SIMPLE HARMONIC MOTION - Puucho

SIMPLE HARMONIC MOTION

PHYSICS

TARGET IIT JEE 2011

XI (J)

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CONTENTS

EXERCISE–I ....................................................................... Page –2

EXERCISE–II ..................................................................... Page –4

EXERCISE–III .................................................................... Page –5

OBJECTIVE QUESTION BANK ...................................... Page –11

ANSWER KEY ..................................................................... Page –23

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Simple Harmonic Motion [2]

EXERCISE–I

Q.1 The figure shows the displacement - time graph of a particle executing

SHM. If the time period of oscillation is 2s, then the equation of motion

is given by x = __________.

5

x

0t(s)

(mm)10

Q.2 A body is in SHM with period T when oscillated from a freely suspended spring. If this spring is cut in

two parts of length ratio 1 : 3 & again oscillated from the two parts separately, then the periods are T1 &

T2 then find T

1/T

2.

Q.3 A body undergoing SHM about the origin has its equation is given by x = 0.2 cos 5t. Find its average

speed from t = 0 to t = 0.7 sec.

Q.4 Two particles A and B execute SHM along the same line with the same amplitude a, same frequency and

same equilibrium position O. If the phase difference between them is = 2 sin–1 (0.9), then find the

maximum distance between the two.

Q.5 The acceleration-displacement (a – x) graph of a particle executing simple

harmonic motion is shown in the figure. Find the frequency of oscillation.

Q.6 A block of mass 0.9 kg attached to a spring of force constant k is lying on

a frictionless floor. The spring is compressed to 2 cm and the block is at

a distance 21 cm from the wall as shown in the figure. When the block

is released, it makes elastic collision with the wall and its period of motion

is 0.2 sec. Find the approximate value of k.

Q.7 A block is kept on a horizontal table. The table is undergoing simple harmonic motion of frequency 3Hz

in a horizontal plane. The coefficient of static friction between block and the table surface is 0.72. Find

the maximum amplitude of the table at which the block does not slip on the surface.

Q.8 A force f =

10 x

+ 2 acts on a particle of mass 0.1 kg, where '

k

' is in m and F in newton. If it

is released from rest at x = 2 m , find :

(a) amplitude ; (b) time period ; (c) equation of motion.

Q.9 Potential Energy (U) of a body of unit mass moving in a one-dimension conservative force field is given

by, U = (x2 – 4x + 3). All units are in S.I.

(i) Find the equilibrium position of the body.

(ii) Show that oscillations of the body about this equilibrium position is simple harmonic motion & find its

time period.

(iii) Find the amplitude of oscillations if speed of the body at equilibrium position is 2 6 m/s.

Q.10 A body is executing SHM under the action of force whose maximum magnitude is 50N. Find the

magnitude of force acting on the particle at the time when its energy is half kinetic and half potential.

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Q.11 The motion of a particle is described by x = 30 sin(t + /6), where x is in cm and t in sec. Potential

energy of the particle is twice of kinetic energy for the first time after t = 0 when the particle is at position

_____ after _____ time.

Q.12 Two blocks A (5kg) and B(2kg) attached to the ends of a spring constant 1120N/m are placed on a

smooth horizontal plane with the spring undeformed. Simultaneously velocities of 3m/s and

10m/s along the line of the spring in the same direction are imparted to A and B then

(a) find the maximum extension of the spring.

(b) when does the first maximum compression occurs after start.

Q.13 Two identical rods each of mass m and length L, are rigidly joined and then suspended in a

vertical plane so as to oscillate freely about an axis normal to the plane of paper passing

through ‘S’ (point of supension). Find the time period of such small oscillations.

Q.14(a) Find the time period of oscillations of a torsional pendulum, if the torsional constant of the wire is

K = 102J/rad. The moment of inertia of rigid body is 10 kg m2 about the axis of rotation.

(b) A simple pendulum of length l = 0.5 m is hanging from ceiling of a car. The car is kept on a horizontal

plane. The car starts accelerating on the horizontal road with acceleration of 5 m/s2. Find the time period

of oscillations of the pendulum for small amplitudes about the mean position.

Q.15 An object of mass 0.2 kg executes SHM along the x-axis with frequency of (25/) Hz. At the point

x = 0.04m the object has KE 0.5 J and PE 0.4 J. The amplitude of oscillation is ________.

Q.16 A body of mass 1 kg is suspended from a weightless spring having force constant 600N/m. Another

body of mass 0.5 kg moving vertically upwards hits the suspended body with a velocity of 3.0m/s and

get embedded in it. Find the frequency of oscillations and amplitude of motion.

Q.17 A body A of mass m1 = 1 kg and a body B of mass m

2 = 4 kg are attached to the ends of a

spring. The body A performs vertical simple harmonic oscillations of amplitude a =1.6 cm

and angular frequency = 25 rad/s. Neglecting the mass of the spring determine the

maximum and minimum values of force the system exerts on the surface on which it rests.

[Take g = 10 m/s2]

Q.18 A spring mass system is hanging from the ceiling of an elevator in equilibrium

Elongation of spring is l. The elevator suddenly starts accelerating downwards

with acceleration g/3 find

(a) the frequency and

(b) the amplitude of the resulting SHM.

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EXERCISE–II

Q.1 A point particle of mass 0.1kg is executing SHM with amplitude of 0.1m. When the particle passes

through the mean position, its K.E. is 8 × 10–3J. Obtain the equation of motion of this particle if the initial

phase of oscillation is 45°.

Q.2 One end of an ideal spring is fixed to a wall at origin O and the axis of spring is parallel to x-axis. A block

of mass m = 1 kg is attached to free end of the spring and it is performing SHM. Equation of position

of block in coordinate system shown is x = 10 + 3sin10t, t is in second and x in cm. Another block of

mass M = 3kg, moving towards the origin with velocity 30cm/c collides with the block performing SHM

at t = 0 and gets struck to it, calculate :

(i) new amplitude of oscillations.

(ii) new equation for position of the combined body.

(iii) loss of energy during collision. Neglect friction.

Q.3 A mass M is in static equilibrium on a massless vertical spring as shown in the

figure. A ball of mass m dropped from certain height sticks to the mass M after

colliding with it. The oscillations they perform reach to height 'a' above the original

level of scales & depth 'b' below it.

(a) Find the constant of force of the spring.; (b) Find the oscillation frequency.

(c) What is the height above the initial level from which the mass m was dropped ?

Q.4 Two identical balls A and B each of mass 0.1 kg are attached to two identical massless springs. The

spring mass system is constrained to move inside a rigid smooth pipe in the form of a circle as in fig. The

pipe is fixed in a horizontal plane. The centres of the ball can move in a circle of radius 0.06m. Each

spring has a natural length 0.06 m and force constant 0.1N/m.Initially both the balls are displaced by an

angle of = /6 radian with respect to diameter PQ of the circle and released from rest

(a) Calculate the frequency of oscillation of the ball B.

(b) What is the total energy of the system.

(c) Find the speed of the ball A when A and B are at the two ends of the

diameter PQ.

Q.5 Two blocks A(2kg) and B(3kg) rest up on a smooth horizontal surface

are connected by a spring of stiffness 120 N/m. Initially the spring is

undeformed. A is imparted a velocity of 2m/s along the line of the spring

away from B. Find the displacement of A t seconds later.

Q.6 The system shown in the figure can move on a smooth surface. The spring is

initially compressed by 6 cm and then released.Find

(a) time period

(b) amplitude of 3 kg block

(c) maximum momentum of 6 kg block

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Q.7 The resulting amplitude A' and the phase of the vibrations

S = A cos(t) +

2tcos

2

A + tcos

4

A +

2

3tcos

8

A= A' cos(t + ) are _____ and

_____ respectively.

Q.8 A spring block (force constant k = 1000 N/m and mass m = 4 kg) system is suspended from the ceiling

of an elevator such that block is initially at rest. The elevator begains to move upwards at t = 0. Acceleration

time graph of the elevator is shown in the figure. Draw the displacement x (from its initial position taking

upwards as positive) vs time graph of the block with respect to the elevator starting from t = 0 to t = 1

sec. Take 2 = 10.

Q.9 A particle of mass m moves in the potential energy U shown above. Find the period of the motion when

the particle has total energy E.

Q.10 A particle of mass 4 kg moves between two points A and B on a smooth horizontal surface under the

action of two forces such that when it is at a point P, the forces are PA2 N and

PB2 N. If the particle

is released from rest at A, find the time it takes to travel a quarter of the way from A to B.

EXERCISE–III

Q.1 A particle free to move along the x-axis has potential energy given by

U(x) = k[1-exp(-x2)] for – < x < +, where k is a positive constant of appropriate dimensions. Then

(A) at point away from the origin, the particle is in unstable equilibrium.

(B) for any finite nonzero value of x, there is a force directed away from the origin.

(C) if its total mechanical energy is k/2, it has its minimum kinetic energy at the origin.

(D) for small displacements from x = 0, the motion is simple harmonic. [JEE’ 99]

Q.2 Three simple harmonic motions in the same direction having the same amplitude a and same period are

superposed. If each differs in phase from the next by 450, then

(A) the resultant amplitude is (1+ 2 )a

(B) the phase of the resultant motion relative to the first is 900.

(C) the energy associated with the resulting motion is (3 + 2 2 ) times the energy associated with any

single motion.

(D) the resulting motion is not simple harmonic. [JEE’ 99]

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Q.3 The period of oscillation of simple pendulum of length L suspended from the roof of a vehicle which

moves without friction down an inclined plane of inclination is given by [JEE’ 2000]

(A) 2 L

gcos(B) 2

L

gsin(C) 2

L

g(D) 2

L

g tan

Q.4 A particle executes simple harmonic motion between x = –A and x = +A. The time taken for it to go

from 0 to A/2 is T1 and to go from A/2 to A is T

2. Then [JEE (Scr)’ 2001]

(A) T1 < T

2(B) T

1 > T

2(C) T

1 = T

2(D) T

1 = 2T

2

Q.5 A diatomic molecule has atoms of masses m1 and m

2. The potential energy of the molecule for the

interatomic separation r is given by V(r) = –A + B(r – r0)2, where r

0 is the equilibrium separation, and A

and B are positive constants. The atoms are compressed towards each other from their equilibrium

positions and released. What is the vibrational frequency of the molecule? [REE’ 2001]

Q.6 A particle is executing SHM according to y = a cos t. Then which of the graphs represents variations

of potential energy : [JEE (Scr)’ 2003]

(A) (I) & (III) (B) (II) & (IV)

(C) (I) & (IV) (D) (II) & (III)

Q.7 Two masses m1 and m

2 connected by a light spring of natural length l

0 is compressed completely and

tied by a string. This system while moving with a velocity v0 along +ve x-axis pass through the origin at

t = 0. At this position the string snaps. Position of mass m1 at time t is given by the equation.

x1 (t) = v

0 t – A (1 – cost)

Calculate :

(a) Position of the particle m2 as a function of time.

(b) l0 in terms of A. [JEE’ 2003]

Q.8 A block P of mass m is placed on a frictionless horizontal surface. Another block Q of same mass is kept

on P and connected to the wall with the help of a spring of spring constant k as shown in the figure. s is

the coefficient of friction between P and Q. The blocks move together performing SHM of amplitude A.

The maximum value of the friction force between P and Q is

(A) kA (B) 2

kA

(C) zero (D) smg [JEE' 2004]

Q.9 A simple pendulum has time period T1. When the point of suspension moves vertically up according to

the equation y = kt2 where k = 1 m/s2 and 't' is time then the time period of the pendulum is T2 then

2

2

1

T

T

is [JEE' 2005 (Scr)]

(A)6

5(B)

10

11(C)

5

6(D)

4

5

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Q.10 A small body attached to one end of a vertically hanging spring is performing SHM

about it's mean position with angular frequency and amplitude a. If at a height y* from

the mean position the body gets detached from the spring, calculate the value of y* so

that the height H attained by the mass is maximum. The body does not interact with the

spring during it's subsequent motion after detachment. (a2 > g). [JEE 2005]

Q.11 Function x =A sin2t + B cos2t + C sin t cos t represents SHM

(A) for any value of A, B and C (except C = 0)

(B) if A = –B; C = 2B, amplitude = 2B

(C) if A = B; C = 0

(D) if A = B; C = 2B, amplitude = B [JEE 2006]

Q.12 A student performs an experiment for determination of

2

2

T

4g

ll1m and he commits an error of l.

For The takes the time of n oscillations with the stop watch of least count T and he commits a human

error of 0.1sec. For which of the following data, the measurement of g will be most accurate?

l T n Amplitude of oscillation

(A) 5 mm 0.2 sec 10 5 mm

(B) 5 mm 0.2 sec 20 5 mm

(C) 5 mm 0.1 sec 20 1 mm

(D) 1 mm 0.1 sec 50 1 mm [JEE 2006]

Q.13 A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4k, respectively

(see figure I). The other ends are attached to identical supports M1 and M2 not attached to the walls.

The springs and supports have negligible mass. There is no friction anywhere. The block B is displaced

towards wall 1 by a small distance x (figure II) and released. The block returns and moves a maximum

distance y towards wall 2. Displacements x and y are measured with respect to the equilibrium position

of the block B. The ratio x

y is [JEE 2008]

Figure :

M2 S2 S1 M1

12

M2 S2 S1 M1

12 x

x

I

II

B

B

(A) 4 (B) 2 (C) 2

1(D)

4

1

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Q.14 Column I gives a list of possible set of parameters measured in some experiments. The variations of the

parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column

I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubbles of the

4 × 4 matrix given in the ORS. [JEE 2008]

Column I Column II

(A) Potential energy of a simple pendulum (y axis) (P)

Ox

y

as a function of displacement (x axis)

(B) Displacement (y axis) as a function of time (Q)

Ox

y

(x axis) for a one dimensional motion at zero or

constant acceleration when the body is moving

along the positive x-direction.

(C) Range of a projectile (y axis) as a function of its (R)

Ox

y

velocity (x axis) when projected at a fixed angle.

(D) The square of the time period (y axis) of a simple (S)

Ox

y

pendulum as a function of its length (x axis)

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Comprehension (3 Questions)

A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of

spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle

of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the

springs and the axle are in a horizontal plane. The unstretched length of each spring is L. The disk is

initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls

without slipping with velocity iVV 00

. The coefficient of friction is µ.

Figure :

2d

d d

V0R

y

x

Q.15 The net external force acting on the disk when its centre of mass is at displacement x with respect to its

equilibrium position is : [JEE 2008]

(A) –kx (B) –2kx (C) –3

kx2(D) –

3

kx4

Q.16 The centre of mass of the disk undergoes simple harmonic motion with angular frequency equal to :

[JEE 2008]

(A) M

k(B)

M

k2(C)

M3

k2(D)

M3

k4

Q.17 The maximum value of V0 for which the disk will roll without slipping is : [JEE 2008]

(A) µgk

M(B) µg

k2

M(C) µg

k

M3(D) µg

k2

M5

Q.18 The x-t graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the

particle at t = 4/3 s is [JEE-2009]

4 8 12 t(s)

x(c

m)

1

0

–1

(A) 22 s/cm

32

3 (B)

22

s/cm32

(C)

22

s/cm32

(D)

22 s/cm32

3

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Q.19 The mass M shown in the figure oscillates in simple harmonic motion with amplitude A. The amplitude of

the point P is [JEE-2009]

M•

k1 k2

P

(A) 2

1

k

Ak(B)

1

2

k

Ak(C)

21

1

kk

Ak

(D) 21

2

kk

Ak

Q.20 A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs

of equal spring constants k. The springs are fixed to rigid supports as shown in the figure, and the rod is

free to oscillate in the horizontal plane. The rod is gently pushed through a small angle in one direction

and released. The frequency of oscillation is [JEE-2009]

(A) M

k2

2

1

(B)

M

k

2

1

(C)

M

k6

2

1

(D)

M

k24

2

1

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OBJECTIVE QUESTION BANKONLY ONE OPTION IS CORRECT.

Take approx. 2 minutes for answering each question.

Q.1 A simple harmonic motion having an amplitude A and time period T is represented by the equation :

y = 5 sin (t + 4) m

Then the values of A (in m) and T (in sec) are :

(A) A = 5; T = 2 (B) A = 10 ; T = 1 (C) A = 5 ; T = 1 (D) A = 10 ; T = 2

Q.2 The maximum acceleration of a particle in SHM is made two times keeping the maximum speed to be

constant. It is possible when

(A) amplitude of oscillation is doubled while frequency remains constant

(B) amplitude is doubled while frequency is halved

(C) frequency is doubled while amplitude is halved

(D) frequency is doubled while amplitude remains constant

Q.3 A stone is swinging in a horizontal circle 0.8 m in diameter at 30 rev / min. A distant horizontal light beam

causes a shadow of the stone to be formed on a nearly vertical wall. The amplitude and period of the

simple harmonic motion for the shadow of the stone are

(A) 0.4 m, 4 s (B) 0.2 m. 2 s (C) 0.4 m, 2 s (D) 0.8 m, 2 s

Q.4 A small mass executes linear SHM about O with amplitude a and period T. Its displacement from O at

time T/8 after passing through O is:

(A) a/8 (B) a/22 (C) a/2 (D) a/2

Q.5 The displacement of a body executing SHM is given by x = A sin (2t + /3). The first time from t = 0

when the velocity is maximum is

(A) 0.33 sec (B) 0.16 sec (C) 0.25 sec (D) 0.5 sec

Q.6 A particle performing SHM is found at its equilibrium at t = 1sec. and it is found to have a speed of

0.25 m/s at t = 2 sec. If the period of oscillation is 6 sec. Calculate amplitude of oscillation

(A) 2

3 m (B)

4

3 m (C)

6

m (D) 8

3

Q.7 The time taken by a particle performing SHM to pass from point A to B where its velocities are same is

2 seconds. After another 2 seconds it returns to B. The time period of oscillation is (in seconds)

(A) 2 (B) 8 (C) 6 (D) 4

Q.8 The angular frequency of motion whose equation is dt

yd4

2

+ 9y = 0 is (y = displacement and t = time)

(A) 4

9(B)

9

4(C)

2

3(D)

3

2

Q.9 Time period of a particle executing SHM is 8 sec. At t = 0 it is at the mean position. The ratio of the

distance covered by the particle in the 1st second to the 2nd second is :

(A) 12

1

(B) 2 (C)

2

1(D) 12

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Q.10 A particle of mass 1 kg is undergoing S.H.M., for which graph between

force and displacement (from mean position) as shown. Its time period,

in seconds, is:

(A) /3 (B) /3 (C) /6 (D)

Q.11 A particle executes SHM on a straight line path. The amplitude of oscillation is 2 cm. When the

displacement of the particle from the mean position is 1 cm, the numerical value of magnitude of acceleration

is equal to the numerical value of magnitude of velocity. The frequency of SHM (in second–1) is:

(A) 2 3 (B) 3

2(C)

2

3(D)

32

1

Q.12 A particle executes SHM with time period T and amplitude A. The maximum possible average velocity

in time 4

Tis

(A) T

A2(B)

T

A4(C)

T

A8(D)

T

A24

Q.13 A particle executes SHM of period 1.2 sec. and amplitude 8 cm. Find the time it takes to travel 3cm

from the positive extremely of its oscillation.

(A) 0.28 sec. (B) 0.32 sec. (C) 0.17 sec. (D) 0.42 sec.

Q.14 A particle performs SHM with a period T and amplitude a. The mean velocity of the particle over the

time interval during which it travels a distance a/2 from the extreme position is

(A) a/T (B) 2a/T (C) 3a/T (D) a/2T

Q.15 A particle moves along the x-axis according to : x = A.[1 + sin t]. What distance does it travel between

t = 0 and t = 2.5/?

(A) 4A (B) 6A (C) 5A (D) None

Q.16 Two particles undergo SHM along parallel lines with the same time period (T) and equal amplitudes. At

a particular instant, one particle is at its extreme position while the other is at its mean position. They

move in the same direction. They will cross each other after a further time

(A) T/8 (B) 3T/8 (C) T/6 (D) 4T/3

Q.17 Two particles are in SHM in a straight line about same equilibrium position. Amplitude A and time period

T of both the particles are equal. At time t=0, one particle is at displacement y1= +A and the other at y

2=

–A/2, and they are approaching towards each other. After what time they cross each other ?

(A) T/3 (B) T/4 (C) 5T/6 (D) T/6

Q.18 Two particles execute SHM of same amplitude of 20 cm with same period along the same line about the

same equilibrium position. The maximum distance between the two is 20 cm. Their phase difference in

radians is

(A) 3

2(B)

2

(C)

3

(D)

4

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Q.19 Two particles A and B perform SHM along the same straight line with the same amplitude ‘a’, same

frequency ‘f’ and same equilibrium position ‘O’. The greatest distance between them is found to be

3a/2. At some instant of time they have the same displacement from mean position. What is the displacement?

(A) a/2 (B) 47a (C) 2a3 (D) 4a3

Q.20 Two pendulums have time periods T and 5T/4. They start SHM at the same time from the mean position.

After how many oscillations of the smaller pendulum they will be again in the same phase:

(A) 5 (B) 4 (C) 11 (D) 9

Q.21 Two particles are in SHM on same straight line with amplitude A and 2A and with same angular frequency

. It is observed that when first particle is at a distance 2A from origin and going toward mean

position, other particle is at extreme position on other side of mean position. Find phase difference

between the two particles

(A) 45° (B) 90° (C) 135° (D) 180°

Q.22 A body performs simple harmonic oscillations along the straight line ABCDE with C as the midpoint of

AE. Its kinetic energies at B and D are each one fourth of its maximum value. If AE = 2R, the distance

between B and D is

(A) 2

R3(B)

2

R(C) R3 (D) R2

Q.23 Two particles P and Q describe simple harmonic motions of same period, same amplitude, along the

same line about the same equilibrium position O. When P and Q are on opposite sides of O at the same

distance from O they have the same speed of 1.2 m/s in the same direction, when their displacements are

the same they have the same speed of 1.6 m/s in opposite directions. The maximum velocity in m/s of

either particle is

(A) 2.8 (B) 2.5 (C) 2.4 (D) 2

Question No. 24 to 26 (3 questions)

The graphs in figure show that a quantity y varies with displacement d in a system undergoing simple

harmonic motion.

(I) (II) (III) (IV)

Which graphs best represents the relationship obtained when y is

Q.24 The total energy of the system

(A ) I (B) II (C) III (D) IV

Q.25 The time

(A ) I (B) II (C) III (D) IV

Q.26 The unbalanced force acting on the system.

(A ) I (B) II (C) III (D) None

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Q.27 A spring mass system preforms S.H.M. If the mass is doubled keeping amplitude same, then the total

energy of S.H.M. will become :

(A) double (B) half (C) unchanged (D) 4 times

Q.28 A mass at the end of a spring executes harmonic motion about an equilibrium position with an amplitude

A. Its speed as it passes through the equilibrium position is V. If extended 2A and released, the speed of

the mass passing through the equilibrium position will be

(A) 2V (B) 4V (C) V

2(D)

V

4

Q.29 A particle starts oscillating simple harmonically from its equilibrium position then, the ratio of kinetic

energy and potential energy of the particle at the time T/12 is : (T = time period)

(A) 2 : 1 (B) 3 : 1 (C) 4 :1 (D) 1 : 4

Q.30 If the potential energy of a harmonic oscillator of mass 2 kg on its equilibrium position is 5 joules and the

total energy is 9 joules when the amplitude is one meter the period of the oscillator (in sec) is:

(A) 1.5 (B) 3.14 (C) 6.28 (D) 4.67

Q.31 A plank with a small block on top of it is under going vertical SHM. Its period is 2 sec. The minimum

amplitude at which the block will separate from plank is :

(A) 2

10

(B)

10

2(C) 2

20

(D)

10

Q.32 Find the ratio of time periods of two identical springs if they are first joined in series & then in parallel

& a mass m is suspended from them :

(A) 4 (B) 2 (C) 1 (D) 3

Q.33 In an elevator, a spring clock of time period TS (mass attached to a spring) and a pendulum clock of time

period TP are kept. If the elevator accelerates upwards

(A) TS well as T

P increases (B) T

S remain same, T

P increases

(C) TS remains same, T

P decreases (D) T

S as well as T

P decreases

Q.34 Two bodies P & Q of equal mass are suspended from two separate massless springs of force constants

k1 & k

2 respectively. If the maximum velocity of them are equal during their motion, the ratio of

amplitude of P to Q is :

(A) 2

1

k

k(B)

1

2

k

k(C)

1

2

k

k(D)

2

1

k

k

Q.35 The springs in fig. A and B are identical but length in A is three times each of that in B.

The ratio of period TA/T

B is

(A) 3 (B) 1/3 (C) 3 (D) 1/3

Q.36 In the figure, the block of mass m, attached to the spring of stiffness k is in contact with the completely

elastic wall, and the compression in the spring is 'e'. The spring is compressed further by 'e' by displacing

the blocktowards left and is then released. If the collision between the block and the wall is completely

elastic then the time period of oscillations of the block will be:

(A) k

m

3

2(B) 2

k

m

(C) k

m

3

(D)

k

m

6

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Q.37 A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is

V(x) = k|x|3 where k is a positive constant. If the amplitude of oscillations is a, then its time period T is

(A) proportional to 1/ a (B) independent of a

(C) proportional to a (D) proportional to a3/2

Q.38 A small bob attached to a light inextensible thread of length l has a periodic time T when

allowed to vibrate as a simple pendulum. The thread is now suspended from a fixed end

O of a vertical rigid rod of length 4

3l(as in figure). If now the pendulum performs periodic

oscillations in this arrangement, the periodic time will be

(A) 4

T3(B)

2

T(C) T (D) 2T

Q.39 A 2 Kg block moving with 10 m/s strikes a spring of constant 2 N/m attached to 2 Kg block at rest

kept on a smooth floor. The time for which rear moving block remain in contact with spring will be

(A) 2 sec (B) 2

1sec

(C) 1 sec (D) 2

1sec

Q.40 In the above question, the velocity of the rear 2 kg block after it separates from the spring will be :

(A) 0 m/s (B) 5 m/s (C) 10 m/s (D) 7.5 m/s

Q.41 A system of two identical rods (L-shaped) of mass m and length l are

resting on a peg P as shown in the figure. If the system is displaced

in its plane by a small angle , find the period of oscillations:

(A) 22

3

l

g (B) 2

2 2

3

l

g(C) 2

2

3

l

g(D) 3

l

3g

Q.42 A ring of diameter 2m oscillates as a compound pendulum about a horizontal axis passing through a

point at its rim. It oscillates such that its centre move in a plane which is perpendicular to the plane of the

ring. The equivalent length of the simple pendulum is

(A) 2m (B) 4m (C) 1.5m (D) 3m

Q.43 A man is swinging on a swing made of 2 ropes of equal length L and in

direction perpendicular to the plane of paper. The time period of the

small oscillations about the mean position is

(A) 2 g2

L(B) 2

g2

L3

(C) 2 g32

L(D)

g

L

Q.44 A rod whose ends are A & B of length 25 cm is hanged in vertical plane. When hanged from point A and

point B the time periods calculated are 3 sec & 4 sec respectively. Given the moment of inertia of rod

about axis perpendicular to the rod is in ratio 9 : 4 at points A and B. Find the distance of the centre of

mass from point A.

(A) 9 cm (B) 5 cm (C) 25 cm (D) 20 cm

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Q.45 A wire frame in the shape of an equilateral triangle is hinged at one vertex so that it can swing freely in a

vertical plane, with the plane of the always remaining vertical. The side of the frame is 3/1 m. The

time period in seconds of small oscillations of the frame will be

(A) 2

(B) 2 (C)

6

(D)

5

Q.46 A circular disc has a tiny hole in it, at a distance z from its center. Its mass is M and radius R (R>z). A

horizontal shaft is passed through the hole and held fixed so that the disc can freely swing in the vertical

plane. For small disturbance, the disc performs SHM whose time period is minimum for z =

(A) R / 2 (B) R / 3 (C) R / 2 (D) R / 3

Q.47 A particle is subjected to two mutually perpendicular simple harmonic motions such that its x and

y coordinates are given by

x = 2 sin t ; y = 2 sin

4t

The path of the particle will be :

(A) an ellipse (B) a straight line (C) a parabola (D) a circle

Q.48 In the figure shown, the spring are connected to the rod at one end and

at the midpoint. The rod is hinged at its lower end. Rotational SHM of

the rod (Mass m, length L) will occur only if

(A) k > mg/3L (B) k > 2mg/3L

(C) k > 2mg/5L (D) k > 0

Q.49 What is the angular frequency of oscillations of the rod in the above problem if k = mg/L?

(A) (3/2).[k/m]1/2 (B) (3/4).[k/m]1/2 (C) [2k/5m]1/2 (D) None

Q.50 The amplitude of the vibrating particle due to superposition of two SHMs,

y1 = sin

t

3

and y2 = sin t is :

(A) 1 (B) 2 (C) 3 (D) 2

Q.51 Two simple harmonic motions y1 = A sin t and y

2 = A cos t are superimposed on a particle of mass m.

The total mechanical energy of the particle is:

(A) 2

1m2A2 (B) m2A2 (C)

4

1m2A2 (D) zero

Q.52 A simple pendulum is oscillating in a lift. If the lift is going down with constant velocity, the time period of

the simple pendulum is T1. If the lift is going down with some retardation its time period is T

2, then

(A) T1 > T

2(B) T

1 < T

2

(C) T1

= T2

(D) depends upon the mass of the pendulum bob

Q.53 Vertical displacement of a plank with a body of mass 'm' on it is varying according to law

y = sin t + 3 cos t. The minimum value of for which the mass just breaks off the plank and the

moment it occurs first after t = 0 are given by: ( y is positive vertically upwards)

(A) g6

2,

2

g (B)

g3

2,

2

g (C)

g

2

3,

2

g (D)

g3

2,g2

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Q.54 Equations y = 2A cos2 t and y = A (sin t + 3 cos t ) represent the motion of two particles.

(A) Only one of these is S.H.M. (B) Ratio of maximum speeds is 2 : 1

(C) Ratio of maximum speeds is 1 : 1 (D) Ratio of maximum accelerations is 1 : 4

Q.55 A block of mass ‘m’ is attached to a spring in natural length of spring constant ‘k’. The other end A of the

spring is moved with a constant velocity v away from the block. Find the maximum extension in the

spring.

(A) k

vm

4

1 2

(B) k

vm 2

(C) k

vm

2

1 2

(D) k

vm2

2

Q.56 In the above question, find the amplitude of oscillation of the block in the reference frame of point A of

the spring.

(A) k

vm

4

1 2

(B) k

vm

2

1 2

(C) k

vm 2

(D) k

vm2

2

Q.57 For a particle acceleration is defined as |x|

ix5a

for x 0 and a

= 0 for x = 0. If the particle is initially

at rest (a, 0) what is period of motion of the particle.

(A) 5/a24 sec. (B) 5/a28 sec. (C) 5/a22 sec. (D) cannot be determined

Q.58 A mass m, which is attached to a spring with spring constant k, oscillates on a horizontal table, with

amplitude A. At an instant when the spring is stretched by 2/A3 , a second mass m is dropped

vertically onto the original mass and immediately sticks to it. What is the amplitude of the resulting

motion?

(A) A2

3(B) A

8

7(C) A

16

13(D) A

3

2

ASSERTION AND REASON

Q.1 Statement-1 : A particle is moving along x-axis. The resultant force F acting on it at position x is given

by F = – ax – b. Where a and b are both positive constants. The motion of this particle is

not SHM.

Statement-2 : In SHM restoring force must be proportional to the displacement from mean position.

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

(C) Statement-1 is true, statement-2 is false.

(D) Statement-1 is false, statement-2 is true.

Q.2 Statement-1 : For a particle performing SHM, its speed decreases as it goes away from the mean

position.

Statement-2 : In SHM, the acceleration is always opposite to the velocity of the particle.





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Q.3 Statement-1 : Motion of a ball bouncing elastically in vertical direction on a smooth horizontal floor is

a periodic motion but not an SHM.

Statement-2 : Motion is SHM when restoring force is proportional to displacement from mean position.





Q.4 Statement-1 : A particle, simultaneously subjected to two simple harmonic motions of same frequency

and same amplitude, will perform SHM only if the two SHM’s are in the same direction.

Statement-2 : A particle, simultaneously subjected to two simple harmonic motions of same frequency

and same amplitude, perpendicular to each other the particle can be in uniform circular

motion.





Q.5 Statement-1 : In case of oscillatory motion the average speed for any time interval is always greater

than or equal to its average velocity.

Statement-2 : Distance travelled by a particle cannot be less than its displacement.





ONE OR MORE THAN ONE OPTION MAY BE CORRECT

Take approx. 3 minutes for answering each question.

Q.1 Part of a simple harmonic motion is graphed in the figure, where y is the displacement from the mean

position. The correct equation describing this S.H.M is

(A) y = 4 cos (0.6t) (B) y = 2 sin

2t

3

10

(C) y = 4 sin

2t

3

10(D) y = 2 cos

2t

3

10

Q.2 Speed v of a particle moving along a straight line, when it is at a distance x from a fixed point on the line

is given by v2 = 108 - 9x2 (all quantities in S.I. unit). Then

(A) The motion is uniformly accelerated along the straight line

(B) The magnitude of the acceleration at a distance 3 cm from the fixed point is 0.27 m/s2.

(C) The motion is simple harmonic about x = 12 m.

(D) The maximum displacement from the fixed point is 4 cm.

Q.3 The displacement-time graph of a particle executing SHM is shown.

Which of the following statements is/are true?

(A) The velocity is maximum at t = T/2

(B) The acceleration is maximum at t = T

(C) The force is zero at t = 3T/4

(D) The potential energy equals the oscillation energy at t = T/2.

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Q.4 A particle is executing SHM with amplitude A, time period T, maximum acceleration ao and maximum

velocity v0. Its starts from mean position at t=0 and at time t , it has the displacement

A/2, acceleration a and velocity v then

(A) t=T/12 (B) a=ao/2 (C) v=v

o/2 (D) t=T/8

Q.5 The amplitude of a particle executing SHM about O is 10 cm. Then:

(A ) When the K .E. is 0.64 of its max. K .E. its displacement is 6cm from O.

(B) When the displacement is 5 cm from O its K.E. is 0.75 of its max.P.E.

(C) Its total energy at any point is equal to i ts maximum K.E.

(D) Its velocity is half the maximum velocity when its displacement is half the maximum displacement.

Q.6 In SHM, acceleration versus displacement (from mean position) graph:

(A) is always a straight line passing through origin and slope –1

(B) is always a straight line passing through origin and slope +1

(C) is a straight l ine not necessarily passing through origin

(D) none of the above

Q.7 For a particle executing S.H.M ., x = displacement from equilibrium position, v = velocity at any instant

and a = acceleration at any instant, then

(A) v-x graph is a circle (B) v-x graph is an ell ipse

(C) a-x graph is a straight line (D) a-v graph is an ellipse

Q.8 A particle starts from a point P at a distance of A /2 from the mean position O & travels towards left as

shown in the figure. If the time period of SHM, executed about O is T and amplitude A then the equation

of motion of particle is :

(A ) x = A sin

6

tT

2(B) x = A sin

6

5t

T

2

(C) x = A cos

6

tT

2(D) x = A cos

3

tT

2

Q.9 The graph plotted between phase angle () and displacement of a particle fromequilibrium position (y) is a sinusoidal curve as shown below. Then the bestmatching is

Column A Column B

(a) K.E. versus phase angle curve (i)

(b) P.E. versus phase angle curve (ii)

(c) T.E. versus phase angle curve (iii)

(d) Velocity versus phase angle curve (iv)

(A) (a)-(i), (b)-(ii), (c)-(iii) & (d)-(iv) (B) (a)-(ii), (b)-(i), (c)-(iii) & (d)-(iv)(C) (a)-(ii), (b)-(i), (c)-(iv) & (d)-(iii) (D) (a)-(ii), (b)-(iii), (c)-(iv) & (d)-(i)

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Q.10 A particle of mass m performs SHM along a straight line with frequency f and amplitude A.

(A) The average kinetic energy of the particle is zero.

(B) The average potential energy is m 2f2A2.

(C) The frequency of ocillation of kinetic energy is 2f.

(D) Velocity function leads acceleration by /2.

Q.11 A system is oscillating with undamped simple harmonic motion. Then the

(A) average total energy per cycle of the motion is its maximum kinetic energy.

(B) average total energy per cycle of the motion is 2

1times its maximum kinetic energy..

(C) root mean square velocity is 2

1times its maximum velocity

(D) mean velocity is 1/2 of maximum velocity.

Q.12 The angular frequency of a spring block system is 0.This system is suspended from the ceiling of an

elevator moving downwards with a constant speed v0. The block is at rest relative to the elevator. Lift is

suddenly stopped. Assuming the downwards as a positive direction, choose the wrong statement:

(A) The amplitude of the block is 0

0v

(B) The initial phase of the block is .

(C) The equation of motion for the block is 0

0v

sin 0t.

(D) The maximum speed of the block is v0.

Q.13 Two particles execute SHM with amplitude A and 2A and angular frequency and 2 respectively. At

t = 0 they starts with some initial phase difference. At, t =

3

2. They are in same phase. Their initial phase

difference is :

(A) 3

(B)

3

2(C)

3

4(D)

Q.14 Two particles are in SHM with same angular frequency and amplitudes A and 2A respectively along

same straight line with same mean position. They cross each other at position A/2 distance from mean

position in opposite direction. The phase between them is :

(A)

4

1sin

6

5 1(B)

4

1sin

6

1(C)

4

1cos

6

5 1(D)

4

1cos

6

1

Q.15 A spring has natural length 40 cm and spring constant 500 N/m. A block of mass 1 kg is attached at one

end of the spring and other end of the spring is attached to ceiling. The block released from the position,

where the spring has length 45 cm.

(A) the block will perform SHM of amplitude 5 cm.

(B) the block will have maximum velocity 530 cm/sec.

(C) the block will have maximum acceleration 15 m/s2

(D) the minimum potential energy of the spring will be zero.

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Q.16 A particle executing a simple harmonic motion of period 2s. When it is at its extreme displacement from

its mean position, it receives an additional energy equal to what it had in its mean position. Due to this ,

in its subsequent motion,

(A) its amplitude will change and become equal to 2 times its previous amplitude

(B) its periodic time will become doubled i.e. 4s

(C) its potential energy will be decreased

(D) it will continue to execute simple harmonic motion of the same amplitude and period as before

receiving the additional energy.

Q.17 A particle is executing SHM of amplitude A, about the mean position x = 0. Which of the following

cannot be a possible phase difference between the positions of the particle at x = + 2A and

x = – 2A .

(A) 75° (B) 165° (C) 135° (D) 195°

Q.18 The figure shows a graph between velocity and displacement

(from mean position) of a particle performing SHM:

(A) the time period of the particle is 1.57s

(B) the maximum acceleration will be 40cm/s2

(C) the velocity of particle is 2 21 cm/s when it is at a distance 1 cm from the mean position.

(D) none of these

Q.19 Two springs with negligible masses and force constant of K1 = 200 Nm–1

and K

2 = 160 Nm–1 are

attached to the block of mass m = 10 kg as shown in the figure. Initially the block is at rest, at the

equilibrium position in which both springs are neither stretched nor compressed. At time t = 0, a sharp

impulse of 50 Ns is given to the block with a hammer.

(A) Period of oscillations for the mass m is3

s.

(B) Maximum velocity of the mass m during its oscillation is 5 ms–1.

(C) Data are insufficient to determine maximum velocity.

(D) Amplitude of oscillation is 0.42 m.

Q.20 A mass of 0.2kg is attached to the lower end of a massless spring of force-constant 200 N/m, the upper

end of which is fixed to a rigid support. Which of the following statements is/are true?

(A) In equilibrium, the spring will be stretched by 1cm.

(B) If the mass is raised till the spring is unstretched state and then released, it will go down by 2cm

before moving upwards.

(C) The frequency of oscillation will be nearly 5 Hz.

(D) If the system is taken to the moon, the frequency of oscillation will be the same as on the earth.

Q.21 The potential energy of a particle of mass 0.1kg, moving along x-axis, is given by U = 5x(x-4)J where

x is in metres. It can be concluded that

(A) the particle is acted upon by a constant force.

(B) the speed of the particle is maximum at x = 2 m

(C) the particle executes simple harmonic motion

(D) the period of oscillation of the particle is /5 s.

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Q.22 Two blocks of masses 3 kg and 6 kg rest on a horizontal smooth

surface. The 3 kg block is attached to a spring with a force constant

k = 900 Nm-1 which is compressed 2 m from beyond the equilibrium

position.The 6 kg mass is at rest at 1m from mean position.3kg mass

strikes the 6 kg mass and the two stick together.

(A) velocity of the combined masses immediately after the collision is 10 ms-1

(B) velocity of the combined masses immediately after thecollision is 5 ms-1

(C) Amplitude of the resulting oscillation is 2 m

(D) Amplitude of the resulting oscillation is 5/2 m.

Q.23 The displacement of a particle varies according to the relation x = 3 sin 100t + 8 cos2 50t . Which of the

following is/are correct about this motion .

(A) the motion of the particle is not S.H.M.

(B) theamplitude of the S.H.M. of the particle is 5 units

(C) the amplitude of the resultant S.H. M. is 73 units

(D) the maximum displacement of the particle from the origin is 9 units .

Q.24 The equation of motion for an oscillating particle is given by x = 3sin(4t) + 4cos(4t), where x is in mm

and t is in second

(A) The motion is simple harmonic (B) The period of oscillation is 0.5 s

(C) The amplitude of oscillation is 5 mm (D) The particle starts its motion from the equilibrium

Q.25 A linear harmonic oscillator of force constant 2 × 106Nm-1 and amplitude 0.01 m has a total mechanical

energy of 160 J. Its

(A) maximum potential energy is 100 J (B) maximum kinetic energy is 100 J

(C) maximum potential energy is 160 J (D) minimum potential energy is zero.

Q.26 The two blocks shown here rest on a frictionless surface. If they are

pulled apart by a small distance and released at t = 0, the time when

1 kg block comes to rest can be

(A) 3

2sec. (B) sec. (C)

2

sec. (D)

9

sec.

Q.27 A disc of mass 3m and a disc of mass m are connected by a massless spring of stiffness k. The heavier

disc is placed on the ground with the spring vertical and lighter disc on top. From its equilibrium position,

the upper disc is pushed down by a distance and released. Then

(A) if > 3mg/k, the lower disc will bounce up

(B) if = 2mg/k, maximum normal reaction from ground on lower disc = 6 mg.

(C) if = 2mg/k, maximum normal reaction from ground on lower disc = 4 mg.

(D) if > 4mg/k, the lower disc will bounce up

Q.28 A particle moves in xy plane according to the law x = a sint and y = a(1-cost) where a and are

constants. The particle traces

(A) a parabola (B) a straight line equallyinclined to x and y axes

(C) a circle (D) a distance proportional to time.

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ANSWER KEY

EXERCISE–I

Q.1 x = 10sin (t + /6) Q.2 1 3 Q.3 2 m/s Q.4 1.8 a

Q.5

2

1Q.6 100 Nm–1 Q.7 2cm

Q.85

11m, (b)

5

sec., (c) x = 0.2 – tcos

5

11 Q.9 (i) x

0 = 2m; (ii) T = 2 p sec.; (iii) 32

Q.10 225 N Q.11 cm610 , sec6

1

3

2sin

1 1

Q.12 (a) 25cm, (b) 3/56 seconds Q.13g18

L172 Q.14 (a) 2 sec, (b) T = 4/15

2 sec

Q.15 0.06m Q.16 10/ Hz, 6

375cm Q.17 60N, 40N

Q.18 (a)L

g

2

1

T

1

, (b)

3

L

EXERCISE–II

Q.1 y = 0.1sin(4t + /4) Q.2 3cm, x = 10 – 3sin5t; E = 0.135J

Q.3 (a)ab

ab

m

mM)c(;

ab

mg2K

, )mM)(ab(

mg2

2

1

Q.4 f =1

; E=42 ×10–5J;v=2×10–2m/s Q.5 0.8t + 0.12 sin 10t

Q.6 (a) 10

sec, (b) 4 cm, (c) 2.40 kg m/s. Q.7

8

A53,

2

1tan 1

Q.8 Q.9 2mg/E22k/m Q.103

s

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EXERCISE–III

Q.1 D Q.2 A, C Q.3 A Q.4 A Q.5f

1=

)mm(B2

mm2

21

21

Q.6 A Q.7 (a) x2 = v

0t + )tcos1(A

m

m

2

1 , (b) l0 = A1

m

m

2

1

Q.8 B Q.9 C Q.10 y* = ag

k

mg2

Q.11 A,B,D Q.12 D

Q.13 C Q.14 (A) P,S (B) Q,R,S (C) S (D) Q Q.15 D Q.16 D

Q.17 C Q.18 D Q.19 D Q.20 C

QUESTION BANK

ONLY ONE OPTION IS CORRECT.

Q.1 A Q.2 C Q.3 C Q.4 D Q.5 A Q.6 A Q.7 B

Q.8 C Q.9 D Q.10 B Q.11 C Q.12 D Q.13 C Q.14 C

Q.15 C Q.16 B Q.17 D Q.18 C Q.19 B Q.20 A Q.21 C

Q.22 C Q.23 D Q.24 A Q.25 D Q.26 D Q.27 C Q.28 A

Q.29 B Q.30 B Q.31 A Q.32 B Q.33 C Q.34 B Q.35 C

Q.36 A Q.37 A Q.38 A Q.39 C Q.40 A Q.41 B Q.42 C

Q.43 B Q.44 D Q.45 D Q.46 C Q.47 A Q.48 C Q.49 A

Q.50 C Q.51 B Q.52 A Q.53 A Q.54 C Q.55 B Q.56 C

Q.57 A Q.58 B

ASSERTION AND REASON

Q.1 D Q.2 C Q.3 A Q.4 D Q.5 A

ONE OR MORE THAN ONE OPTION MAY BE CORRECT

Q.1 B Q.2 B Q.3 B, C, D Q.4 A, B Q.5 A, B, C Q.6 D

Q.7 B,C,D Q.8 B; D Q.9 B Q.10 B; C Q.11 A; C Q.12 B

Q.13 B Q.14 A Q.15 B; C; D Q.16 A Q.17 C Q.18 A; B; C

Q.19 A; B Q.20 A; B;C; D Q.21 B; C; D Q.22 A; C

Q.23 B; D Q.24 A; B; C Q.25 B; C Q.26 A; B; C

Q.27 B; D Q.28 C; D

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