20mL by adding alcohol. Now, 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film we can calculate the thickness of the film which will give us the size of oleic acid molecule.

Read the passage carefully and answer the following questions:

What will be the volume of oleic acid in one drop of this solution?

The volume of oleic acid in one drop is 1/400mL

### In an experiment to estimate the size of a molecule of oleic acid, 1mL of oleic acid is dissolved in 19mL of alcohol. Then 1mL of this solution is diluted to 20mL by adding alcohol. Now, 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film we can calculate the thickness of the film which will give us the size of oleic acid molecule.

Read the passage carefully and answer the following questions:

a) What would be the volume of oleic acid in each mL of solution prepared?

b) How will you calculate the volume of n drops of this solution of oleic.

a) 1 mL of oleic acid is found in every 20 mL of oleic acid. This signifies that 1/20 mL of oleic acid is present in each mL of solution. Adding alcohol dilutes 1 mL of this solution to 20 mL. As a...

### In an experiment to estimate the size of a molecule of oleic acid, 1mL of oleic acid is dissolved in 19mL of alcohol. Then 1mL of this solution is diluted to 20mL by adding alcohol. Now, 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film we can calculate the thickness of the film which will give us the size of oleic acid molecule.

Read the passage carefully and answer the following questions:

a) Why do we dissolve oleic acid in alcohol?

b) What is the role of lycopodium powder?

a) Because oleic acid does not dissolve in water, it is dissolved in alcohol. b) When oleic acid is introduced, lycopodium powder clears the circular area. This makes it possible to measure the area...

### An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that T = k/R √r3/g where k is a dimensionless constant and g is acceleration due to gravity.

Kepler's third law states that, $T^{2} \propto a^{3}$ i.e., square of time period $\left(T^{2}\right)$ of a satellite revolving around a planet, is proportional to the cube of the radius of the...

### If the velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

We will us Principle of homogeneity for solving this problem. $\begin{array}{l} {[\mathrm{h}]=\left[\mathrm{ML}^{2}...

### In the expression , E, m, l and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.

We have, $P=El^{2}m^{-5}G^{-2}$ E is the energy having dimension $ML^{2}T{-2}$ m is the mass having dimension [M] L is the angular momentum having dimension $[ML^{2}T^{1}]$ G is the gravitational...

### A physical quantity X is related to four measurable quantities a, b, c and d as follows: . The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4%, respectively. What is the percentage error in quantity X? If the value of X calculated on the basis of the above relation is 2.763, to what value should you round off the result.

The given physical quantity is $X=a^{2}b^{3}c^{5/2}d^{-2}$ Percentage error in X is given as (∆x/x)(100) Percentage error in a is given as (∆a/a)(100) = 1% Percentage error in b is given as...

### The volume of a liquid flowing out per second of a pipe of length l and radius r is written by a student as where P is the pressure difference between the two ends of the pipe and η is coefficient of viscosity of the liquid having dimensional formula . Check whether the equation is dimensionally correct.

Dimension of the given physical quantity is as follows, [V] = dimension of volume/dimension of time $=[L^{3}]/[T]$ $=[M^{-1}T^{-2}]$ LHS $=[L^{3}T^{-1}]$ RHS $=[L^{3}T^{-1}]$ LHS = RHS Hence, the...

### A new system of units is proposed in which unit of mass is α kg, unit of length β m and unit of time γ s. How much will 5 J measure in this new system?

Let Q be the physical quantity as $n_{1}u_{1}=n_{2}u_{2}$ Let $M_{1}$, $L_{1}$, $T_{1}$ be the units of mass, length, and time for the first system. and $M_{2}$,$L_{2}$,$T_{2}$ be the units of mass,...

### If the unit of force is 100 N, unit of length is 10 m and unit of time is 100 s, what is the unit of mass in this system of units?

Force [F] is given as 100 N Length [L] is given as 10 m Time [t] is given as 100 s $[F]=[MLT^{2}]$ Substituting the values, we get M = 105 kg

### During a total solar eclipse-the moon almost entirely covers the sphere of the sun. Write the relation between the distances and sizes of the sun and moon.

$R_{me}=$ distance of the moon from the earth $R_{se}=$ distance of the sun from the moon $A_{sun}=$ area of the sun $A_{moon}=$ area of the moon...

### The vernier scale of a travelling microscope has 50 divisions which coincide with 49 main scale divisions. If each main scale division is 0.5 mm, calculate the minimum inaccuracy in the measurement of distance.

The minimum inaccuracy in the measurement of distance = (1/50)(0/5)mm = 0.01 mm

### The distance of a galaxy is of the order of m. Calculate the order of magnitude of time taken by light to reach us from the galaxy.

The distance of the galaxy is given as 1025m Speed of light is known as $3\times 10^{8}m/s$ Time taken is t = distance/speed $=3.33\times 10^{16}s$

### Which of the following time measuring devices is most precise?

(a) A wall clock.

(b) A stopwatch.

(c) A digital watch.

(d) An atomic clock.

Give the reason for your answer.

Correct option is d) an atomic clock as it measures up to one second.

### From parallax measurement, the sun is found to be at a distance of about 400 times the earth-moon distance. Estimate the ratio of sun-earth diameters.

The ratio of sun-earth diameter is $D_{sun}/D_{earth}=100$

### (a) The earth-moon distance is about 60 earth radius. What will be the diameter of the earth (approximately in degrees) as seen from the moon?

(b) Moon is seen to be of (½)° diameter from the earth. What must be the relative size compared to the earth?

(a) Because the distance between the moon and the earth is greater than the radius of the earth, it is considered as an arc. Let the length of the arc be $R_{e}$ Distance between the moon and the...

### Name the device used for measuring the mass of atoms and molecules.

Mass spectrograph is a tool for determining the mass of atoms and molecules.

### The radius of an atom is of the order of 1 Å and radius of nucleus is of the order of fermi. How many magnitudes higher is the volume of the atom as compared to the volume of the nucleus?

Radius of atom is given as $1 Å = 10^{-10} m$ Radius of nucleus is known as $1 fermi = 10^{-15} m$ Volume of atom will be $4/3\pi Ra^{3}$ Volume of nucleus will be $4/3\pi Rn^{3}$...

### Why do we have different units for the same physical quantity?

Because physical quantities differ from location to place, we have several units for the same physical quantity.

### Which of the following are not a unit of time?

a) second

b) parsec

c) year

d) light year

Correct answers are b) parsec and d) light year

### Which of the following ratios express pressure?

a) Force/area

b) Energy/volume

c) Energy/area

d) Force/volume

Correct answers are a) force/area and b) energy/volume

### If Planck’s constant (h) and speed of light in vacuum (c) are taken as two fundamental quantities, which of the following can, also, be taken to express length, mass, and time in terms of the three chosen fundamental quantities?

a) mass of the electron

b) universal gravitational constant

c) charge of the electron

d) mass of proton

Correct answers are a) mass of electron b) universal gravitational constant and d) mass of proton

### Photon is quantum of radiation with energy E = hv where v is frequency and h is Planck’s constant. The dimensions of h are the same as that of:

a) linear impulse

b) angular impulse

c) linear momentum

d) angular momentum

Correct options are b) angular impulse and d) angular momentum

### If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity?

a)

b)

c)

d)

e)

Correct answer is d) $PR-Q^{2}/R$ and e) $(R + Q)/P$

### On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:

a) y = a sin 2πt/T

b) y = a sin vt

c) y = a/T sin (t/a)

d) y = a√2 [sin (2 πt/T) – cos (2πt/T)]

Correct answers are b) y = a sin vt and c) y = a/T sin (t/a)

### If momentum (P), area (A), and time (T) are taken to be fundamental quantities, then energy has the dimensional formula

a)

b)

c)

d)

Correct answer is d) $P^{1}A^{1/2}T^{-1}$

### Young’s modulus of steel is . When expressed in CGS units of , it will be equal to:

a)

b)

c)

d)

Correct answer is c) $1.9\times 10^{12}$ Both Dyne and Newton are force units. While Dyne is measured in the C-G-S (Centimeter – Gram – Second) system, Newton is measured in the contemporary SI...

### The mean length of an object is 5 cm. Which of the following measurements is most accurate?

a) 4.9 cm

b) 4.805 cm

c) 5.25 cm

d) 5.4 cm

Correct answer is a) 4.9 cm

### Which of the following measurements is most precise?

a) 5.00 mm

b) 5.00 cm

c) 5.00 m

d) 5.00 km

Correct answer is a) 5.00 mm

### You measure two quantities as A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m. We should report the correct value for √AB as:

a) 1.4 m ± 0.4 m

b) 1.41 m ± 0.15 m

c) 1.4 m ± 0.3 m

d) 1.4 m ± 0.2 m

Correct answer is d) 1.4 m ± 0.2 m

### Measure of two quantities along with the precision of the respective measuring instrument is:

A = 2.5 m/s ± 0.5 m/s

B = 0.10 s ± 0.01 s

The value of AB will be

a) (0.25 ± 0.08) m

b) (0.25 ± 0.5) m

c) (0.25 ± 0.05) m

d) (0.25 ± 0.135) m

Correct answer is a) (0.25 ± 0.08) m Here, $\mathrm{A}=2.5 \mathrm{~ms}^{-1} \pm 0.5 \mathrm{~ms}^{-1}, \mathrm{~B}=0.10 \mathrm{~s} \pm 0.01 \mathrm{~s}$ $\mathrm{AB}=\left(2.5...

### Which of the following pairs of physical quantities does not have the same dimensional formula?

a) work and torque

b) angular momentum and Planck’s constant

c) tension and surface tension

d) impulse and linear momentum.

Correct answer is c) Tension and surface tension. Tension has the dimension: $[MLT^-2]$ Surface Tension has the dimension: $[ML^0T^-2]$

### The length and breadth of a rectangular sheet are 16.2 cm and 10.1 cm respectively. The area of the sheet inappropriate significant figures and error is:

a) 164 ± 3

b) 163.62 ± 2.6

c) 163.6 ± 2.6

d) 163.62 ± 3

Correct answer is a) 164 ± 3 $cm^{2}$ Error in product of quantities: Suppose $x=a \times b$ Let, $\Delta a$ be the absolute error in measurement of a, $\Delta b$ be the absolute error in...

### The numbers 2.745 and 2.735 on rounding off to 3 significant figures will give:

a) 2.75 and 2.74

b) 2.74 and 2.73

c) 2.75 and 2.73

d) 2.74 and 2.74

Correct answer is d) 2.74 and 2.74 By convention, we apply the following criteria for rounding off measurements: 1. If the dropped digit is less than 5, the preceding digit remains unaffected. 2. If...

### The mass and volume of a body are 4.237 g and respectively. The density of the material of the body in correct significant figures is:

a)

b)

c)

d)

Correct answer is c) $1.7g/cm^{3}$

### The sum of the numbers 436.32, 227.2, and 0.301 inappropriate significant figures is:

a) 663.821

b) 664

c) 663.8

d) 663.82

Correct answer is c) 663.8

### The number of significant figures in 0.06900 is:

a) 5

b) 4

c) 2

d) 3

The correct answer is b) 4 The number of zeroes to the left of a non-zero integer is not considered relevant, whereas the number of zeroes to the right of a non-zero number is.

### A mass attached to a spring is free to oscillate, with angular velocity , in a horizontal plane without friction or damping. It is pulled to a distance and pushed towards the centre with a velocity at time Determine the amplitude of the resulting oscillations in terms of the parameters and [Hint: Start with the equation and note that the initial velocity is negative.]

The angular velocity of the spring be $\omega$ $x=a \cos (\omega t+\theta)$ At $t=0, x=x_{0}$ On Substituting these values in the above equation we get, $\mathrm{x}_{0}=\mathrm{A} \cos \theta-(1)$...

### A body describes simple harmonic motion with an amplitude of and a period of . Find the acceleration and velocity of the body when the displacement is 0

Amplitude is given as $=5 \mathrm{~cm}=0.05 \mathrm{~m}$ Time period is given as $=0.2 \mathrm{~s}$ When the displacement is $y$, then acceleration is given as $A=-\omega^{2} y$ Velocity is given as...

### A body describes simple harmonic motion with an amplitude of and a period of . Find the acceleration and velocity of the body when the displacement is

(a)

(b)

Amplitude is given as $=5 \mathrm{~cm}=0.05 \mathrm{~m}$ Time period is given as $=0.2 \mathrm{~s}$ When the displacement is $y$, then acceleration is given as $A=-\omega^{2} y$ Velocity is given as...

### A circular disc of mass is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be s. The radius of the disc is . Determine the torsional spring constant of the wire. (Torsional spring constant is defined by the relation , where is the restoring couple and the angle of twist).

Mass of the circular disc is given as $10 \mathrm{~kg}$ Period of torsional oscillation is given as $1.5 \mathrm{~s}$ Radius of the disc is given as $15 \mathrm{~cm}=0.15 \mathrm{~m}$ Restoring...

### Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

The mass of the particle executing simple harmonic motion is $m$. The particle's displacement at a given time $t$ is given by $x=A \sin \omega t$ Velocity of the particle is given as...

### You are riding in an automobile of mass . Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by during one complete oscillation. Estimate the values of (a) the spring constant and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports .

(a) Mass of the automobile is given as $=3000 \mathrm{~kg}$ The suspension sags by a length of $15 \mathrm{~cm}$ Decrease in amplitude $=50 \%$ during one complete oscillation If each spring's...

### An air chamber of volume V has a neck area of cross-section into which a ball of mass just fits and can move up and down without any friction. Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Figure]

Solution: Volume of the air chamber is given as $\mathrm{V}$ Cross-sectional area of the neck is given as $\mathrm{A}$ Mass of the ball be $m$ The ball is fitted in the neck at position given as...

### One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

Area of cross-section of the U-tube is given as $A$ Density of the mercury column is given as $\rho$ Acceleration due to gravity is given as $g$ Restoring force, F = Weight of the mercury column of...

### A cylindrical piece of cork of density of base area A and height h floats in a liquid of density . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period where is the density of cork. (Ignore damping due to viscosity of the liquid)

Base area of the cork is given as $=\mathrm{A}$ Height of the cork is given as $h$ Density of the liquid is given as $\rho_{1}$ Density of the cork is given as $\rho$ In equilibrium: Weight of the...

### A simple pendulum of length I and having a bob of mass is suspended in a car. The car is moving on a circular track of radius with a uniform speed If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?

The centripetal acceleration supplied by the circular motion of the car, as well as the acceleration due to gravity, will be felt by the bob of the basic pendulum. Acceleration due to gravity is...

### Answer the following questions:

(a) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give the correct time during the free fall?

(b) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?

(a) Wristwatches work on the principle of spring action and are not affected by gravity's acceleration. As a result, the time on the watch will be accurate. (b) The cabin's acceleration owing to...

### Answer the following questions:

(a) Time period of a particle in SHM depends on the force constant and mass of the particle: . A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?

(b) The motion of a simple pendulum is approximately simple harmonic for small-angle oscillations. For larger angles of oscillation, a more involved analysis shows that is greater than Think of a qualitative argument to appreciate this result.

(a) The spring constant $k$ is proportional to the mass in the case of a simple pendulum. The numerator ($m$) and denominator ($d$) will cancel each other out. As a result, the simple pendulum's...

### The acceleration due to gravity on the surface of moon is . What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is ? (g on the surface of earth is )

Acceleration due to gravity on the surface of moon is given as $g^{\prime}=1.7 \mathrm{~m} \mathrm{~s}^{-2}$ Acceleration due to gravity on the surface of earth is given as $g=9.8 \mathrm{~m}...

### The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of . If the piston moves with simple harmonic motion with an angular frequency of , what is its maximum speed?

Angular frequency of the piston is given as $\omega=200 \mathrm{rad} / \mathrm{min}$ Stroke is given as$=1.0 \mathrm{~m}$ Amplitude is given as $A=1.0 / 2$ $=0.5 \mathrm{~m}$ The maximum speed...

### Figure (a) shows a spring of force constant k clamped rigidly at one end and a mass attached to its free end. A force applied at the free end stretches the spring. Figure (b) shows the same spring with both ends free and attached to a mass at either end. Each end of the spring in Fig. (b) is stretched by the same force F.

(a) What is the maximum extension of the spring in the two cases? (b) If the mass in Fig. (a) and the two masses in Fig. (b) is released, what is the period of oscillation in each case? Solution:...

### Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: ( is in cm and is in .

(a) (b)

(a)$x=3 \sin (2 m t+\pi / 4)$ $=-3 \cos (2 \pi t+\pi / 4+\pi / 2)$ $=-3 \cos (2 \pi t+3 \pi / 4)$ $=-3 \cos (2 \pi t+3 \pi / 4)$ On comparing with the standard equation $A \cos (\omega t+\Phi)$, we...

### Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: ( is in cm and is in .

(a)

(b)

(a) $x=-2 \sin (3 t+\pi / 3)$ $=2 \cos (3 t+\pi / 3+\pi / 2)$ $=2 \cos (3 t+5 \pi / 6)$ On comparing the above equation with the standard equation, $x=A \cos (\omega t+\Phi)$, Amplitude will be $A=2...

### The given figures correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.

Obtain the corresponding simple harmonic motions of the $x$-projection of the radius vector of the revolving particle $P$, in each case. Solution: (a) Time period is given as $t=2 \mathrm{~s}$...

### In the given figure, let us take the position of mass when the spring is unstreched as , and the direction from left to right as the positive direction of -axis. Give as a function of time for the oscillating mass if at the moment we start the stopwatch (t , the mass is at the maximum compressed position. In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

Solution: The body is in the left position at maximal compression, with an initial phase of $3 \pi / 2$ rad. Then, $x=a \sin (\omega t+3 \pi / 2)$ $=-a \cos \omega t$ $=-2 \cos 20 t$ As a result,...

### In the given figure, let us take the position of mass when the spring is unstreched as , and the direction from left to right as the positive direction of -axis. Give as a function of time for the oscillating mass if at the moment we start the stopwatch (t , the mass is

(a) at the mean position,

(b) at the maximum stretched position.

Solution: Distance travelled by the mass sideways is given as $a=2.0 \mathrm{~cm}$ Angular frequency of oscillation can be calculated as, $\omega=\sqrt{k} / \mathrm{m}$ $=\sqrt{1200 / 3}$...

### A spring having with a spring constant is mounted on a horizontal table as shown in Fig. 14.24. A mass of is attached to the free end of the spring. The mass is then pulled sideways to a distance of and released.

Determine the maximum speed of the mass. Solution: Spring constant is given as $\mathrm{k}=1200 \mathrm{~N} \mathrm{~m}^{-1}$ Mass is given as $\mathrm{m}=3 \mathrm{~kg}$ Displacement is given as...

### A spring having with a spring constant is mounted on a horizontal table as shown in Fig. 14.24. A mass of is attached to the free end of the spring. The mass is then pulled sideways to a distance of and released.

Determine<br>(i) the frequency of oscillations, (ii) maximum acceleration of the mass Solution: Spring constant is given as $\mathrm{k}=1200 \mathrm{~N} \mathrm{~m}^{-1}$ Mass is given as...

### A spring balance has a scale that reads from 0 to . The length of the scale is A body suspended from this balance, when displaced and released, oscillates with a period of S. What is the weight of the body?

Maximum mass that the scale can rea is given as $M=50 \mathrm{~kg}$ Maximum displacement of the spring = Length of the scale, $I=20 \mathrm{~cm}$ $=0.2 \mathrm{~m}$ Time period is given as $T=0.6...

### The motion of a particle executing simple harmonic motion is described by the displacement function, If the initial (t ) position of the particle is and its initial velocity is , what are its amplitude and initial phase angle? The angular frequency of the particle is . If instead of the cosine function, we choose the sine function to describe the SHM: , what are the amplitude and initial phase of the particle with the above initial conditions. Solution:

At positlon, t = 0, The given function is $x(t)=A \cos (\omega t+\phi).....(1)$ $\begin{array}{l} 1=A \cos (\omega \times 0+\phi)=A \cos \phi \\ A \cos \phi=1 \end{array}$ Differentiating equation...

### Which of the following relationships between the acceleration a and the displacement of a particle involve simple harmonic motion?

(a)

(b)

The condition of SHM is stated below, Acceleration is directly proportional to negative of displacement of particle. If ' $\mathrm{a}$ ' is the acceleration, $\mathrm{x}$ is the displacement Then,...

### Which of the following relationships between the acceleration a and the displacement of a particle involve simple harmonic motion?

(a)

(b)

Condition of SHM is stated below, Acceleration is directly proportional to negative of displacement of particle. If ' $\mathrm{a}$ ' is the acceleration, $\mathrm{x}$ is the displacement Then, for...

### A particle is in linear simple harmonic motion between two points, A and B, apart. Take the direction from to as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at away from A going towards , and

(b) at away from B going towards .

(a) Positive, Positive, Positive From the end $A$, the particle is travelling toward point 0. This motion is going from $A$ to $B$, which is the standard positive direction. As a result, the...

### A particle is in linear simple harmonic motion between two points, A and B, apart. Take the direction from to as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at the mid-point of AB going towards A,

(b) at away from B going towards

(a) Negative, Zero, Zero A basic harmonic motion is being performed by the particle. The particle's mean location is denoted by $O$. Its highest velocity is at the mean position $O$. Because the...

### A particle is in linear simple harmonic motion between two points, A and B, apart. Take the direction from to as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at the end ,

(b) at the end B

(a) Zero, Positive, Positive Points A and B are the path's two ends, with A-B=10cm and'O' being the path's halfway. Between the end locations, a particle moves in a linear simple harmonic motion....

### Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant):

(a)

(b)

(a) $\exp \left(-\omega^{2} \mathrm{t}^{2}\right)$ It's a one-time exponential function that doesn't repeat. As a result, the motion is non-periodic. (b) The given function $1+\omega t+\omega^{2}...

### Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant):

(a)

(b)

(a) $3 \cos [4 \pi-2 \omega t]=3 \cos [2 \omega t-\pi / 4]$ The equation can be written in the form $\cos (\omega t+\phi)$. It is S.H.M with the period $2 \pi / 2 \omega=\pi / \omega$ (b) $\cos...

### Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant):

(a)

(b)

(a) $\sin \omega t-\cos \omega t$ =2[(1/2)sinωt-(1/2)cosωt] =\sqrt{2}[(1 / \sqrt{2}) \sin \omega t-(1 / \sqrt{2}) \cos \omega t] $=\sqrt{2}[\sin \omega t \times \cos (\pi / 4)-\cos...

### The figures depicts plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?

(a) Because the motion is repeated in only one position, the depicted graph does not illustrate periodic motion. The full motion during one period must be repeated successively for a periodic...

### The figures depicts plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?

(a) Motion is not periodic since it does not repeat itself after a set length of time. (b) The following graph depicts a periodic motion that repeats every 2 seconds.

### Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?

(a) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.

(b) general vibrations of a polyatomic molecule about its equilibrium position.

(a) Simple harmonic motion (b) SHM is not periodic, although general vibrations of a polyatomic molecule about its equilibrium position are. The inherent frequencies of a polyatomic molecule are...

### Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?

(a) the rotation of earth about its axis.

(b) motion of an oscillating mercury column in a U-tube.

(a) The earth's rotation is not a to-and-fro motion around a fixed point. As a result, it is regular but not S.H.M. (b) Simple harmonic motion

### Which of the following examples represents periodic motion?

(a) A hydrogen molecule rotating about its centre of mass.

(b) An arrow released from a bow.

(a) The rotation of a hydrogen molecule around its mass centre is periodic. This is due to the fact that when a hydrogen molecule spins about its centre of mass, it returns to the same location...

### Which of the following examples represents periodic motion?

(a) A swimmer completing one (return) trip from one bank of a river to the other and back.

(b) A freely suspended bar magnet displaced from its N-S direction and released.

(a) The motion of the swimmer is not regular. A swimmer's move between the sides of a river is to and fro. It does not, however, have a set duration. This is because the swimmer's back and forth...

### Given below are densities of some solids and liquids. Give rough estimates of the size of their atoms:

[Hint: Assume the atoms to be ‘tightly packed’ in a solid or liquid phase, and use the known value of Avogadro’s number. You should, however, not take the actual numbers you obtain for various atomic sizes too literally. Because of the crudeness of the tight packing approximation, the results only indicate that atomic sizes are in the range of a few .

If $r$ is the radius of the atom then the volume of each atom will be $(4 / 3) \pi r^{3}$ Volume of all the substance will be $=(4 / 3) \pi r^{3} \times N=M / \rho$ $M=$ atomic mass of the substance...

### A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have a uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres

where refer to number density at heights and respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:

where is the density of the suspended particle, and ‘, that of surrounding medium. [ is Avogadro’s number, and the universal gas constant.] [Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]

Law of atmosphere states that, $\mathrm{n}_{2}=\mathrm{n}_{1} \exp \left[-\mathrm{mg}\left(\mathrm{h}_{2}-\mathrm{h}_{1}\right) / \mathrm{k}_{\mathrm{B}} T\right]$ According to Archimedes principle,...

### From a certain apparatus, the diffusion rate of hydrogen has an average value of 1. The diffusion of another gas under the same conditions is measured to have an average rate of . Identify the gas.

[Hint: Use Graham’s law of diffusion: , where are diffusion rates of gases 1 and 2 , and and their respective molecular masses. The law is a simple consequence of kinetic theory.]

Rate of diffusion of hydrogen is given as $R_{1}=28.7 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ Rate of diffusion of another gas is given as $R_{2}=7.2 \mathrm{~cm}^{3} \mathrm{~s}^{-1}$ According to...

### A metre long narrow bore held horizontally (and closed at one end) contains a long mercury thread, which traps a column of air. What happens if the tube is held vertically with the open end at the bottom?

Length of the narrow bore is given as $L=1 \mathrm{~m}=100 \mathrm{~cm}$ Length of the mercury thread is given as $\mid=76 \mathrm{~cm}$ Length of the air column between mercury and the closed end,...

### Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at atm and temperature . Take the radius of a nitrogen molecule to be roughly A. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of ).

Mean free path is given as $1.11\times10^{-7}$ Collision frequency is given as $4.58\times10^{9}s^{-1}$ Successive collision time ≅ 500 x (Collision time) Pressure inside the cylinder containing...

### At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at (atomic mass of , of ).

Temperature of the helium atom is given as $T_{\mathrm{He}}=-20^{\circ} \mathrm{C}=253 \mathrm{~K}$ Atomic mass of argon is given as $M_{\text {Ar }}=39.9 u$ Atomic mass of helium is given as $M_{H...

### Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not, in which case is the largest?

All three vessels are the same size and have the same capacity. As a result, the pressure, volume, and temperature of each gas are the same. The three vessels will each contain an equal quantity of...

### Estimate the average thermal energy of a helium atom at the temperature of 10 million kelvin (the typical core temperature in the case of a star).

At temperature, $\mathrm{T}=10^{7} \mathrm{~K}$, we have Average thermal energy will be $=(3 / 2) \mathrm{kT}$ $=(3 / 2) \times 1.38 \times 10^{-23} \times 10^{7}$ We get, $=2.07 \times 10^{-16}...

### Estimate the average thermal energy of a helium atom at

(i) room temperature ,

(ii) the temperature on the surface of the Sun (6000 K)

(i) At room temperature the temperature is $T=27^{\circ} \mathrm{C}=300 \mathrm{~K}$ Average thermal energy will be $=(3 / 2) \mathrm{kT}$ Where, $k$ is the Boltzmann constant having value $1.38...

### Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity at a temperature of and 1 atm pressure.

Volume of the room is given as $V=25.0 \mathrm{~m}^{3}$ Temperature of the room is given as $T=27^{0} \mathrm{C}=300 \mathrm{~K}$ Pressure in the room will be $P=1 \mathrm{~atm}=1 \times 1.013...

### An air bubble of volume rises from the bottom of a lake deep at a temperature of To what volume does it grow when it reaches the surface, which is at a temperature of 35 C?

Volume of the air bubble is given as $\mathrm{V}_{1}=1.0 \mathrm{~cm}^{3}$ $=1.0 \times 10^{-6} \mathrm{~m}^{3}$ Air bubble rises to height given as $d=40 \mathrm{~m}$ Temperature at a depth of 40m...

### An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm and a temperature of After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to . Estimate the mass of oxygen taken out of the cylinder , molecular mass of .

Volume of gas is given as $V_{1}=30$ litres $=30 \times 10^{-3} \mathrm{~m}^{3}$ Gauge pressure is given as $\mathrm{P}_{1}=15 \mathrm{~atm}=15 \times 1.013 \times 10^{5} \mathrm{P}$ a Temperature...

### The figure shows plot of PV/T versus for of oxygen gas at two different temperatures.

(a) What is the value of PV/T where the curves meet on the $y$-axis? (b) If we obtained similar plots for $1.00 \times 10^{-3}$ kg of hydrogen, would we get the same value of PV/T at the point where...

### The figure shows plot of PV/T versus for of oxygen gas at two different temperatures.

(a) What does the dotted plot signify? (b) Which is true: $\mathbf{T}_{1}>\mathbf{T}_{2}$ or $\mathbf{T}_{1}<\mathbf{T}_{2}$ ? Solution: (a) The dotted plot is perpendicular to the x-axis,...

### 7.1. The centre of mass of which of the following is located outside the body? a) a pencil b) a shotput c) a dice d) a bangle

Answer: The correct option is d) a bangle. Explanation: The mass of a bangle is distributed along the circumference of the ring shape of the bangle as it is a hollow shape. The center of mass is...

### Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP: 1 atmospheric pressure, Show that it is litres.

We know, The ideal gas equation is given as: $P V=n R T$ Where, $R$ is the universal gas constant having value $8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ $\mathrm{n}$ is the number of...

### Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 A.

Diameter of an oxygen molecule is given as $d=3 \AA$ Radius will be, $r=d / 2$ $r=3 / 2=1.5 \AA=1.5 \times 10^{-8} \mathrm{~cm}$ Actual volume occupied by 1 mole of oxygen gas at STP is given as...

### Two narrow bores of diameters 3.0 mm and 6.0 mm are joined together to form a U-tube open at both ends. If the U-tube contains water, what is the difference in its levels in the two limbs of the tube? Surface tension of water at the temperature of the experiment is 7.3 × 10–2 N m–1. Take the angle of contact to be zero and density of water to be 1.0 × 103 kg m–3 (g = 9.8 m s–2).

Answer : According to the question, the diameter of the first bore is d1 = 3.0 mm = 3 × 10-3 m The radius of the first bore is r1 = 3/2 = 1.5 x 10-3 m. The diameter of the second bore is d2 =6mm The...

### Mercury has an angle of contact equal to 140° with soda lime glass. A narrow tube of radius 1.00 mm made of this glass is dipped in a trough containing mercury. By what amount does the mercury dip down in the tube relative to the liquid surface outside ? Surface tension of mercury at the temperature of the experiment is 0.465 N m–1. Density of mercury = 13.6 × 103 kg m–3

Answer : According to the question, the density of mercury is ρ =13.6 × 103 kg/m3 Acceleration due to gravity, g = 9.8 m/s2 The angle of the contact between mercury and soda-lime glass is θ = 140°...

### In Millikan’s oil drop experiment, what is the terminal speed of an uncharged drop of radius 2.0 × 10–5 m and density 1.2 × 103 kg m–3. Take the viscosity of air at the temperature of the experiment to be 1.8 × 10–5 Pa s. How much is the viscous force on the drop at that speed? Neglect buoyancy of the drop due to air.

Answer According to the question, the acceleration due to gravity is g = 9.8 m/s2 The radius of the uncharged drop is r = 2.0 × 10-5 m The density of the uncharged drop is ρ = 1.2 × 103 kg m-3 The...

### A plane is in level flight at constant speed and each of its wings has an area of 25 m2. If the speed of the air is 180 km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg/m3), g = 9.8 m/s2

Answer : Area of the wings of the plane, A=2×25=50 m2 Speed of air over the lower wing, V1 =180km/h= 180 x (5/18) = 50 m/s Speed of air over the upper wing, V2 =234km/h= 234 x (5/18) = 65 m/s...

### (a) What is the largest average velocity of blood flow in an artery of radius 2×10–3m if the flow must remain lanimar? (b) What is the corresponding flow rate ? (Take viscosity of blood to be 2.084 × 10–3 Pa s).

Answer : According to the question, the radius of the vein is r = 2 × 10-3 m And the diameter of the vein is d = 2 × 1 × 10-3 m = 2 × 10-3 m The viscosity of blood is given by η = 2.08 x 10-3 m The...

### During a blood transfusion, the needle is inserted in a vein where the gauge pressure is 2000 Pa. At what height must the blood container be placed so that blood may just enter the vein? [Use the density of whole blood from Table 10.1]

Answer - According to the question, the density of whole blood is ρ = 1.06 × 103 kg m-3 Gauge pressure is P = 2000 Pa Acceleration due to gravity is g = 9.8 m/s2 =P/ρg =200/(1.06 x 103 x 9.8)...

### Two vessels have the same base area but different shapes. The first vessel takes twice the volume of water that the second vessel requires to fill upto a particular common height. Is the force exerted by the water on the base of the vessel the same in the two cases ? If so, why do the vessels filled with water to that same height give different readings on a weighing scale ?

Answer : The pressure and hence the force applied on the two vessels will be the same because the base area is the same. When the walls of the vessel are not perpendicular to the base, force is also...

### A manometer reads the pressure of a gas in an enclosure as shown in Fig. 10.25 (a) When a pump removes some of the gas, the manometer reads as in Fig. 10.25 (b) The liquid used in the manometers is mercury and the atmospheric pressure is 76 cm of mercury. (a) Give the absolute and gauge pressure of the gas in the enclosure for cases (a) and (b), in units of cm of mercury. (b) How would the levels change in case (b) if 13.6 cm of water (immiscible with mercury) is poured into the right limb of the manometer? (Ignore the small change in the volume of the gas).

Answer ; (a) For diagram (a): According to the question, atmospheric pressure, P0 = 76 cm of Hg Gauge pressure is the difference in mercury levels between the two arms. The gauge pressure is 20 cm...

### A tank with a square base of area 1.0 m2 is divided by a vertical partition in the middle. The bottom of the partition has a small-hinged door of area 20 cm2. The tank is filled with water in one compartment, and an acid (of relative density 1.7) in the other, both to a height of 4.0 m. compute the force necessary to keep the door close.

Answer : According to the given, the area of the hinged door is a = 20 cm2= 20 × 10-4 m The base area of given tank A = 2 m2 and the density of water is ρ1 = 103 kg/m3and the density of acid is ρ2 =...

### A truck parked outside a petrol pump blows a horn of frequency in still air. The Wind then starts blowing towards the petrol pump at Calculate the wavelength, speed, and frequency of the horn’s sound for a man standing at the petrol pump. Is this situation completely identical to a situation when the observer moves towards the truck at sand the air is still?

For the standing observer: Frequency is given as $\mathrm{v}_{\mathrm{H}}=200 \mathrm{~Hz}$ Velocity of sound is given as $v=340 \mathrm{~m} / \mathrm{s}$ Speed of the wind is given as $v_{w}=20...

### A man standing at a certain distance from an observer blows a horn of frequency in still air.

(a) Find the horn’s frequency for the observer when the man (i) runs towards him at (ii) runs away from him at .

(b) Find the speed of sound in both the cases.

[Speed of sound in still air is

Frequency of the horn is given as $\mathrm{v}_{\mathrm{H}}=200 \mathrm{~Hz}$ Velocity of the man is given as $\mathrm{v}_{\mathrm{T}}=20 \mathrm{~m} / \mathrm{s}$ Velocity of sound is given as...