Answer: According to the question, the length is: l = 20 cm =0.2 m and the frequency is, v = 1237.5 Hz We know that the velocity of sound in air = 330 m/s The following expression is: l = λ/4 λ = 4l...
A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this wire when a tension of 2.06 × 104 N is applied?
Answer: According to the question, the length is, l = 12 m and the total mass, M = 2.10 kg Mass per unit length is given by7 m = M/l = 2.1/12 and we know that tension si: T = 2.06 × 104 N Therefore,...
The equilibrium constant expression for a gas reaction is, Write the balanced chemical equation corresponding to this expression.
Answer: The equilibrium constant, Kc, is defined as the product of the equilibrium concentrations of products over the equilibrium concentrations of reactants, each raised to the power of the...
A mixture of mol of mol of and of is introduced into a reaction vessel at . At this temperature, the equilibrium constant, for the reaction is Is the reaction mixture at equilibrium? If not, what is the direction of the net reaction?
Answer: The given reaction is: $\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NH}_{3}(\mathrm{~g})$ The given concentration of various species is...
A sample of is placed in a flask at a pressure of . At equilibrium, the partial pressure of is .What is for the given equilibrium?
Answer: The initial concentration of $HI$ is $0.2 atm$. It has a partial pressure of $0.04 atm$ when it is in equilibrium with the surrounding environment. The pressure of $HI$ drops by...
For the following equilibrium, at Both the forward and reverse reactions in the equilibrium are elementary bimolecular reactions. What is , for the reverse reaction?
Answer: Kp and Kc are equilibrium constants for reversible reactions. The equilibrium constant Kp is stated in terms of atmospheric pressure, whereas Kc is expressed in terms of concentrations...
Write the expression for the equilibrium constant, for each of the following reactions: (i) (ii)
Answer: The equilibrium constant, Kc, is defined as the product of the equilibrium concentrations of products over the equilibrium concentrations of reactants, each raised to the power of the...
A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fixed point.
Find the equation of the line passing through the point of intersection of 2x + y = 5 and x + 3y + 8 = 0 and parallel to the line 3x + 4y = 7.
Given lines are \[\begin{array}{*{35}{l}} 2x\text{ }+\text{ }y\text{ }=\text{ }5\text{ }\ldots \ldots 1 \\ x\text{ }+\text{ }3y\text{ }=\text{ }-8\text{ }\ldots \ldots 2 \\ \end{array}\] Firstly,...
When two waves of almost equal frequencies v1 and v2 reach at a point simultaneously, what is the time interval between successive maxima?
Answer: Beats are created when two waves of nearly identical frequency arrive at the same location at the same time. It is given that the two frequencies are almost equal, that is v1 = v2. And for...
At what temperatures will the speed of sound in air be 3 times its value at 0 degreesC?
Answer: Speed of sound in air is $ v=\sqrt{\frac{\gamma RT}{M}} $ where 'T' represents the absolute temperature. since both γ and M are constant, we can write: $ \therefore v\alpha \sqrt{T} $ $...
A sitar wire is replaced by another wire of same length and material but of three times earlier radius. If the tension in the wire remains the same, by what factor will the frequency change?
Answer: The wire is stretched both from the end so the frequency of stretched wire is as follows: $ v=\frac{n}{2L}\sqrt{\frac{T}{m}} $ As the number of harmonic n, length L, and tension (T) are kept...
The displacement of an elastic wave is given by the function y = 3 sin ωt + 4 cos ωt where y is in cm and t is in second. Calculate the resultant amplitude.
Answer: According to the question, y = 3 sin ωt + 4 cos ωt Let’s suppose that 3 = a cos φ and 4 = a sin φ Then, the equation becomes: y = a cos φ ωt + a sin φ ωt y = a sin (ωt + φ) tan φ = 4/3 φ =...
An organ pipe of length L open at both ends is found to vibrate in its first harmonic when sounded with a tuning fork of 480 Hz. What should be the length of a pipe closed at one end, so that it also vibrates in its first harmonic with the same tuning fork?
Answer: Because the medium, frequency, and number of harmonics are the same in open and closed pipes, the number of nodes and (wave) will be the same in both circumstances. When the harmonic is...
A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. Under what conditions would the tuning fork still be is resonance with the wire?
Answer: When a sonometer vibrates, the length of the wire is doubled. The sonometer will resonate at 2L if the tuning fork resonates at L. The frequency of the sonometer is expressed using the...
A tuning fork A, marked 512 Hz, produces 5 beats per second, where sounded with another unmarked tuning fork B. If B is loaded with wax the number of beats is again 5 per second. What is the frequency of the tuning fork B when not loaded?
Answer: When the tuning fork B is loaded with wax, the tuning fork's frequency falls below its original frequency. Let's say that when the tuning fork is marked 512 Hz, the tuning fork B is marked...
Which of the following statements are true for a stationary wave?
a) every particle has a fixed amplitude which is different from the amplitude of its nearest particle b) all the particles cross their mean position at the same time c) all the particles are...
A train, standing in a station yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with a speed of 10 m/s. Given that the speed of sound in still air is 340 m/s
a) the frequency of sound as heard by an observer standing on the platform is 400 Hz b) the speed of sound for the observer standing on the platform is 350 m/s c) the frequency of sound as heard by...
The transverse displacement of a string is given by y(x,t) = 0.06 sin (2πx/3) cos (120πt). All the points on the string between two consecutive nodes vibrate with
a) same frequency b) same phase c) same energy d) different amplitude Answer: The correct answers are a) same frequency b) same phase d) different amplitude Explanation: This is the standing wave...
During propagation of a plane progressive mechanical wave
a) all the particles are vibrating in the same phase b) amplitude of all the particles is equal c) particles of the medium executes SHM d) wave velocity depends upon the nature of the medium Answer:...
Speed of sound waves in a fluid depends upon
a) directly on the density of the medium b) square of bulk modulus of the medium c) inversely on the square root of density d) directly on the square root of bulk modulus of the medium Answer: The...
The displacement of a string is given by y(x,t) = 0.06 sin (2πx/3) cos (120πt) where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0 × 10-2kg.
a) it represents a progressive wave of frequency 60 Hz b) it represents a stationary wave of frequency 60 Hz c) it is the result of the superposition of two waves of wavelength 3 m, frequency 60 Hz...
A transverse harmonic wave on a string is described by y (x,t) = 3.0 sin (36t + 0.018x + π/4) where x and y are in cm and t is in s. The positive direction of x is from left to right
a) the wave is travelling from right to left b) the speed of the wave is 20 m/s c) frequency of the wave is 5.7 Hz d) the least distance between two successive crests in the wave is 2.5 cm Answer:...
A train whistling at constant frequency is moving towards a station at a constant speed V. The train goes past a stationary observer on the station. The frequency n’ of the sound as heard by the observer is plotted as a function of time t. Identify the expected curve
Answer: The correct option is c) Explanation: Apparent Frequency can be expressed in a variety of ways. Assume that the observer (O) and the source (S) are both traveling in the same direction down...
A string of mass 2.5 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, the disturbance will reach the other end in
a) one second b) 0.5 second c) 2 seconds d) data given is insufficient Answer: The correct option is b) 0.5 second Explanation: According to the question, m=2.50 kg, l = 20 m and T=200N $ \mu...
Equation of a plane progressive wave is given by y = 0.6 sin 2π(t-x/2). On reflection from a denser medium, its amplitude becomes 2/3 of the amplitude of the incident wave
a) y = 0.6 sin 2π(t+x/2) b) y = -0.4 sin 2π(t+x/2) c) y = 0.4 sin 2π(t+x/2) d) y = -0.4 sin 2π(t-x/2) Answer: The correct option is b) y = -0.4 sin 2π(t+x/2) Explanation: The amplitude of the...
A sound wave is passing through the air column in the form of compression and rarefaction. In consecutive compressions and rarefactions,
a) density remains constant b) Boyle’s law is obeyed c) bulk modulus of air oscillates d) there is no transfer of heat Answer: The correct option is d) there is no transfer of heat Explanation:...
Which of the following statements are true for wave motion?
a) mechanical transverse waves can propagate through all mediums b) longitudinal waves can propagate through solids only c) mechanical transverse waves can propagate through solids only d)...
With the propagation of longitudinal waves through a medium, the quantity transmitted is
a) matter b) energy c) energy and matter d) energy, matter, and momentum Answer: The correct option is b) energy The transfer of energy across a medium is caused by longitudinal waves propagating...
Change in temperature of the medium changes
a) frequency of sound waves b) amplitude of sound waves c) wavelength of sound waves d) loudness of sound waves Answer: The correct option is c) wavelength of sound waves Explanation: The velocity...
Speed of sound wave in air
a) is independent of temperature b) increases with pressure c) increases with an increase in humidity d) decreases with an increase in humidity Answer: The correct option is c) increases with an...
Sound waves of wavelength λ travelling in a medium with a speed of v m/s enter into another medium where its speed is 2v m/s. The wavelength of sound waves in the second medium is
a) λ b) λ/2 c) 2 λ d) 4 λ Answer: The correct answer is c) 2 λ Explanation: According to the question, the wavelength of sound waves in the first medium, λ = u/v ...
Water waves produced by a motorboat sailing in water are
a) neither longitudinal nor transverse b) both longitudinal and transverse c) only longitudinal d) only transverse Answer: The correct option is b) both longitudinal and transverse Let's...
A simple pendulum of time period 1s and length l is hung from a fixed support at O, such that the bob is at a distance H vertically above A on the ground. The amplitude is θo. The string snaps at θ = θ0/2. Find the time taken by the bob to hit the ground. Also, find the distance from A where bob hits the ground. Assume θ0 to be small so that sin θo, θo and cos θ0 1.
Answer: At t = t1 and θ= θ0/a We have, T = 1 sec θ0/2 = θ0 cos 2π/T t1 d θ/dt = – θ0 2π sin 2πt At t = 1/6, that is when θ = θ0/2 d θ/dt = – θ0π√3 v/l = – θ0π√3 v = -θ0π√3l θ0l [1/2 – π√6H/g] gives...
A tunnel is dug through the centre of the earth. Show that a body of mass ‘m’ when dropped from rest from one end of the tunnel will execute simple harmonic motion.
Answer: Let g’ be the acceleration due to gravity of the earth inside the earth g’ = g(1-d/R) R-d = y g’ = g y/R F = -mg y/R F is proportional to (-y) The motion is SHM when the body is in a tunnel....
One end of a V-tube containing mercury is connected to a suction pump and the other end to the atmosphere. The two arms of the tube are inclined to horizontal at an angle of 45o each. A small pressure difference is created between two columns when the suction pump is removed. Will the column of mercury in V-tube execute simple harmonic motion? Neglect capillary and viscous forces. Find the time period of oscillation.
Answer: Let h0 be the initial height of the columns and dx represent the element that needs to be considered Then the mass is given by: dm = A.dx.ρ Potential energy on the left of dm = (dm)gh Total...
A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute SHM with a time period where m is mass of the body and ρ is the density of the liquid.
Answer: Let us consider that the verticle displacement at the equilibrium position is $ {{x}_{0}} $ At equilibrium mg = Buoyant Force = $ A{{x}_{0}}\rho g $ When it is displaced further by a...
A body of mass m is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand so that the spring is neither stretched nor compressed. Suddenly the support of the hand is removed. The lowest position attained by the mass during oscillation is 4 cm below the point, where it was held in hand.
a) what is the amplitude of oscillation? b) find the frequency of oscillation? Answer: The mass m will oscillate between the lowest and the highest point, which is where it was held in hand. As a...
A person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of 2 s-1 and an amplitude 5 cm. A weighing machine on the platform gives the persons weight against time.
a) will there be any change in weight of the body, during the oscillation? b) if the answer to part a) is yes, what will be the maximum and minimum reading in the machine and at which position?...
Consider a pair of identical pendulums, which oscillate with equal amplitude independently such that when one pendulum is at its extreme position making an angle of 2o to the right with the vertical, the other pendulum makes an angle of 1o to the left of the vertical. What is the phase difference between the pendulums?
Answer: We can write the following set of equations: θ1 = θ0 sin (wt + δ1) ...............................(1) θ2 = θ0 sin (wt + δ2) ...
A mass of 2 kg is attached to the spring of spring constant 50 N/m. The block is pulled to a distance of 5 cm from its equilibrium position at x = 0 on a horizontal frictionless surface from rest at t = 0. Write the expression for its displacement at anytime t.
Answer: $ \omega =\sqrt{\frac{k}{m}}=5 $ Therefore, $ x=5\sin 5t $
A body of mass m is situated in a potential field U(x) = U0 (1 – cos αx) when U0 and α are constants. Find the time period of small oscillations.
Answer: According to the question, we have: U =U0 (1−cosαx) We know taht, F = −dU / dx $ F=-\frac{d}{dx}\left[ {{U}_{0}}(1-\cos \alpha x) \right] $ $ F=-{{U}_{0}}\alpha \sin \alpha x $ As it is...
Find the displacement of a simple harmonic oscillator at which its PE is half of the maximum energy of the oscillator.
Answer: we know that the mean position of the oscillator at x with PE is = 1/2 kx2 Therefore, PE = -m2x2 When PE is maximum, KE = 0 at x = A We get, E = 1/2 mω2A2 PE = 1/2 total energy x2 = 1/2 A2 x...
Show that the motion of a particle represented by y = sin ωt – cos ωt is simple harmonic with a period of 2π/ω.
Answer: A function that represents SHM is written as sin (2π/T t + ϕ) We have, y = sin ωt – cos ωt y = √2 sin (ωt – π/4) The standard equation of SHM has: y = a sin (2π/T t + ϕ) Therefore, T =...
Find the time period of mass M when displaced from its equilibrium position and then released for the system shown in the figure.
Answer: y represents the distance across which the mass is pulled. The spring's total extension is y + y = 2y. The initial mean location of equilibrium = x0 In the spring, the net extension is equal...
The length of a second’s pendulum on the surface of the earth is 1 m. What will be the length of a second’s pendulum on the moon?
Answer: We know that the expression for the time period of a simple pendulum is T = 2π√l/g Where l represents the length of the pendulum and, g denotes the acceleration due to gravity Therefore,...
Draw a graph to show the variation of PE, KE, and total energy of a simple harmonic oscillator with displacement.
Answer:
Show that for a particle executing SHM, velocity, and displacement have a phase difference of π/2.
Answer: We know that the equation for the displacement of SHM is x = a cos ꞷt where velocity can be determined as: v = dx/dt = -aꞷ sin ꞷt Here, the phase displacement is ϕ1 = ꞷt And the phase...
In figure, what will be the sign of the velocity of the point P’, which is the projection of the velocity of the reference particle P. P is moving in a circle of radius R in the anticlockwise direction.
Answer: P' is the perpendicular of the particle Pat's velocity vector at time t. When the particle moves from P to P1, its foot moves from P' to Q, which is away from the positive axis. As a result,...
What is the ratio between the distance travelled by the oscillator in one time period and amplitude?
Answer: According to the question, the distance traveled by the oscillator in one time period = 4A Where A represents the amplitude of the oscillation Therefore, the required ratio becomes = 4A/A =...
What is the ratio of maximum acceleration to the maximum velocity of a simple harmonic oscillator?
Answer: We know that the equation for the displacement of SHM is x = a sin (ꞷt + ϕ) And the velocity of the particle is given by- v = dx/dt = d a sin (ꞷt + ϕ)/dt Expression for the maximum velocity...
When will the motion of a simple pendulum be simple harmonic?
Answer: Consider a basic pendulum with a restoring torque and O as the fixed point. The restoring force is given as F = -mg sin θ θ is so small that θ = arc/radius = x/l As a result, F = – mg x/l As...
What are the two basic characteristics of a simple harmonic motion?
Answer: The two primary properties of a SHM are as follows: a) Acceleration and displacement are proportionate in magnitude. b) The acceleration is in the direction of the mean position, while the...
Two identical springs of spring constant K are attached to a block of mass m and to fixed supports as shown in the figure. When the mass is displaced from equilibrium position by a distance x towards the right, find the restoring force.
Answer: When mass m is pushed from equilibrium by a distance x to the right, spring B is squeezed by a distance x, and the force (kx) is applied to mass m to the left. However, spring A will stretch...
Displacement versus time curve for a particle executing SHM is shown. Identify the points marked at which
i) velocity of the oscillator is zero ii) speed of the oscillator is maximum Answer: (i) The oscillator's velocity is zero when the points A, C, E, and G are at their extreme positions. (ii) The...
A body is performing SHM. Then its
a) average total energy per cycle is equal to its maximum kinetic energy b) average kinetic energy per cycle is equal to half of its maximum kinetic energy c) mean velocity over a complete cycle is...
A particle is in linear simple harmonic motion between two points A and B, 10 cm apart. Taken the direction from A to B as the +ve direction and choose the correct statements.
a) the sign of velocity, acceleration, and force on the particle when it is 3 cm away from A going towards B are positive b) the sign of velocity of the particle at C going towards O is negative c)...
A body is performing SHM. Then its
a) average total energy per cycle is equal to its maximum kinetic energy b) average kinetic energy per cycle is equal to half of its maximum kinetic energy c) mean velocity over a complete cycle is...
The displacement time graph of a particle executing SHM is shown in the figure. Which of the following statement is/are true?
a) the force is zero at t = 3T/4 b) the acceleration is maximum at t = 4T/4 c) the velocity is maximum at t = T/4 d) the PE is equal to KE of oscillation t = T/2 Answer: The correct options are a)...
Which of the following statements is/are true for a simple harmonic oscillator?
a) force acting is directly proportional to the displacement from the mean position and opposite to it b) motion is periodic c) acceleration of the oscillator is constant d) the velocity is periodic...
Displacement versus time curve for a particle executing SHM is shown in the figure. Choose the correct statements.
a) phase of the oscillator is same at t = 0 s and t = 2 s b) phase of the oscillator is the same at t = 2 s and t = 6 s c) phase of the oscillator is the same at t = 1 s and t = 7 s d) phase of the...
The motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is
a) simple harmonic motion b) non-periodic motion c) periodic motion d) periodic but not SHM Answer: The correct options are a) simple harmonic motion b) non-periodic motion Explanation: Allow the...
The rotation of the earth about its axis is
a) periodic motion b) simple harmonic motion c) periodic but not simple harmonic motion d) non-periodic motion Answer: The correct answers are a) periodic motion c) periodic but not simple harmonic...
When a mass m is connected individually to two springs S1 and S2, the oscillation frequencies are v1 and v2. If the same mass is attached to the two springs are shown in the figure, the oscillation frequency would be
a) v1 + v2 b) c) d) Answer: The correct option is b) Explanation: The spring constants for both the springs have been used in the expressions below: $ {{v}_{1}}=\frac{1}{2\pi...
A particle executing SHM has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s2. The period of oscillation is
a) πs b) π/2 s c) 2 πs d) π/t s Answer: The correct option is a) πs Explanation: It isgiven that the maximum velocity is $ {{v}_{\max }}=A\omega =30cm/s $ And the maximum acceleration is as follows:...
Figure shows the circular motion of a particle. The radius of the circle, the period, sense of revolution, and the initial position are indicated on the figure. The simple harmonic motion of the x-projection of the radius vector of the rotating particle P is
a) x(t) = B sin (2πt/30) b) x(t) = B cos (πt/15) c) x(t) = B sin (πt/15 + π/2) d) x(t) = B cos (πt/15 + π/2) Answer: The correct option is a) x(t) = B sin (2πt/30) Explanation: We have, $ x(t)=A\sin...
Explain the formation of a chemical bond.
Answer: “A chemical bond is an attractive force that binds chemical elements together.” Many theories exist for forming chemical bonds, including valence shell electron pair repulsion, electronic,...
Four pendulums A, B, C, and D are suspended from the same elastic support as shown in the figure. A and C are of the same length, while B is smaller than A and D is larger than A. If A is given a transverse displacement,
a) D will vibrate with maximum amplitude b) C will vibrate with maximum amplitude c) B will vibrate with maximum amplitude d) all four will oscillate with equal amplitude Answer: The correct option...
The displacement of a particle varies with time according to the relation y = a sin ωt + b cos ωt
a) the motion is oscillatory but not SHM b) the motion is SHM with amplitude a + b c) the motion is SHM with amplitude a2 + b2 d) the motion is SHM with amplitude √a2 + b2 Answer: The correct option...
A particle is acted simultaneously by mutually perpendicular simple harmonic motions x = a cos ωt and y = a sin ωt. The trajectory of motion of the particle will be
a) an ellipse b) a parabola c) a circle d) a straight line Answer: The correct option is c) a circle Given x=acosωt (i) y=asinωt ...
Motion of an oscillating liquid column in a U-tube is
a) periodic but not simple harmonic b) non-periodic c) simple harmonic and time period is independent of the density of the liquid d) simple harmonic and time period is directly proportional to the...
The relation between acceleration and displacement of four particles are given below:
a) ax = +2x b) ax = +2x2 c) ax = -2x2 d) ax = -2x Which one of the particles is executing simple harmonic motion? Answer: The correct option is d) ax = -2x Explanation: The acceleration in simple...
If A = {x : x ∈ W, x < 2}, B = {x : x ∈ N, 1 < x < 5}, C = {3, 5} find(i) A × (B ∩ C)(ii) A × (B ∪ C)
Given, A = {x: x ∈ W, x < 2}, B = {x : x ∈N, 1 < x < 5} C = {3, 5}; W is the set of whole numbers A = {x: x ∈ W, x < 2} = {0, 1} B = {x : x ∈N, 1 < x < 5} = {2, 3, 4} (i) (B∩C) =...
Let A = {–1, 2, 3} and B = {1, 3}. Determine(i) B × B(ii) A × A
Given, A = {–1, 2, 3} and B = {1, 3} (i) B × B {1, 3} ×{1, 3} So, B × B = {(1, 1), (1, 3), (3, 1), (3, 3)} As a result, the Cartesian product is {(1, 1), (1, 3), (3, 1), (3, 3)} (ii) A × A {–1, 2,...
The displacement of a particle is represented by the equation y = sin3 ωt. The motion is
a) non-periodic b) periodic but not simple harmonic c) simple harmonic with period 2π/ω d) simple harmonic with period π/ω Answer: The correct option is c) simple harmonic with period 2π/ω The...
The displacement of a particle is represented by the equation y = 3 cos(π/4 – 2ωt). The motion of the particle is
a) simple harmonic with period 2p/w b) simple harmonic with period π/ω c) periodic but not simple harmonic d) non-periodic Answer: The correct option is b) simple harmonic with period π/ω...
Consider a rectangular block of wood moving with a velocity vo in a gas at temperature T and mass density ρ. Assume the velocity is along the x-axis and the area of cross-section of the block perpendicular to vo is A. Show that the drag force on the block is
, where m is the mass of the gas molecule. Answer: Let ρm represent the number of molecules per unit volume Then the expression for the change in momentum by a molecule on front side is = 2m (v +...
Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universals set (s) for all the three sets A, B and C(i) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}(ii) {1, 2, 3, 4, 5, 6, 7, 8}
(i) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} As a result, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set...
A box of 1.00 m3 is filled with nitrogen at 1.50 atm at 300 K. The box has a hole of an area 0.010 mm2. How much time is required for the pressure to reduce by 0.10 atm, if the pressure outside is 1 atm?
Answer: Volume of the box = 1 m3 = V1 Initial pressure P1 = 1.5 atm Final pressure P2 = 1.4 atm Air pressure Pa = 1 atm Initial temperature T1 = 300 K Final temperature T2 = 300 K Area of the hole =...
Ten small planners are flying at a speed of 150 km/h in total darkness in an air space that is 20 × 20 × 1.5 km3 in volume. You are in one of the planes, flying at random within this space with no way of knowing where the other planes are. On the average about how long a time will elapse between near collision with your plane. Assume for this rough computation that a safety region around the plane can be approximated by a sphere of radius 10 m.
Answer: We know that, Time = distance/speed Number of particles per unit volume v = N/volume n = 0.0167 km-3 d = 10 × 10-3 km v = 150 km/hr Therefore, we get: time = 225 hrs
Which of the following are examples of the null set(i) Set of odd natural numbers divisible by 2(ii) Set of even prime numbers
(i) Because odd numbers are not divisible by two, a set of odd natural numbers divisible by two is a null set. (ii) Because 2 is an even prime number, the set of even prime numbers is not a null...
Consider an ideal gas with the following distribution of speeds
Speed (m/s) % of molecules 200 10 400 20 600 40 800 20 1000 10 a) calculate Vrms and hence T (m = 3.0 × 10-26 kg) b) if all the molecules with speed 1000 m/s escape from the system, calculate new...
Which of the following sets are finite or infinite(i) The set of months of a year(ii) 1, 2, 3 …
(i) Because there are 12 items in the set of months of a year, it is a finite set. (ii) Because it contains an unlimited amount of natural numbers, 1, 2, 3,... is an infinite set.
State whether the following set is finite or infinite: The set of circles passing through the origin (0, 0)
Because an infinite number of circles can pass through the origin (0, 0), the set of circles travelling through the origin is endless.
In the following, state whether A = B or not:(i) A = {2, 4, 6, 8, 10}; B = {x: x is positive even integer and x ≤ 10}(ii) A = {x: x is a multiple of 10}; B = {10, 15, 20, 25, 30 …}
(i) A = {2, 4, 6, 8, 10}; B = {x: x is a positive even integer and x ≤ 10} = {2, 4, 6, 8, 10} So, A = B (ii) A = {x: x is a multiple of 10} B = {10, 15, 20, 25, 30 …} We know that 15 ∈ B but 15 ∉ A....
From the sets given below, select equal sets:A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14}, D = {3, 1, 4, 2}, E = {–1, 1}, F = {0, a}, G = {1, –1}, H = {0, 1}
A = {2, 4, 8, 12}; B = {1, 2, 3, 4}; C = {4, 8, 12, 14} D = {3, 1, 4, 2}; E = {–1, 1}; F = {0, a} G = {1, –1}; H = {0, 1} We know, 8 ∈ A, 8 ∉ B, 8 ∉ D, 8 ∉ E, 8 ∉ F, 8 ∉ G, 8 ∉ H A ≠ B, A ≠ D, A ≠...
Explain why
a) there is no atmosphere on moon b) there is a fall in temperature with altitude Answer: a) The moon has no atmosphere since the gravitational pull is minimal and the Vrms is bigger on the moon,...
Answer: According to the question, the final KE of the gas is 0 The change in KE is as follows: ∆K= 1/2 (nm)v2 ∆T is the change in the temperature ∆U = nCv∆T ∆K = ∆U Making use of the expression, we...
Calculate the number of degrees of freedom of molecules of hydrogen in 1 cc of hydrogen gas at NTP.
Answer: According to the question, the volume occupied by the molecules of a gas is 22400 cc and the number of molecules in 1 cc of hydrogen are 2.688 × 1019 We know that the hydrogen has a total...
A balloon has 5.0 g mole of helium at 7 degrees C. Calculate
a) the number of atoms of helium in the balloon b) the total internal energy of the system Answer: We know that the average KE per molecule is 3/2kT No.of moles of helium are n = 5 g mole We have T...
When air is pumped into a cycle tyre the volume and pressure of the air in the tyre both are increased. What about Boyle’s law in this case?
Answer: We have, PV = P(m/ρ) = constant P/ ρ = constant Volume = m/ ρ where m is constant When air is pushed into the cycle's tyre, the mass of the air increases as the number of molecules...
We have 0.5 g of hydrogen gas in a cubic chamber of size 3 cm kept at NTP. The gas in the chamber is compressed keeping the temperature constant till a final pressure of 100 atm. Is one justified in assuming the ideal gas law, in the final state?
Answer: We know that volume of 1 molecule = 4/3 πr3 = 4.20 × 10-30 m3 And the number of moles in 0.5 g H2 gas = 0.25 mole Volume of H2 molecule in 0.25 mole = 1.04×6.023× 10+23-30 = 6.264 ×...
A gas mixture consists of molecules of tyres A, B, and C with masses mA > mB > mC. Rank the three types of molecules in decreasing order of
a) average KE b) rms speeds Answer: a) From the above result, the pressure and temperature are the same, therefore, KEc > KEb > KEa b) When P and T are constant, (Vrms)c > (Vrms)b >...
The container shown in the figure has two chambers, separated by a partition, of volumes V1 = 2.0 litre and V2 = 3.0 litre. The chambers contain μ1 = 4.0 and μ2 = 5.0 moles of a gas at pressure p1 = 1.00 atm and p2 = 2.00 atm. Calculate the pressure after the partition is removed and the mixture attains equilibrium.
V1 V2 μ1 μ2 p1 p2 Answer: We know that for an ideal gas, PV = μRT The two equations that we can write are: P1V1 = μ1R1T1 P2V2 = μ2R2T2 P1 = 1 atm, P2 = 2 atm and V1 = 2L, V2 = 3L Also, T1 = T = T2...
Calculate the ratio of the mean free paths of the molecules of two gases having molecular diameters 1 A and 2 A. The gases may be considered under identical conditions of temperature, pressure, and volume.
Answer: We know that that we can write: I α (1/d2) d1 = 1Å d2 = 2Å Therefore, l1 : l2 = 4 : 1 The ratio of the mean free paths of the molecules is 4:1
A gas mixture consists of 2.0 moles of oxygen and 4.0 moles of neon a temperature T. Neglecting all vibrational modes, calculate the total internal energy of the system. (Oxygen has two rotational modes.)
Answer: We know that oxygen has 5 degrees of freedom. Therefore, energy per mole = (5/2)RT Therefore, for 2 moles of oxygen, energy = 5RT Neon has 3 degrees of freedom. Therefore, energy per mole...
Two molecules of a gas have speeds of 9 × 106 m/s and 1 × 106 m/s respectively. What is the root mean square speed of these molecules?
Answer: Expression for root mean square velocity is as follows: $ {{v}_{rms}}=\sqrt{\frac{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+......+v_{n}^{2}}{n}} $ For two meolecules, the formula reduces to: $...
The molecules of a given mass of a gas have root mean square speeds of 100 m/s at 27oC and 1.00 atmospheric pressure. What will be the root mean square speeds of the molecules of the gas at 127oC and 2.0 atmospheric pressure?
Answer: According to the question, Vrms = 100 m/s T1 = 300 K and T2 = 400 K Expression for root mean square velocity is Vrms = √3RT/M Upon substituting the given values, we get Vrms = 115.4...
The volume of a given mass of a gas at 27 degrees C, 1 atm is 100 cc. What will be its volume at 327oC?
Answer: According to the question, T1 = 27oC = 300 K and the volume V1 = 100 cm3 It is known that V is proportional to T So, we can write V/T = constant Or, V1/T1 = V2/T2 Upon re-arranging, we have...
Calculate the number of atoms in 39.4 g gold. Molar mass of gold is 197 g/mole.
Answer: According to the question, the molar mass = mass of Avagadro's number of atoms = 6.023 × 1023 atoms Mass of gold is m = 39.4 g and the molar mass of gold is M = 197 g/mol We know that 197 g...
Solve for x, the inequalities in
Solution: Hence, \[\begin{array}{*{35}{l}} 1\text{ }\le \text{ }y\text{ }<\text{ }2 \\ \Rightarrow ~1\text{ }\le \text{ }\left| x-\text{ }\text{ }2 \right|\text{ }<\text{ }2 \\ \end{array}\]...
If arg (z – 1) = arg (z + 3i), then find x – 1 : y. where z = x + iy
Let \[z\text{ }=\text{ }x\text{ }+\text{ }iy\] Given that, \[arg\text{ }\left( z-\text{ }\text{ }1 \right)\text{ }=\text{ }arg\text{ }\left( z\text{ }+\text{ }3i \right)\] \[\Rightarrow ~arg\text{...
If the real part of ( z̅ + 2)/ ( z̅ – 1) is 4, then show that the locus of the point representing z in the complex plane is a circle.
Let z = x + iy Now, \[\Rightarrow ~{{x}^{2}}~+\text{ }x\text{ }\text{ }-2\text{ }+\text{ }{{y}^{2}}~=\text{ }4\text{ }\left( {{x}^{2}}~\text{ }-2x\text{ }+\text{ }1\text{ }+\text{ }{{y}^{2}}...
If a = cos θ + i sin θ, find the value of
SOLUTION: a = cos θ + i sin θ
When an ideal gas is compressed adiabatically, its temperature rises: the molecules on the average have more kinetic energy than before. The kinetic energy increases,
a) because of collisions with moving parts of the wall only b) because of collisions with the entire wall c) because the molecules gets accelerated in their motion inside the volume d) because of...
Prove 2 + 4 + 6 + …+ 2n = n^2 + n for all natural numbers n.
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }2\text{ }+\text{ }4\text{ }+\text{ }6\text{ }+\text{ }\ldots \text{ }+\text{ }2n\text{ }=\text{ }n^2\text{ }+\text{...
Which of the following diagrams depicts ideal gas behaviour?
Answer: The correct answer is a) c)
The quantum numbers of six electrons are given below. Arrange them in order of increasing energies. If any of these combination(s) has/have the same energy lists: n = 4, l = 2, ml = –2 , ms = –1/2 n = 3, l = 2, ml= 1 , ms = +1/2 n = 4, l = 1, ml = 0 , ms = +1/2 n = 3, l = 2, ml = –2 , ms = –1/2 n = 3, l = 1, ml = –1 , ms= +1/2 n = 4, l = 1, ml = 0 , ms = +1/2
The 4d, 3d, 4p, 3d, 3p, and 4p orbitals are home to electrons 1, 2, 3, 4, 5, and 6. (respectively). Ranking these orbitals in the increasing order of energies: (3p) < (3d) < (4p) < (4d).
Prove 2n < (n + 2)! for all natural number n.
As indicated by the inquiry, \[P\left( n \right)\text{ }is\text{ }2n\text{ }<\text{ }\left( n\text{ }+\text{ }2 \right)!\] In this way, subbing various qualities for n, we get, \[P\left( 0...
The unpaired electrons in Al and Si are present in 3p orbital. Which electrons will experience more effective nuclear charge from the nucleus?
The net positive charge acting on an electron in an atom's orbital with more than one electron is known as the nuclear charge. The nuclear charge increases as the atomic number increases. Silicon...
In a diatomic molecules, the rotational energy at a given temperature
a) obeys Maxwell’s distribution b) have the same value for all molecules c) equals the translational kinetic energy for each molecule d) is 2/3rd the translational kinetic energy for each molecule...
Diatomic molecules like hydrogen have energies due to both translational as well as rotational motion. From the equation in kinetic theory pV = 2/3 E, E is
a) the total energy per unit volume d) only the translational part of energy because rotational energy is very small compared to the translational energy c) only the translational part of the energy...
Prove n(n^2 + 5) is divisible by 6, for each natural number n.
As per the inquiry, \[P\left( n \right)\text{ }=\text{ }n\left( n^2\text{ }+\text{ }5 \right)\] is distinguishable by 6. Along these lines, subbing various qualities for n, we get, \[P\left( 0...
Prove n^3 – n is divisible by 6, for each natural number n ≥ 2.
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }n^3\text{ }-\text{ }n\] is distinct by 6. Along these lines, subbing various qualities for n, we get, \[P\left( 0 \right)\text{...
ABCDEFGH is a hollow cube made of an insulator. Face ABCD has a positive charge on it. Inside the cube, we have ionized hydrogen. The usual kinetic theory expression for pressure
a) will be valid b) will not be valid since the ions would experience forces other than due to collision with the walls c) will not be valid since collisions with walls would not be elastic d) will...
An inflated rubber balloon contains one mole of an ideal gas, has a pressure p, volume V, and temperature T. If the temperature rises to 1.1T and the volume is increased to 1.05V, the final pressure will be
a) 1.1 p b) p c) less than p d) between p and 1.1 Answer: The correct option is d) between p and 1.1 Explanation:
Prove For any natural number n, x^n – y^n is divisible by x – y, where x integers with x ≠ y.
As indicated by the inquiry, \[P\left( n \right)\text{ }=\text{ }xn\text{ }\text{ }yn\] is detachable by \[x\text{ }\text{ }y,\text{ }x\] whole numbers with\[x\text{ }\ne \text{ }y\] . In this way,...
Prove For any natural number n, 7^n – 2^n is divisible by 5.
As indicated by the inquiry, \[~P\left( n \right)\text{ }=\text{ }7n\text{ }\text{ }2n\] is distinct by 5. In this way, subbing various qualities for n, we get, \[~P\left( 0 \right)\text{ }=\text{...
A vessel of volume V contains a mixture of 1 mole of hydrogen and 1 mole of oxygen. Let f1(v)dv, denote the fraction of molecules with speed between v and (v + dv) with f2(v)dv similarly for oxygen. Then
a) f1(v) + f2(v) = f(v) obeys the Maxwell’s distribution law b) f1(v), f2(v) will obey the Maxwell’s distribution law separately c) neither f1(v) nor f2(v) will obey the Maxwell’s distribution law...
List gases which are responsible for greenhouse effect.
The major gases that cause greenhouse effect are: 1) Chlorofluorocarbons (CFCs) 2) Methane (CH4) 3) Carbon dioxide (CO2) 4) Nitrous oxide (NO) 5) Water(H2O) 6) Ozone (O3)
1 mole of H2 gas is contained in a box of volume V = 1.00 m3 at T = 300 K. The gas is heated to a temperature of T = 3000 K and the gas gets converted to a gas of hydrogen atoms. The final pressure would be
a) same as the pressure initially b) 2 times the pressure initially c) 10 times the pressure initially d) 20 times the pressure initially Answer: The correct option is d) 20 times the pressure...
Carbon monoxide gas is more dangerous than carbon dioxide gas. Why?
Carbon dioxide (CO2) and carbon monoxide (CO) are both produced when various fuels are burned. In nature, carbon monoxide is harmful, but carbon dioxide is non-toxic. Because carbon monoxide forms a...
Volume versus temperature graphs for a given mass of an ideal gas are shown in the figure at two different values of constant pressure. What can be inferred about the relation between P1 and P2?
a) P1 > P2 b) P1 = P2 c) P1 < P2 d) data is insufficient Answer: The correct option is a) P1 > P2 Explanation: When the pressure of an ideal gas is constant, Chale's law is obeyed, i.e. V ∝...
A cylinder containing an ideal gas is in a vertical position and has a piston of mass M that is able to move up or down without friction. If the temperature is increased,
a) both p and V of the gas will change b) only p will increase according to Charle’s law c) V will change but not p d) p will change but not V Answer: The correct option is c) V will change but not...
State which of the following statements are true and which are false. Justify your answer. (i) 35 ∈ {x | x has exactly four positive factors}. (ii) 128 ∈ {y | the sum of all the positive factors of y is 2y}
Solution: (i) The statement is true As per the question, $35 \in\{x \mid x$ has exactly four positive factors $\}$ 1, 5, 7, 35 are the possible positive factors of 35 So, 35 belongs to provided set...
Boyle’s law is applicable for an
a) adiabatic process b) isothermal process c) isobaric process d) isochoric process Answer: The correct option is b) isothermal process Explanation: In an isothermal process, when the temperature...
1 mole of an ideal gas is contained in a cubical volume V, ABCDEFGH at 300 K. One face of the cube (EFGH) is made up of a material which totally absorbs any gas molecule incident on it. At any given time,
a) the pressure on EFGH would be zero b) the pressure on all the faces will the equal c) the pressure of EFGH would be double the pressure on ABCD d) the pressure on EFGH would be half that on ABCD...
If Y = {x | x is a positive factor of the number 2p – 1 (2p – 1), where 2p – 1 is a prime number}. Write Y in the roaster form.
Solution: As per the question, $Y=\left\{x \mid x\right.$ is a positive factor of the number $2^{p-1}\left(2^{p}-1\right)$, in which $2^{p}-1$ is a prime number $\}$. Roster form, 1 and p itself are...
A cubic vessel (with faces horizontal + vertical) contains an ideal gas at NTP. The vessel is being carried by a rocket which is moving at a speed of 500 m/s in the vertical direction. The pressure of the gas inside the vessel as observed by us on the ground
a) remains the same because 500 m/s is very much smaller than vrms of the gas b) remains the same because the motion of the vessel as a whole does not affect the relative motion of the gas molecules...
Calculate the wavelength, frequency and wavenumber of a light wave whose period is 2.0 × 10–10 s.
Frequency of the light wave $\nu$ = $\frac{1}{Period} \frac{1}{Period}$ $=\frac{1}{2.0\times 10^{-10}\, s} =5.0\times 10^{9}\, s^{-1 }$ Wavelength of the light wave$\lambda=c\nu$ Where, c denotes...
Prove each of the statements in 4n – 1 is divisible by 3, for each natural number n.
As per the inquiry, \[P\left( n \right)\text{ }=\text{ }4n\text{ }\text{ }1\] is separable by 3. Thus, subbing various qualities for n, we get, \[P\left( 0 \right)\text{ }=\text{ }40\text{...
How many neutrons and protons are there in the following nuclei?
\({}_{6}^{13}C\): Mass number of carbon-13 = 13 Atomic number of carbon = Number of protons in one carbon atom = 6 Therfore, total number of neutrons in 1 carbon atom = Mass number – Atomic number =...
Give an example of a statement P(n) which is true for all n. Justify your answer.
As indicated by the inquiry, P(n) which is valid for all n. Let P(n) be, ⇒ P(k) is valid for all k. In this manner, P(n) is valid for all n.
(i) Calculate the number of electrons which will together weigh one gram. (ii) Calculate the mass and charge of one mole of electrons.
1 electron weighs 9.109*10-31 kg. Therefore, number of electrons that weigh 1 g (10-3 kg) = 1.098*1027 electrons (ii) Mass of one mole of electrons = NA* mass of one electron =...
Consider one mole of a perfect gas in a cylinder of unit cross-section with a piston attached. A spring is attached to the piston and to the bottom of the cylinder. Initially the spring is unstretched and the gas is in equilibrium. A certain amount of heat Q is supplied to the gas causing an increase of volume from V0 to V1.
c) using the first law of thermodynamics, write down the relation between Q, Pa, V, V0, and k. Answer: c) The relation between Q, Pa, V, V0, and k is as follows: dQ = dU + dW where, dU = Cv (T – T0)...
Consider one mole of a perfect gas in a cylinder of unit cross-section with a piston attached. A spring is attached to the piston and to the bottom of the cylinder. Initially the spring is unstretched and the gas is in equilibrium. A certain amount of heat Q is supplied to the gas causing an increase of volume from V0 to V1.
a) what is the initial pressure of the system? b) what is the final pressure of the system? Answer: a) Pa is the initial pressure of the system inside the cylinder b) The final pressure of the...
Consider that an ideal gas is expanding in a process given by P = f(V), which passes through a point (V0, P0). Show that the gas is absorbing heat at (P0, V0) if the slope of the curve P = f(V) is larger than the slope of the adiabat passing through (P0, V0).
Answer: The slope of the graph as given by: $ \left( {{V}_{0}},{{P}_{0}} \right)={{\left( \frac{dP}{dV} \right)}_{{{V}_{0}},{{P}_{0}}}} $ Making use of the above relation, we can determine that $...
A cycle followed by an engine is shown in the figure. Find heat exchanged by the engine, with the surroundings for each section of the cycle considering Cv = (3/2)R.
AB: constant volume BC: constant pressure CD: adiabatic DA: constant pressure Answer: a) AB: constant volume b) BC: constant pressure c) CD: adiabatic QCD = 0 d) DA: constant pressure, there is...
A cycle followed by an engine is shown in the figure.
A to B: volume constant B to C: adiabatic C to D: volume constant D to A: adiabatic VC = VD = 2VA = 2VB c) what is the work done by the engine in one cycle in terms of PA, PB, VA? d) what is the...
A cycle followed by an engine is shown in the figure.
A to B: volume constant B to C: adiabatic C to D: volume constant D to A: adiabatic VC = VD = 2VA = 2VB a) in which part of the cycle heat is supplied to the engine from outside? b) in which part of...
Consider a P-V diagram in which the path followed by one mole of a perfect gas in a cylindrical container is shown in the figure.
c) given the internal energy for one mole of gas at temperature T is (3/2)RT, find the heat supplied to the gas when it is taken from state 1 to 2 with V2 = 2V1. Answer: We know that the expression...
Consider a P-V diagram in which the path followed by one mole of a perfect gas in a cylindrical container is shown in the figure.
a) find the work done when the gas is taken from state 1 to state 2 b) what is the ratio of temperature T1/T2 if V2 = 2V1 Answer: According to the question, PV1/2 = K = constant a) Expression for...
Convert the following into basic units: (i) 28.7 pm (ii) 15.15 pm (iii) 25365 mg
(i) 28.7 pm $1 pm = 10^{ -12 } \; m$ $28.7 pm = 28.7 \times 10^{ -12 } \; m$ $= 2.87 \times 10^{ -11 } \; m$ (ii) 15.15 pm $1 pm = 10^{ -12 } \; m$ $15.15 pm = 15.15 \times 10^{ -12 } \; m$...
A welding fuel gas contains carbon and hydrogen only. Burning a small sample of it in oxygen gives 3.38 g carbon dioxide, 0.690 g of water and no other products. A volume of 10.0 L (measured at STP) of this welding gas is found to weigh 11.6 g. Find: (i) Empirical formula (ii) Molar mass of the gas, and (iii) Molecular formula
(i) Empirical formula 1 mole of $CO_{ 2 }$ contains 12 g of carbon Therefore, 3.38 g of $CO_{ 2 }$ will contain carbon = $\frac{ 12 \; g }{ 44 \; g } \; \times 3.38 \; g$ = 0.9217 g 18 g of...
A person driving a car suddenly applies the brakes on seeing a child on the road ahead. If he is not wearing a seat belt, he falls forward and hits his head against the steering wheel. Why?
If a person is not wearing a seat belt and abruptly hits the brakes, he will fall forward and bang his head against the steering wheel because his upper body will continue to move in the same...
The position time graph of a body of mass 2 kg is as given in the figure. What is the impulse on the body at t – 0 sec and t = 4 sec.
Mass of body, m = 2 kg Time, t = 0 Initial velocity, v1 = 0 From graph, we know that t ≥ 0 to t ≤ 4 which is a straight line. The velocity of the body is constant v2 = tan θ = 3/4 = 0.75 m/s At t ≥...
A person of mass 50 kg stands on a weighing scale on a lift. If the lift is descending with a downward acceleration of 9 m/s2 what would be the reading of the weighing scale?
When the lift lowers with an acceleration a, the apparent weight on the weighing scale decreases. W' denotes the apparent weight. Therefore, W' = R = (mg – ma) = m(g – a) As a result, W' = 50(10-9)...
A girl riding a bicycle along a straight road with a speed of 5 m/s throws a stone of mass 0.5 kg which has a speed of 15 m/s with respect to the ground along her direction of motion. The mass of the girl and the bicycle is 50 kg. Does the speed of the bicycle change after the stone is thrown? What is the change in speed, if so?
Given, m1 = 50 kg m2 = 0.5 kg u1 = 5 m/s u2 = 5 m/s v1 = ? v2 = 15 m/s The law of conservation of momentum states that Final momentum equals initial momentum. u1 = m1v1 + m2v2 (m1 + m2) We obtain by...
Two billiard balls A and B, each of mass 50 g and moving in opposite directions with speed of 5 m/s each, collide and rebound with the same speed. If the collision lasts for 10-3 seconds, which of the following statements are true? a) the impulse imparted to each ball is 0.25 kg.m/s and the force on each ball is 250N b) the impulse imparted to each ball is 0.25 kg.m/s and the force exerted on each ball is 25 × 10-5 N c) the impulse imparted to each ball is 0.5 Ns d) the impulse and the force on each ball are equal in magnitude and opposite in direction
The correct answer is: c) the impulse imparted to each ball is 0.5 Ns d) the impulse and the force on each ball are equal in magnitude and opposite in direction
In the figure, a body A of mass m slides on a plane inclined at angle θ1 to the horizontal and μ1 is the coefficient of friction between A and the plane. A is connected by a light string passing over a frictionless pulley to another body B, also of mass m, sliding on a frictionless plane inclined at angle θ2 to the horizontal. Which of the following statements are true?;a) A will never move up the plane b) A will just start moving up the plane when c) for A to move up the plane, θ2 must always be greater than θ1 d) B will always slide down with constant speed
The correct answer is: b) A will just start moving up the plane when \(\mu =\frac{\sin {{\theta }_{2}}-\sin {{\theta }_{1}}}{\cos {{\theta }_{1}}}\) c) for A to move up the plane, θ2 must always be...
Mass m1 moves on a slope making an angle θ with the horizontal and is attached to mass m2 by a string passing over a frictionless pulley as shown in the figure. The coefficient of friction between m1 and the sloping surface is μ. Which of the following statements are true?;a) if m2 > m1 sin θ, the body will move up the plane b) if m2 > m1(sin θ + μ cos θ), the body will move up the plane c) if m2 < m1 (sin θ + μ cos θ), the body will move up the plane d) if m2 < m1 (sin θ – μ cos θ), the body will move down the plane
The correct answer is: b) if m2 > m1(sin θ + μ cos θ), the body will move up the plane d) if m2 < m1 (sin θ – μ cos θ), the body will move down the plane
In figure, the coefficient of friction between the floor and the body B is 0.1. The coefficient of friction between the bodies B and A is 0.2. A force F is applied as shown on B. The mass of A is m/2 and of B is m. Which of the following statements are true?;a) the bodies will move together if F = 0.25 mg b) the body A will slip with respect to B if F = 0.5 mg c) the bodies will move together if F = 0.5 mg d) the bodies will be at rest if F = 0.1 mg e) the maximum value of F for which the two bodies will move together is 0.45 mg
The correct answer is: a) the bodies will move together if F = 0.25 mg b) the body A will slip with respect to B if F = 0.5 mg d) the bodies will be at rest if F = 0.1 mg e) the maximum value of F...
The motion of a particle of mass m is given by x = 0 for t < 0 sec, x(t) = A sin 4p t for 0 < t < (1/4) sec, and x = 0 for t > (1/4) sec. Which of the following statements is true? a) the force at t = (1/8) sec on the particle is -16π2Am b) the particle is acted upon by on impulse of magnitude 4π2Am at t = 0 sec and t = (1/4) sec c) the particle is not acted upon by any force d) the particle is not acted upon by a constant force e) there is no impulse acting on the particle
The correct answer is: a) the force at t = (1/8) sec on the particle is -16π2Am b) the particle is acted upon by on impulse of magnitude 4π2Am at t = 0 sec and t = (1/4) sec d) the particle is not...
A car of mass m starts from rest and acquires a velocity along the east a in two seconds. Assuming the car moves with uniform acceleration, the force exerted on the car is a) mv/2 eastward and is exerted by the car engine b) mv/2 eastward and is due to the friction on the tyres exerted by the road c) more than mv/2 eastward exerted due to the engine and overcomes the friction of the road d) mv/2 exerted by the engine
The correct answer is b) mv/2 eastward and is due to the friction on the tyres exerted by the road
A body with mass 5 kg is acted upon by a force F=-3i+4j N. If its initial velocity at t = 0 is v=6i-12j m/s, the time at which it will just have a velocity along the y-axis is a) never b) 10 s c) 2 s d) 15 s
The correct answer is b) 10 s
A body of mass 2 kg travels according to the law x(t) = p(t) + qt2 + rt3 where p = 3 m/s, q = 4 m/s2, and r = 5 m/s3. The force acting on the body at t=2 seconds is a) 136 N b) 134 N c) 158 N d) 68 N
The correct answer is a) 136 N
Conservation of momentum in a collision between particles can be understood from a) conservation of energy b) Newton’s first law only c) Newton’s second law only d) both Newton’s second and third law
The correct answer is d) both Newton’s second and third law
In the previous problem, the magnitude of the momentum transferred during the hit is a) zero b) 0.75 kg.m/s c) 1.5 kg.m/s d) 14 kg.m/s
The correct option is c) 1.5 kg.m/s
A cricket ball of mass 150 g has an initial velocity u=3i+4j m/s and a final velocity v=-(3i+4j) m/s after being hit. The change in momentum a) zero (b)-(0.45i+0.6j) (c)-(0.9i+1.2j)) (d)-5(i+j)
The correct answer is (c) -(0.9i+1.2j)
A metre scale is moving with uniform velocity. This implied a) the force acting on the scale is zero, but a torque about the centre of mass can act on the scale b) the force acting on the scale is zero and the torque acting about the centre of mass of the scale is also zero c) the total force acting on it need not be zero but the torque on it is zero d) neither the force nor the torque needs to be zero
The correct answer is b) the force acting on the scale is zero and the torque acting about the centre of mass of the scale is also zero
A ball is travelling with uniform translator motion. This means that a) it is at rest b) the path can be a straight line or circular and the ball travels with uniform speed c) all parts of the ball have the same velocity and the velocity is constant d) the centre of the ball moves with constant velocity and the ball spins about its centre uniformly
The correct option is c) all parts of the ball have the same velocity and the velocity is constant
A balloon filled with helium rises against gravity increasing its potential energy. The speed of the balloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy? You can neglect the viscous drag of air and assume that the density of air is constant.
The net buoyant force Equals vpg when the dragging viscous force of the air on the balloon is ignored. Where v is the volume of air that has been displaced. The upward net density is denoted by p....
Two identical steel cubes collide head-on face to face with a speed of 10 cm/s each. Find the maximum compression of each. Young’s modulus for steel = Y = 2 × 1011 N/m2.
Y = stress/strain Y = FL/A∆L WD = F∆L KE = 5 × 10-4 J WD = KE ∆L = 5 × 10-7 m
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...
A rocket accelerates straight up by ejecting gas downwards. In a small time interval ∆t, it ejects a gas of mass ∆m at a relative speed u. Calculate KE of the entire system at t + ∆t and t and show that the device that ejects gas does work = (1/2) ∆m u2 in this time interval.
M is the rocket's mass at any given moment t. The rocket's velocity is v. The mass of the gas expelled during the time interval t is m. As a result, K = 1/2 u2∆m
A curved surface as shown in the figure. The portion BCD is free of friction. There are three spherical balls of identical radii and masses. Balls are released from one by one from A which is at a slightly greater height than C. with the surface AB, ball 1 has large enough friction to cause rolling down without slipping; ball 2 has a small friction and ball 3 has a negligible friction. a) for which balls is total mechanical energy conserved? b) which ball can reach D? c) for balls which do not reach D, which of the balls can reach back A?
a) For ball 1 the total mechanical energy is conserved b) Ball 1 reaches D c) Ball 3 reaches back A
A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle of 30o by a force of 10 N parallel to the inclined surface. The coefficient of friction between the block and the incline is 0.1. If the block is pushed up by 10 m along the incline, calculate a) work done against gravity b) work done against the force of friction c) increase in potential energy d) increase in kinetic energy e) work done by an applied force
a) Work against gravity equals mgh 5 m= h 50 J WD against gravity b) The work done against the friction force is fs = 53 J. d) WD against gravity = 50 J increase in PE d) The system's increase in KE...
On complete combustion, a litre of petrol gives off heat equivalent to 3 × 107 J. In a test drive a car weighing 1200 kg, including the mass of driver, runs 15 km per litre while moving with a uniform speed on a surface and air to be uniform, calculate the force of friction acting on the car during the test drive, if the efficiency of the car engine were 0.5.
car engine Efficiency = 0.5 Energy given by the car with 1 litre of petrol = 1.5 × 107 WD = 1.5 × 107 f = 103 N
An adult weighing 600 N raises the centre of gravity of his body by 0.25 m while taking each step of 1 m length in jogging. If he jogs for 6 km, calculate the energy utilized by him in jogging assuming that there is no energy loss due to friction of ground and air. Assuming that the body of the adult is capable of converting 10% of energy intake in the form of food, calculate the energy equivalents of food that would be required to compensate energy utilized for jogging.
The energy used up is given as = mgh mg = 600 N h = 0.25m No.of steps in 6 km = 6000 steps Energy used in 6000 m = (6000)(600)(0.25)J Energy utilized in jogging = 9 × 104 J
An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of 50,000 kg is moving with a speed of 36 km/h when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If 90% of the energy of the wagon is lost due to friction, calculate the spring constant.
KE = 1/2 mv2 m = 50000 kg v = 10 m/s KE = 2500000J KE of spring = 10% of the KE wagon K = 5 × 105 N/m
Suppose the average mass of raindrops is 3.0 × 10^(-5) kg and their average terminal velocity 9 m/s. Calculate the energy transferred by rain to each square meter of the surface at a place which receives 100 cm of rain in a year.
Energy transferred by the rain to the surface of the earth = 1/2 mv2 The velocity of the rain = 9 m/s Mass = (volume)(density) = 1000 kg Energy transferred by 100 cm rainfall = 1/2 mv2 =...
Two pendulums with identical bobs and lengths are suspended from a common support such that in rest position the two bobs are in common. One of the bobs is released after being displaced by 10o so that it collides elastically head-on with the other bob. a) describe the motion of two bobs b) draw a graph showing variation in energy of either pendulum with time for 0 ≤ t ≤ 2T where T is the period of each pendulum
A raindrop of mass 1.00 g falling from a height of 1 km hits the ground with a speed of 50 m/s. Calculate a) the loss of PE of the drop b) the gain in KE of the drop c) is the gain in KE equal to loss of PE? If not why?
a) PE at the highest point = 10 J b) Gain in KE = 1/2 mv2 = 1.250 J c) Gain in KE is not equal to the PE
The bob A of a pendulum released from horizontal to the vertical hits another bob B of the same mass at rest on a table as shown in the figure. If the length of the pendulum is 1 m, calculate a) the height to which bob A will rise after collision b) the speed with which bob B starts moving. Neglect the size of the bobs and assume the collision to be elastic.
a) After the impact, bob A does not rise much because the PE of bob A is converted to KE and the momentum is transferred to bob B. (B) The speed of bob B is calculated as the sum of bob A's KE and...
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 $24.36 \times 10^{2}s$ $G=6.67 \times 10^{-11}Nm^{2}kg^{-1}$...
Consider a one-dimensional motion of a particle with total energy E. There are four regions A, B, C, and D in which the relation between potential energy V, kinetic energy (K) and total energy is as given below: Region A: V > E Region B: V < E Region C: K > E Region D: V > K State with reason in each case whether a particle can be found in the given region or not.
E = V + K and V > E for area A, implying that the KE is negative and therefore this is not feasible. K = E – V and V E for area B, implying that both energies are larger than zero. V = E – K and...
A ball of mass m, moving with a speed 2v0 collides inelastically with an identical ball at rest. Show that a) for a head-on collision, both the balls move forward b) for a general collision, the angle between the two velocities of scattered balls is less than 90o.
a) Let v1 and v2 be the velocities of the two balls after the collision. According to the law of conservation of momentum, mv0 = mv1 + mv2 v2 = v1 + 2ev0 e < 1 b) Using the law of conservation of...
A graph of potential energy V(x) versus x is shown in the figure. A particle of energy E0 is executing motion in it. Draw graph of velocity and kinetic energy versus x for one complete cycle AFA.
From the given graph of KE versus x From the below graph of velocity versus x
A bob of mass m suspended by a light string of length L is whirled into a vertical circle as shown in the figure. What will be the trajectory of the particle if the string is cut at a) point B b) point C c) point X
a) When the string is severed at point B, the particle's tangential velocity will be vertically downward, and the bob will travel in the same direction. b) When the string is severed at point C, the...
Two bodies of unequal mass are moving in the same direction with equal kinetic energy. The two bodies are brought to rest by applying retarding force of the same magnitude. How would the distance moved by them before coming to rest compare?
KE1 = KE2 WD1 = WD2 F1s1 = F2s2 F1 = F2 s1 = s2
Give an example of a situation in which an applied force does not result in a change in kinetic energy.
The kinetic energy of work done in a circular motion remains unchanged.
The average work done by a human heart while it beats once is 0.5 J. Calculate the power used by heart if it beats 72 times in a minute.
P = WD/time WD is one beat of heart = 0.5 J WD in 72 beats = 36 J P = WD/t = 0.6 W
Calculate the power of a crane in watts, which lifts a mass of 100 kg to a height of 10 min 20 sec.
P = WD/time = Fs cos θ/t = mgh cos θ/t h = 10 m t = 20 sec F = mg = 1000 Therefore, P = 500 Watts
In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls a) kinetic energy b) total linear momentum Give a reason for your answer in each case.
Because there is no non-conservative force, the kinetic energy and total linear momentum of the billiard balls are preserved.
A body is moved along a closed loop. Is the work done in moving the body necessarily zero? If not, state the condition under which work done over a closed path is always zero.
When the conservative force acts on the body during motion, the work done by the moving body is zero. When a non-conservative force acts on a moving body, the work done by the body is not zero.
A body falls towards earth in the air. Will its total mechanical energy be conserved during the fall? Justify.
The free-falling body's total mechanical energy is not preserved since it is utilised to overcome the frictional force of the air molecules.
Calculate the work done by a car against gravity in moving along a straight horizontal road. The mass of the car is 400 kg and the distance moved is 2m.
WD = Fs cos θ WD = Fs cos 90o = 0 Hence, the work done by the car against the gravity is zero.
A body is being raised to a height h from the surface of the earth. What is the sign of work done by a) applied force b) gravitational force
a) The applied force produces positive work. b) The gravitational pull produces negative work.
Why is electrical power required at all when the elevator is descending? Why should there be a limit on the number of passengers in this case?
In the event of an elevator falling, the number of passengers is limited because it is not a free fall and descends at a constant speed.
A rough inclined plane is placed on a cart moving with a constant velocity u on horizontal ground. A block of mass M rests on the incline. Is any work done by a force of friction between the block and incline? Is there then a dissipation of energy?
The block is clearly tilted on the plane in the illustration above. There is no work done since there is no displacement and no waste of energy.
Two blocks M1 and M2 having equal mass are free to move on a horizontal frictionless surface. M2 is attached to a massless spring as shown in the figure. Initially, M2 is at rest and M1 is moving toward M2 with speed v and collides head-on with M2.;a) while spring is fully compressed all the KE of M1 is stored as PE of spring b) while spring is fully compressed the system momentum is not conserved, though final momentum is equal to the initial momentum c) if spring is massless, the final state of the M1 is the state of rest d) if the surface on which blocks are moving has friction, then a collision cannot be elastic
c) if spring is massless, the final state of the M1 is a state of rest d) if the surface on which blocks are moving has friction, then a collision cannot be elastic
A bullet of mass m fired at 30o to the horizontal leaves the barrel of the gun with a velocity v. The bullet hits a soft target at a height h above the ground while it is moving downward and emerges out with half the kinetic energy it had before hitting the target. Which of the following statements are correct in respect of bullet after it emerges out of the target? a) the velocity of the bullet will be reduced to half its initial value b) the velocity of the bullet will be more than half of its earlier velocity c) the bullet will continue to move along the same parabolic path d) the bullet will move in a different parabolic path e) the bullet will fall vertically downward after hitting the target f) the internal energy of the particles of the target will increase
b) the velocity of the bullet will be more than half of its earlier velocity d) the bullet will move in a different parabolic path f) the internal energy of the particles of the target will...
A man, of mass m, standing at the bottom of the staircase, of height L, climbs it and stands at its top. a) work done by all forces on man is zero b) work done by all the force on man is zero c) work done by the gravitational force on man is mgL d) the reaction force from a step does not do work because the point of application of the force does not move while the force exists
b) work done by all the force on man is zero d) the reaction force from a step does not do work because the point of application of the force does not move while the force exists
A cricket ball of mass 150 g moving with a speed of 126 km/h hits at the middle of the bat, held firmly at its position by the batsman. The ball moves straight back to the bowler after hitting the bat. Assuming that collision between ball and bat is completely elastic and the two remain in contact for 0.001 sec, the force that the batsman had to apply to hold the bat firmly at its place would be a) 10.5 N b) 21 N c) 1.05 × N d) N
c) 1.05 × 104 N
Which of the diagrams in figure correctly shows the change in kinetic energy of an iron sphere falling freely in a lake having sufficient depth to impart it a terminal velocity?
(b
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 $1/2 mgR$
Show the nature of the following graph for a satellite orbiting the earth.
TE vs orbital radius R
Total energy of the satellite is $-GMm/2R$
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...
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 $0^{o}$.
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.