PHYSICS Class
II IIT-JEE Achiever 2016-17 Intensive Revision Program Worksheet – 14
Topics
Simple Harmonic Motion
Date:
20-12-2016
Subjective Questions 2
1. A particle moves under the force F = (x − 6x) N, where x is in metres. For small displacements from the origin what is the force constant in the simple harmonic motion approximation? A 2. At x = , what fraction of the mechanical energy is potential energy? What fraction is kinetic? Assume 2 potential energy to be zero at mean position. 3. A body executing SHM covers a, b in successive seconds starting from one end. Find its amplitude. 4. A particle executing SHM moves from one end to the other. Its distances from the mid-point of its path, 2π at successive seconds, are observed to be x 1, x2, x3. Prove that its time period is x + x3 cos −1 1 2x 2 2
5. A body of mass 200 g is in equilibrium at x = 0 under the influence of a force F = ( −100 x + 10x )N. (a) If the body is displaced a small distance from equilibrium, what is the period of its oscillations? (b) If the amplitude is 4.0 cm, by how much do we error in assuming that F = −kx at the end points of the motion. 2
6. A point moves along the x-axis according to the equation x = a sin ωt −
π . Find the amplitude, 4
period, velocity and projection vx as a function of x. 7. A particle of mass m is located in a unidimensional potential field where the potential energy of the particle depends on the coordinates as U(x) = U 0(1 − cos ax), U0 are a constants. Find the period of small oscillations that the particle performs about the equilibrium position. 2
2
2
8. The speed v of a particle moving along x-axis is given by, v = 8bx − x − 12b , where b is a constant. Find amplitude of oscillations. 9. A small mass m is fastened to a vertical wire which is under tension T as shown in the figure. What will be the frequency of vibration of the mass m if it is displaced laterally a slight distance and then released?
10. A uniform board of length L and weight W is balanced on a fixed semicircular cylinder of radius r as shown in the figure. If the plank is tilted slightly from its equilibrium position, determine its period of oscillation. 11. A uniform cylindrical pulley of mass M and radius R can freely rotate about the horizontal axis O. The free end of a thread, lightly wound on the pulley, carries a block of weight A. At a certain angle α, it
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counterbalances a point mass m, fixed at the rim of the pulley. Find the frequency of small oscillations of the arrangement. 12. A uniform board is placed on two spinning wheels as shown in figure. The axes of the wheels are separated by a distance l = 20 cm. The coefficient of friction between the board and the wheels is K = 0.18. Explain that in this case the board performs harmonic oscillations. Find the period of these oscillations.
13. A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period. 14. A thin rod of length L and mass M is free to rotate in a vertical plane about a horizontal axis passing through the point O where one end of the rod is pivoted as shown in the figure. The other end of the rod is connected to a vertical massless spring of force constant k. The lower end of the vertical spring is rigidly fixed to the ground at G. When the rod is in equilibrium position it is parallel to the ground. When the rod is slightly rotated from its equilibrium position and released. (a) Find its time period T of small oscillations. (b) What will be the maximum linear speed of the displaced end of the rod if the amplitude of oscillations is taken θ0?
15. A uniform rod of mass m and length l is suspended through a light wire of length l and torsional constant k as shown in the figure. Find the time period if t he system makes
(a) small oscillations in the vertical plane about the suspension point and (b) angular oscillations in the horizontal plane about the centre of the rod Multiple choice questions with one correct alternative
16. Two cars A and B depart simultaneously from the same position and in same direction on a straight road. −1
−2
A starts with initial velocity 2 m s and acceleration 2 m s . While B starts with initial velocity 2 m s
−1
−2
and acceleration 4 m s . The driver of car A hears a sound of frequency 352 Hz emitted by car B after 10 s after the start. Find the actual frequency of the sound as emitted by B (Take velocity of the −1
sound = 330 m s ) (A) 271 Hz
(B) 371 Hz
(C) 550 Hz
(D) 713 Hz
17. In a sonometer wire, the tension is maintained by suspending a 50.7 kg mass from the free end of the 3
wire. The suspended mass has a volume of 0.0075 m . The fundamental frequency of the wire is 260 Hz. If the suspended mass is completely submerged in water, the fundamental frequency will become (A) 200 Hz
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(B) 220 Hz
(C) 230 Hz
2
(D) 240 Hz
18. The vibrating portion of a wire which is stretched with a weight of 6.48 kg weighs 0.5 g. When sounding in fundamental note, it is found to give 20 beats in 5 seconds, with vibrating tuning fork of frequency 256. If the length of the wire is slightly decreased, the note emitted by it is observed to be in unison with that of the fork. The original length of the wire is (A) 1 m
(B) 0.5 m
(C) 1.5 m
(D) 2 m
19. The time period of a particle in SHM is 8 seconds. At t = 0, it is at the mean position. The ratio of the distances traveled by it in the first and second is 1 1 (A) (B) 2 2
(C)
1 3
(D)
1 2 −1
20. A vertical U-tube of uniform cross-section contains water upto a height of 30 cm. If the water on one side is depressed and then released, its motion up and down the two sides of the tube is simple harmonic. The time period of this SHM is nearly (A) 4 s
(B) 3 s
(C) 2 s
(D) 1 s
21. A cubical body (side 0.1 m and mass 0.002 kg) floats in water. It is pressed and then released so that it oscillates vertically. The time period is (A) 0.01 s
(B) 0.02 s
(C) 0.03 s
(D) 0.04 s
22. A copper wire is held at the two ends by rigid support. At 30 °C the wire is just taut, with negligible tension. The speed of transverse waves in this wire at 10 °C is (Y = 1.3 × 10 Nm− , α = 1.7 × 10 − °C− 11
2
5
1
and ρ = 9 × 10 kg m− ) 3
(A) 70 ms
−1
3
−1
(B) 30 ms
(C) 90 ms
−1
−1
(D) 120 ms
23. A wave pulse starts propagating in +x direction along a non-uniform wire of length L under a tension T with mass per unit length given as µ = µ0 + αx, where µ0 and α are constants. The time taken by the pulse to travel from the lighter end (x = 0) to the heavier end is 2 2 ( −µ 0 + αL)3/ 2 + µ 30/ 2 (µ0 + αL)3/ 2 − µ 30/ 2 (A) (B) 3α T 3α T 2 2 (µ0 + αL)3/ 2 + µ 30 / 2 (µ 0 + αL) 3/ 2 − µ 30/ 2 (C) (D) 2α T 2α T 24. Figure shows three identical springs A, B, C. When a 4 kg weight is hung on A, it descends by 1 cm. When a 6 kg weight is hung on C, it will descend by (A) 1.5 cm (B) 3.0 cm (C) 4.5 cm (D) 6.0 cm
25. A simple pendulum of length l has a time period T for small oscillation. A fixed obstacle is placed directly below the point of suspension, so that only the lower quarter of the string continues oscillations. The pendulum is released from rest at a certain point O. The period of oscillation (assuming small angles) will be
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(A) T T (B) 4 3T (C) 4 T (D) 1+ 3 2
(
)
26. The radius of steel wire A is twice that of B and the tension in A is half that in B. If transverse waves of same frequency are generated in these two steel wires, then the ratio of the velocities of waves in A and B is (A) 1 : 2
(B) 1 : 2
(C) 1 : 2 2
(D) 3 : 2 2
27. Two sources A and B are sounding notes of frequency 680 Hz. A listener moves from A to B with a −1
constant velocity u. If the speed of sound is 340 m s , the value of u so that he hears 10 beats per second is (A) 2.0 m s
−1
(B) 2.5 m s
−1
(C) 3.0 m s
−1
(D) 3.5 m s
−1
−1
28. A wire having a linear density 0.1 kg m is kept under a tension 490 N. It is observed that it resonates at a frequency of 400 Hz and the next higher frequency 450 Hz. The length of wire is (A) 0.4 m
(B) 0.7 m
(C) 0.6 m
(D) 0.49 m
Read the passage given below and answer the following questions b choosing the correct alternative
A string of natural length 2 l can just support certain weight, when it is stretched till its whole length is 3 l. One end of the string is now attached to a point on a smooth horizontal table and the same weight is attached to the other end and can move on the table. When the weight is pulled out to any distance and let go, the string becomes slack again after some time. 29. If f is the force required to produce unit strain in the string, then weight required to increase the length from 2l to 3l is (A) f
(B)
f 3
(C)
f 2
(D)
f 4
30. The period of oscillation of the mass attached to the string is (A) T = 2π
l
g
l
(B) T = π
g
a
(C) T = 2π
2g
(D) T = 2π
31. Time interval in which string becomes slack is (A) π
l
g
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(B)
π
l
2
g
(C) 2π
4
l
g
(D)
π
l
2
2g
2a g
!PP 32. A mass m is suspended by mass of two coiled spring which have the same length in unscratched condition as in figure. Their constant are k 1 and k 2 respectively. When set into vertical vibrations. the period will be
m (A) 2π k k 1 2
k (B) 2π m 1 k 2
m (C) 2π k − k 1 2
33. The equation of a damped simple harmonic motion is m
d2 x dt
2
+b
dx dt
m (D) 2π k1 + k 2
+ kx = 0. Then the angular frequency
of oscillation is 1
1
k b − m 4m 2 2
b 2 k − m 4m 2
2
(A) ω =
(B) ω =
1
k b 2 (C) ω = − m 4m
k b2 (D) ω = − m 4m
2
34. The angular velocity and the amplitude of a simple pendulum is ‘ ω’ and ‘a’ respectively. At displacement X from the mean position if its kinetic energy is T and potential energy is V, then the ratio of T to V is (A)
X 2 ω2 a 2 − X 2 ω2
(B)
X2 a 2 − X2
(C)
( a 2 − X 2ω2 ) X 2 ω2
(D)
(a 2 − X2 ) X2
35. A particle is oscillating in SHM. What fraction of total energy is kinetic when the particle is at the mean position (A is the amplitude of oscillation) 3 2 4 (A) (B) (C) 4 4 7
(D)
A 2
from
5 7
36. If ‘x’, ‘v’ and ‘a’ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then which of the following does not change with time aT aT aT (A) a 2T 2 + 4π2v 2 (B) (C) (D) x 2πv v 37. 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 (A) (C)
3 32
π
2
2
π cm/s
(B)
−π 32
2
32
c m / s2
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(D) −
5
2
c m / s2 3
32
π 2c m / s 2
Read the passage given below and answer the following questions b choosing the correct alternative
Two identical blocks P and Q have equal masses and are connected to two identical springs as shown. Initially the springs are unstretched. The block P A is moved to left by and Q by A to the right side from equilibrium 2 position. Both the blocks are released simultaneously and they undergo perfectly inelastic collision. In perfectly inelastic collision between two blocks, the momentum remains conserved and blocks stick together. Initially, time period of both the blocks was T. Angular frequency of spring block system is ω =
k m
and
maximum speed of particle in SHM is ωA, where A is the amplitude of oscillation. 38. The energy of oscillation of combined mass is T (A) T (B) 2
(C) 2T
39. The amplitude of the combined mass is A (A) 3A (B) 2
(C)
40. The energy of oscillation of combined mass is kA 2 2 (A) kA (B) 2
(C)
2A 3
(D)
kA 2 16
(D)
(D)
T 2
A 4 kA 2 8
Read the passage given below and answer questions b choosing the correct alternative
Longitudinal standing waves can be produced in columns of air in pipes. The closed end of the pipe is a node. The open end of a pipe is always an antinode. The lowest natural frequency is called fundamental frequency f 0 and has wavelength λ0. A harmonic is an integral multiple of fundamental frequency (nf) with n = 1, 2, 3 … For a complete cycle of a wave of wavelength λ, the distance between any two consecutive nodes or any two consecutive antinodes is half the wave length. i.e.
λ 2
(Assume v = 340 ms
−1
)
41. The auditory canal of the outer ear is 3 cm, closed at one end by the ear drum. The fundamental frequency associated with air column is 2
(A) 1.1 × 10 Hz
3
(B) 2.8 × 10 Hz
3
(C) 5.6 × 10 Hz
4
(D) 1.7 × 10 Hz
42. A pipe resonates at 60 Hz, 100 Hz, and 140 Hz. How long is the pipe? (A) 1.4 m
(B) 2.8 m
(C) 4.3 m
(D) 8.5 m
43. A pipe of length L is closed at both the ends. What is its fundamental wavelength? L (A) (B) L (C) 2L (D) 4L 2
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