QUESTION FOR SHORT ANSWER

Q.1 Suppose we have a block of unknown mass and a spring of unknown force constant. Show how we canpredict the period of oscillation of this block-spring system simply by measuring the extension of thespring produced by attaching the block to it.

Q.2 How are each of the following properties of a simple harmonic oscillator affected by doubling the amplitude:period, force constant, total mechanical energy, maximum velocity, maximum acceleration?

Q.3 A person stands on a bathroom-type scale, which rests on a platform suspended by a large spring. Thewhole system executes simple harmonic motion in a vertical direction. Describe the variation in scalereading during a period of motion.

Q.4 Predict by qualitative arguments whether a pendulum oscillating with large amplitude will have a periodlonger or shorter than the period for oscillations with small amplitude. (Consider extreme cases.)

Q.5 If taken to the Moon, will there be any change in the frequency of oscillation of a torsional pendulum? Asimple pendulum? A spring-block oscillator? A physical pendulum?

Q.6 How can a pendulum be used to trace out a sinusoidal curve?

Q.7 During SHM, are the displacement and velocity ever in the same direction? The velocity and acceleration?The displacement and the acceleration?

ONLY 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 π(t + 4) mThen 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 displacement of a body executing SHM is given by x = A sin (2πt + π/3). The first time from t = 0when the velocity is maximum is(A) 0.33 sec (B) 0.16 sec (C) 0.25 sec (D) 0.5 sec

Q.3 The maximum acceleration of a particle in SHM is made two times keeping the maximum speed to beconstant. 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.4 The potential energy of a simple harmonic oscillator of mass 2 kg in its mean position is 5 J. If its totalenergy is 9J and its amplitude is 0.01 m, its time period would be(A) π/10 sec (B) π/20 sec (C) π/50 sec (D) π/100 sec

Q.5 A plank with a small block on top of it is under going vertical SHM. Its period is 2 sec. The minimumamplitude at which the block will separate from plank is :

(A) 2

10

π(B)

10

2π(C) 2

20

π(D)

10

π

Q.6 Time period of a particle executing SHM is 8 sec. At t = 0 it is at the mean position. The ratio of thedistance covered by the particle in the 1st second to the 2nd second is :

(A) 12

1

+(B) 2 (C)

2

1(D) 12 +

Q.7 The angular frequency of motion whose equation is 2dt

yd4

2

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

(A) 4

9(B)

9

4(C)

2

3(D)

3

2

Q.8 Two particles are in SHM in a straight line. 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 y2= –A/2, and they are approachingtowards each other. After what time they cross each other ?(A) T/3 (B) T/4 (C) 5T/6 (D) T/6

Q.9 Two particles execute SHM of same amplitude of 20 cm with same period along the same line about thesame equilibrium position. The maximum distance between the two is 20 cm. Their phase difference inradians is

(A) 3

2π(B)

2

π(C)

3

π(D)

4

π

Q.10 Two particles P and Q describe simple harmonic motions of same period, same amplitude, along thesame line about the same equilibrium position O. When P and Q are on opposite sides of O at the samedistance from O they have the same speed of 1.2 m/s in the same direction, when their displacements arethe same they have the same speed of 1.6 m/s in opposite directions. The maximum velocity in m/s ofeither particle is(A) 2.8 (B) 2.5 (C) 2.4 (D) 2

Q.11 A particle performs SHM with a period T and amplitude a. The mean velocity of the particle over thetime 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.12 A body performs simple harmonic oscillations along the straight line ABCDE with C as the midpoint ofAE. Its kinetic energies at B and D are each one fourth of its maximum value. If AE = 2R, the distancebetween B and D is

(A) 2

R3(B)

2

R(C) R3 (D) R2

Q.13 A graph of the square of the velocity against the square of the acceleration of a given simple harmonicmotion is

(A) (B) (C) (D)

Question No. 14 to 16 (3 questions)The graphs in figure show that a quantity y varies with displacement d in a system undergoing simpleharmonic motion.

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

Which graphs best represents the relationship obtained when y is

Q.14 The total energy of the system(A) I (B) II (C) III (D) IV

Q.15 The time(A) I (B) II (C) III (D) IV

Q.16 The unbalanced force acting on the system.(A) I (B) II (C) III (D) IV

Q.17 A small mass executes linear SHM about O with amplitude a and period T. Its displacement from O attime T/8 after passing through O is:(A) a/8 (B) a/2√2 (C) a/2 (D) a/√2

Q.18 A particle executes SHM of period 1.2 sec and amplitude 8 cm. Find the time it takes to travel 3cm fromthe positive extremity of its oscillation.(A) 0.28 sec (B) 0.32 sec (C) 0.17 sec (D) 0.42 sec

Q.19 A particle executes SHM on a straight line path. The amplitude of oscillation is 2 cm. When thedisplacement of the particle from the mean position is 1 cm, the numerical value of magnitude of accelerationis equal to the numerical value of magnitude of velocity. The frequency of SHM (in second–1) is:

(A) 2π 3 (B) 3

2π(C)

π2

3(D)

32

1

π

Q.20 A stone is swinging in a horizontal circle 0.8 m in diameter at 30 rev / min. A distant horizontal light beamcauses a shadow of the stone to be formed on a nearly vertical wall. The amplitude and period of thesimple 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.21 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.22 In an elevator, a spring clock of time period TS (mass attached to a spring) and a pendulum clock of timeperiod TP are kept. If the elevator accelerates upwards(A) TS well as TP increases (B) TS remain same, TP increases(C) TS remains same, TP decreases (D) TS as well as TP decreases

Q.23 A man is swinging on a swing made of 2 ropes of equal length L and indirection perpendicular to the plane of paper. The time period of thesmall oscillations about the mean position is

(A) 2π g2

L(B) 2π

g2

L3

(C) 2π g32

L(D) π

g

L

Q.24 Two bodies P & Q of equal mass are suspended from two separate massless springs of force constantsk1 & k2 respectively. If the maximum velocity of them are equal during their motion, the ratio ofamplitude 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.25 Vertical displacement of a plank with a body of mass 'm' on it is varying according to law

y = sin ωt + 3 cos ωt. The minimum value of ω 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 π(B)

g3

2,

2

g π(C)

g

2

3,

2

g π(D)

g3

2,g2

π

Q.26 A ring of diameter 2m oscillates as a compound pendulum about a horizontal axis passing through apoint at its rim. It oscillates such that its centre move in a plane which is perpendicular to the plane of thering. The equivalent length of the simple pendulum is(A) 2m (B) 4m (C) 1.5m (D) 3m

Q.27 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.28 A particle of mass m moves in a one-dimensional potential energy U(x) = –ax2 + bx4, where 'a' and 'b'are positive constants. The angular frequency of small oscillations about the minima of the potentialenergy is equal to

(A) b2

aπ (B)

m

a2 (C)

m

a2(D)

m2

a

Q.29 A tunnel is dug in the earth across one of its diameter. Two masses ‘m’ & ‘2m’ are dropped from theends of the tunnel. The masses collide and stick to each other and perform S.H.M. Then amplitude ofS.H.M. will be: [R = radius of the earth](A) R (B) R / 2 (C) R / 3 (D) 2R / 3

Q.30 Two particles undergo SHM along parallel lines with the same time period (T) and equal amplitudes. Ata particular instant, one particle is at its extreme position while the other is at its mean position. Theymove 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.31 A heavy brass sphere is hung from a light spring and is set in vertical small oscillation with a period T. Thesphere is now immersed in a non-viscous liquid with a density 1/10th the density of the sphere. If thesystem is now set in vertical S.H.M., its period will be(A) (9/10)T (B) (9/10)2T (C) (10/9) T (D) T

Q.32 A simple pendulum is oscillating in a lift. If the lift is going down with constant velocity, the time period ofthe simple pendulum is T1. If the lift is going down with some retardation its time period is T2, then(A) T1 > T2

(B) T1 < T2

(C) T1 = T2

(D) depends upon the mass of the pendulum bob

Q.33 A system of two identical rods (L-shaped) of mass m and length l areresting on a peg P as shown in the figure. If the system is displaced in its plane by a small angle θ, find the period of oscillations:

(A) 2π2

3

l

g (B) 2π

2 2

3

l

g(C) 2π

2

3

l

g (D) 3πl

3g

Q.34 A particle starts oscillating simple harmonically from its equilibrium position then, the ratio of kineticenergy 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.35 A spring mass system preforms S.H.M. If the mass is doubled keeping amplitude same, then the totalenergy of S.H.M. will become :(A) double (B) half (C) unchanged (D) 4 times

Q.36 A mass at the end of a spring executes harmonic motion about an equilibrium position with an amplitudeA. Its speed as it passes through the equilibrium position is V. If extended 2A and released, the speed ofthe mass passing through the equilibrium position will be

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

2(D)

V

4

Q.37 A particle of mass m moves in the potential energy U shown above.The period of the motion when the particle has total energy E is

(A) 2mg/E24k/m2 +π (B) k/m2π

(C) 2mg/E22k/m +π (D) 2mg/E22

Q.38 A 2 Kg block moving with 10 m/s strikes a spring of constant π2 N/m attached to 2 Kg block at restkept 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.39 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.40 A particle is subjected to two mutually perpendicular simple harmonic motions such that its x andy coordinates are given by

x = 2 sin ωt ; y = 2 sin

π+ω

4t

The path of the particle will be :(A) an ellipse (B) a straight line (C) a parabola (D) a circle

Q.41 The amplitude of the vibrating particle due to superposition of two SHMs,

y1 = sin ωπ

t+

3 and y2 = sin ω t is :

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

Q.42 Two simple harmonic motions y1 = A sin ωt and y2 = A cos ωt are superimposed on a particle of mass m.The total mechanical energy of the particle is:

(A) 2

1mω2A2 (B) mω2A2 (C)

4

1mω2A2 (D) zero

ONE OR MORE THAN ONE OPTION MAY BE CORRECTTake approx. 3 minutes for answering each question.

Q.1 Two particles are in SHM on same straight line with amplitude A and 2A and with same angular frequency

ω. 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.2 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.3 Speed v of a particle moving along a straight line, when it is at a distance x from a fixed point on the lineis 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.4 A block is placed on a horizontal plank. The plank is performing SHM along a vertical line with amplitudeof 40cm. The block just loses contact with the plank when the plank is momentarily at rest. Then:(A) the period of its oscillations is 2π/5 sec.(B) the block weights on the plank double its weight, when the plank is at one of the positions ofmomentary rest.(C) the block weights 1.5 times its weight on the plank halfway down from the mean position.(D) the block weights its true weight on the plank, when velocity of the plank is maximum.

Q.5 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.

Q.6 The potential energy of a particle of mass 0.1kg, moving along x-axis, is given by U = 5x(x-4)J wherex 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 π/5 s.

Q.7 A particle is executing SHM with amplitude A, time period T, maximum acceleration ao and maximumvelocity v0. Its starts from mean position at t=0 and at time t , it has the displacementA/2, acceleration a and velocity v then(A) t=T/12 (B) a=ao/2 (C) v=vo/2 (D) t=T/8

Q.8 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 its maximum K.E.(D) Its velocity is half the maximum velocity when its displacement is half the maximum displacement.

Q.9 The displacement of a particle varies according to the relation x = 3 sin 100t + 8 cos2 50t . Which of thefollowing 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.10 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 line not necessarily passing through origin(D) none of the above

Q.11 A particle moves in xy plane according to the law x = a sinωt and y = a(1-cosωt) where a and ω areconstants. 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.

Q.12 For a particle executing S.H.M., x = displacement from equilibrium position, v = velocity at any instantand a = acceleration at any instant, then(A) v-x graph is a circle (B) v-x graph is an ellipse(C) a-x graph is a straight line (D) a-v graph is an ellipse

Q.13 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.14 Equations y = 2A cos2 ωt and y = A (sin ωt + 3 cos ωt ) 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.15 A particle starts from a point P at a distance of A/2 from the mean position O & travels towards left asshown in the figure. If the time period of SHM, executed about O is T and amplitude A then the equationof motion of particle is :

(A) x = A sin

π+

π

6t

T

2(B) x = A sin

π+

π

6

5t

T

2

(C) x = A cos

π+

π

6t

T

2(D) x = A cos

π+

π

3t

T

2

Q.16 A mass of 0.2kg is attached to the lower end of a massless spring of force-constant 200 N/m, the upperend 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 2cmbefore 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.17 The angular frequency of a spring block system is ω0.This system is suspended from the ceiling of anelevator moving downwards with a constant speed v0. The block is at rest relative to the elevator. Lift issuddenly stopped. Assuming the downwards as a positive direction, choose the wrong statement:

(A) The amplitude of the block is 0

0v

ω

(B) The initial phase of the block is π.

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

0v

ω sin ω0t.

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

Q.18 A cylindrical block of density ρ is partially immersed in a liquid of density 3ρ. The plane surface of theblock remains parallel to the surface of the liquid. The height of the block is 60 cm. The block performsSHM when displaced from its mean position. [Use g = 9.8 m/s2](A) the maximum amplitude is 20 cm. (B) the maximum amplitude is 40 cm(C) the time period will be 2π/7 seconds. (D) none

Q.19 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.20 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 π2f2A2.(C) The frequency of ocillation of kinetic energy is 2f.(D) Velocity function leads acceleration by π/2.

Q.21 A linear harmonic oscillator of force constant 2 × 106Nm-1 and amplitude 0.01 m has a total mechanicalenergy 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.22 The graph plotted between phase angle (φ) 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)

Q.23 Two springs with negligible masses and force constant of K1 = 200 Nm–1 and K2 = 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 theequilibrium position in which both springs are neither stretched nor compressed. At time t = 0, a sharpimpulse of 50 Ns is given to the block with a hammer.

(A) Period of oscillations for the mass m is3

πs.

(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.

Answer Key

ONLY ONE OPTION IS CORRECT.Q.1 A Q.2 A Q.3 C Q.4 D Q.5 A

Q.6 D Q.7 C Q.8 D Q.9 C Q.10 D

Q.11 C Q.12 C Q.13 D Q.14 A Q.15 D

Q.16 D Q.17 D Q.18 C Q.19 C Q.20 C

Q.21 B Q.22 C Q.23 B Q.24 B Q.25 A

Q.26 C Q.27 A Q.28 B Q.29 C Q.30 B

Q.31 D Q.32 A Q.33 B Q.34 B Q.35 C

Q.36 A Q.37 C Q.38 C Q.39 A Q.40 A

Q.41 C Q.42 B

ONE OR MORE THAN ONE OPTION MAY BE CORRECT

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

Q.5 B, C, D Q.6 B, C, D Q.7 A, B Q.8 A, B, C

Q.9 B, D Q.10 D Q.11 C, D Q.12 B, C, D

Q.13 A, B, C Q.14 C Q.15 D Q.16 A, B, C, D

Q.17 B Q.18 A, C Q.19 A, C Q.20 B, C

Q.21 B, C Q.22 B Q.23 A, B