Chapter 14 Oscillations

Question 14.1. The displacement of a particle is represented by the equation $y = 3 \cos\left(\frac{\pi}{4} - 2\omega t\right)$. The motion of the particle is

(a) simple harmonic with period $2\pi/\omega$.

(b) simple harmonic with period $\pi/\omega$.

(c) periodic but not simple harmonic.

(d) non-periodic.

Question 14.2. The displacement of a particle is represented by the equation $y = \sin^3\omega t$. The motion is

(a) non-periodic.

(b) periodic but not simple harmonic.

(c) simple harmonic with period $2\pi/\omega$.

(d) simple harmonic with period $\pi/\omega$.

Question 14.3. The relation between acceleration and displacement of four particles are given below:

(a) $a_x = +2x$.

(b) $a_x = +2x^2$.

(c) $a_x = –2x^2$.

(d) $a_x = –2x$.

Which one of the particles is executing simple harmonic motion?

Question 14.4. 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 density of the liquid.

Question 14.5. A particle is acted simultaneously by mutually perpendicular simple hormonic motions $x = a \cos\omega t$ and $y = a \sin\omega t$. The trajectory of motion of the particle will be

(a) an ellipse.

(b) a parabola.

(c) a circle.

(d) a straight line.

Question 14.6. The displacement of a particle varies with time according to the relation $y = a \sin\omega t + b \cos\omega t$.

(a) The motion is oscillatory but not S.H.M.

(b) The motion is S.H.M. with amplitude a + b.

(c) The motion is S.H.M. with amplitude $a^2 + b^2$.

(d) The motion is S.H.M. with amplitude $\sqrt{a^2 + b^2}$.

Question 14.7. Four pendulums A, B, C and D are suspended from the same elastic support as shown in Fig. 14.1. 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 the four will oscillate with equal amplitude.

Question 14.8. Figure 14.2. 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) $x(t) = B \sin\left(\frac{2\pi t}{30}\right)$.

(b) $x(t) = B \cos\left(\frac{\pi t}{15}\right)$.

(c) $x(t) = B \sin\left(\frac{\pi t}{15} + \frac{\pi}{2}\right)$.

(d) $x(t) = B \cos\left(\frac{\pi t}{15} + \frac{\pi}{2}\right)$.

Question 14.9. The equation of motion of a particle is $x = a \cos(\alpha t)^2$. The motion is

(a) periodic but not oscillatory.

(b) periodic and oscillatory.

(c) oscillatory but not periodic.

(d) neither periodic nor oscillatory.

Question 14.10. A particle executing S.H.M. has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s². The period of oscillation is

(a) $\pi$ s.

(b) $\frac{\pi}{2}$ s.

(c) $2\pi$ s.

(d) $\frac{t}{\pi}$ s.

Question 14.11. When a mass m is connected individually to two springs $S_1$ and $S_2$, the oscillation frequencies are $\nu_1$ and $\nu_2$. If the same mass is attached to the two springs as shown in Fig. 14.3, the oscillation frequency would be

(a) $\nu_1 + \nu_2$.

(b) $\sqrt{\nu_1^2 + \nu_2^2}$.

(c) $\left(\frac{1}{\nu_1} + \frac{1}{\nu_2}\right)^{-1}$.

(d) $\sqrt{\nu_1^2 - \nu_2^2}$.

Question 14.12. The rotation of earth about its axis is

(a) periodic motion.

(b) simple harmonic motion.

(c) periodic but not simple harmonic motion.

(d) non-periodic motion.

Question 14.13. 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 S.H.M.

Question 14.14. Displacement vs. time curve for a particle executing S.H.M. is shown in Fig. 14.4. 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 same at t = 2 s and t = 6 s.

(c) Phase of the oscillator is same at t = 1 s and t = 7 s.

(d) Phase of the oscillator is same at t = 1 s and t = 5 s.

Question 14.15. Which of the following statements is/are true for a simple harmonic oscillator?

(a) Force acting is directly proportional to 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.

Question 14.16. The displacement time graph of a particle executing S.H.M. is shown in Fig. 14.5. Which of the following statement is/are true?

(a) The force is zero at $t = \frac{3T}{4}$.

(b) The acceleration is maximum at $t = \frac{4T}{4}$.

(c) The velocity is maximum at $t = \frac{T}{4}$.

(d) The P.E. is equal to K.E. of oscillation at $t = \frac{T}{2}$.

Question 14.17. A body is performing S.H.M. 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 equal to $\frac{2}{\pi}$ times of its maximum velocity.

(d) root mean square velocity is $\frac{1}{\sqrt{2}}$ times of its maximum velocity.

Question 14.18. A particle is in linear simple harmonic motion between two points A and B, 10 cm apart (Fig. 14.6). Take 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) The sign of velocity, acceleration and force on the particle when it is 4 cm away from B going towards A are negative.

(d) The sign of acceleration and force on the particle when it is at point B is negative.

Question 14.19. Displacement versus time curve for a particle executing S.H.M. is shown in Fig. 14.7. Identify the points marked at which (i) velocity of the oscillator is zero, (ii) speed of the oscillator is maximum.

Question 14.20. Two identical springs of spring constant K are attached to a block of mass m and to fixed supports as shown in Fig. 14.8. When the mass is displaced from equilllibrium position by a distance x towards right, find the restoring force

Question 14.21. What are the two basic characteristics of a simple harmonic motion?

Question 14.22. When will the motion of a simple pendulum be simple harmonic?

Question 14.23. What is the ratio of maxmimum acceleration to the maximum velocity of a simple harmonic oscillator?

Question 14.24. What is the ratio between the distance travelled by the oscillator in one time period and amplitude?

Question 14.25. In Fig. 14.9, 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 anticlockwise direction.

Question 14.26. Show that for a particle executing S.H.M, velocity and displacement have a phase difference of $\pi/2$.

Question 14.27. Draw a graph to show the variation of P.E., K.E. and total energy of a simple harmonic oscillator with displacement.

Question 14.28. The length of a second’s pendulum on the surface of Earth is 1m. What will be the length of a second’s pendulum on the moon?

Question 14.29. Find the time period of mass M when displaced from its equilibrium positon and then released for the system shown in Fig 14.10.

Question 14.30. Show that the motion of a particle represented by $y = \sin \omega t – \cos \omega t$ is simple harmonic with a period of $2\pi/\omega$.

Question 14.31. Find the displacement of a simple harmonic oscillator at which its P.E. is half of the maximum energy of the oscillator.

Question 14.32. A body of mass m is situated in a potential field $U(x) = U_0 (1-\cos\alpha x)$ when $U_0$ and $\alpha$ are constants. Find the time period of small oscillations.

Question 14.33. A mass of 2 kg is attached to the spring of spring constant 50 Nm⁻¹. The block is pulled to a distance of 5cm 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.

Question 14.34. 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 2° to the right with the vertical, the other pendulum makes an angle of 1° to the left of the vertical. What is the phase difference between the pendulums?

Question 14.35. A person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of 2.0 s⁻¹ and an amplitude 5.0 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 answer to part (a) is yes, what will be the maximum and minimum reading in the machine and at which position?

Question 14.36. 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 4cm below the point, where it was held in hand.

(a) What is the amplitude of oscillation?

(b) Find the frequency of oscillation?

Question 14.37. 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 S.H.M. with a time period. $T = 2\pi \sqrt{\frac{m}{A\rho g}}$ where m is mass of the body and $\rho$ is density of the liquid.

Question 14.38. One end of a V-tube containing mercury is connected to a suction pump and the other end to atmosphere. The two arms of the tube are inclined to horizontal at an angle of 45° 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.

Question 14.39. 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.

Question 14.40. 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 (Fig. 14.11). The amplitude is $\theta_o$. The string snaps at $\theta = \theta_0 / 2$. Find the time taken by the bob to hit the ground. Also find distance from A where bob hits the ground. Assume $\theta_0$ to be small so that $\sin\theta_0 \approx \theta_0$ and $\cos\theta_0 \approx 1$.