Physics

### Temperature dependence of resistivity ρ(T) of semiconductors, insulators, and metals is significantly based on the following factors: a) number of charge carriers can change with temperature T b) time interval between two successive collisions can depend on T c) length of material can be a function of T d) mass of carriers is a function of T

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

### 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...

### 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...

### 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 communicationa) calculate height of such a satelliteb) 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 Rb) 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 ofa) 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 gravitya) are always at the same point for any size of the bodyb) are always at the same point only for spherical bodiesc) can never be at the same pointd) is close to each other for objects, say of sizes less than 100 me) 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 objectb) none of the three laws of Kepler will be validc) only the third law will become invalidd) 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 changeb) we will become hotter after billions of yearsc) we will be going around but not strictly in closed orbitsd) 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 validb) only the third law will be validc) the second law will not changed) 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 magnitudesa) walking on ground would become more difficultb) the acceleration due to gravity on earth will not changec) raindrops will fall much fasterd) 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-lawa) planets will not have elliptic orbitsb) circular orbits of planets is not possiblec) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolicd) 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 altitudeb) acceleration due to gravity increases with increasing depthc) acceleration due to gravity increases with increasing latituded) 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 massb) that the gravitational mass and inertial mass are equal is an experimental resultc) that the acceleration due to gravity on earth is the same for all bodies is due to the equality of gravitational mass and inertial massd) 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. Theya) will not move around the sun since they have very small masses compared to the sunb) will move in an irregular way because of their small masses and will drift away outer spacec) will move around the sun in closed orbits but not obey Kepler’s lawsd) 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 moona) will be elliptical will not be strictly elliptical because the total gravitational force on it is not centralc) is not elliptical but will necessarily be a closed curved) 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 becausea) the solar cells and batteries in satellites run outb) the laws of gravitation predict a trajectory spiralling inwardsc) of viscous forces causing the speed of the satellite and hence height to gradually decreased) 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 systema) the torque is zerob) the torque causes the earth to spinc) the rigid body result is not applicable since the earth is not even approximately a rigid bodyd) 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 woulda) be similarly trueb) not be true because the force between earth and mercury is not inverse square lawc) not be true because the major gravitational force on mercury is due to sund) 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 gravitya) will be directed towards the centre but not the same everywhereb) will have the same value everywhere but not directed towards the centrec) will be same everywhere in magnitude directed towards the centred) 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...

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### 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...

### Which of the following pairs of physical quantities does not have the same dimensional formula?a) work and torqueb) angular momentum and Planck’s constantc) tension and surface tensiond) 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 2.74b) 2.74 and 2.73c) 2.75 and 2.73d) 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 number of significant figures in 0.06900 is:a) 5b) 4c) 2d) 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 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...

### 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 atmosphereswhere 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...

### 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...