Chapter 14 - OSCILLATIONS

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.

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.

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?

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?

Inextensible
string PAHOP=l(b) If answer to part (a) is yes, what will be the maximum and
minimum reading in the machine and at which position?

Consider a pair of identical pendulums, which oscillate with equal
amplitude independently such that when one pendulum is at
its extreme position making an angle of 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?

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.

A body of mass m is situated in a potential field U(x) = U0 (1-cos αx)
when U0 and α are constants. Find the time period of small
oscillations.

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

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

Show that the motion of a particle represented by
y = sinω t – cos ω t is simple harmonic with a period of 2π/ω.

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?

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

Show that for a particle executing S.H.M, velocity and
displacement have a phase difference of π/2.

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

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

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

When will the motion of a simple pendulum be simple harmonic?

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

Using the correspondence of S. H. M. and uniform circular motion, find
displacement, velocity, amplitude, time period and frequency of a particle
executing SH.M?

The acceleration due to gravity on the surface of moon is 1.7 m/s². What is the

time period of simple pendulum on moon if its time period on the earth is 3.5s?

Two linear simple harmonic motions of equal amplitudes and angular frequency w
and 2 w are impressed on a particle along axis X and Y respectively. If the initial
phase difference between them is 2/p, find the resultant path followed by theparticle?

A particle is executing S H M of amplitude 4 cm and T = 4 sec. find the time taken
by it to move from positive extreme position to half of its amplitude?

A simple pendulum is executing Simple harmonic motion with a time T. If the
length of the pendulum is increased by 21 %. Find the increase in its time period?

The time period of a body suspended by a spring is T. What will be the new time
period if the spring is cut into two equal parts and 1) the body is suspended by one
part. 2) Suspended by both parts in parallel?

Give some practical examples of S. H. M?

Can a simple pendulum experiment be done inside a satellite?

Is the motion of a simple pendulum strictly simple harmonic?

The kinetic energy of a particle executing S.H.M. is 16J when it is in its mean
position. If the amplitude of oscillations is 25cm and the mass of the particle is
5.12kg. Calculate the time period of oscillations?

What is ratio of frequencies of the vertical oscillations when two springs of
spring constant K are connected in series and then in parallel?

A particle executing SH.M has a maximum displacement of 4 cm and its

acceleration at a distance of 1 cm from its mean position is 3 cm/s². What will be

its velocity when it is at a distance of 2 cm from its mean position?

What is Simple pendulum? Find an expression for the time period and frequency
of a simple pendulum?

An 8 kg body performs S.H.M. of amplitude 30 cm. The restoring force is 60N,
when the displacement is 30cm. Find: - a) Time period b) the acceleration c)
potential and kinetic energy when the displacement is 12cm?

The bob of a vibrating simple pendulum is made of ice. How will the period of
swing will change when the ice starts melting?

The springs of spring factor k, 2k, k respectively are connected in parallel to a
mass m. If the mass = 0.08kg m and k = 2N|m, then find the new time period?

The soldiers marching on a suspended bridge are advised to go out of steps.
Why?

A simple harmonic oscillator is represented by the equation : Y = 0.40 Sin
(440t+0.61)
Y is in metres
t is in seconds
Find the values of 1) Amplitude 2) Angular frequency 3) Frequency of oscillation
4) Time period of oscillation, 5) Initial phase.

At what distance from the mean position, is the kinetic energy in a simple
harmonic oscillator equal to potential energy?

The girl sitting on a swing stands up. What will be the effect on periodic time of
swing?

A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x₀ and pushed towards the centre with a velocity v₀ at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x₀ and v₀. [Hint: Start with the equation x = a cos (ωt+θ) and note that the initial velocity is negative.]

A body describes simple harmonic motion with amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm, (b) 3 cm, (c) 0 cm.

A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant α is defined by the relation J = –α θ, where J is the restoring couple and θ the angle of twist).

Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant k and (b) the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.

An air chamber of volume V has a neck area of cross section a into which a ball of massm just fits and can move up and down without any friction (Fig.14.33). Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.33].

One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?

(c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?
(d) What is the frequency of oscillation of a simple pendulum mounted in a cabinthat is freely falling under gravity?

The acceleration due to gravity on the surface of moon is 1.7 ms^–2. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s? (g on the surface of earth is 9.8 ms^–2)

(a) x = –2 sin (3t + π/3)
(b) x = cos (π/6 – t)
(c) x = 3 sin (2πt + π/4)
(d) x = 2 cos πt

In Exercise 14.9, let us take the position of mass when the spring is unstreched as x =0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is
(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

 spring having with a spring constant 1200 N m^–1 is mounted on a horizontal table as shown in Fig. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.

A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body?

The motion of a particle executing simple harmonic motion is described by the displacement function,
x (t) = A cos (ωt + φ).
If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s^–1. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.

Which of the following relationships between the acceleration a and the displacement xof a particle involve simple harmonic motion?
(a) a = 0.7x
(b) a = –200x²
(c) a = –10x
(d) a = 100x³

A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is
(a) at the end A,
(b) at the end B,
(c) at the mid-point of AB going towards A,
(d) at 2 cm away from B going towards A,
(e) at 3 cm away from A going towards B, and
(f) at 4 cm away from B going towards A.

 Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant):
(a) sin ωt – cos ωt
(b) sin³ ωt
(c) 3 cos (π/4 – 2ωt)
(d) cos ωt + cos 3ωt + cos 5ωt
(e) exp (–ω²t²)

 Figure 14.27 depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?

Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(a) the rotation of earth about its axis.
(b) motion of an oscillating mercury column in a U-tube.
(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.
(d) general vibrations of a polyatomic molecule about its equilibrium position.

Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of a river to the other and back.
(b) A freely suspended bar magnet displaced from its N-S direction and released.
(c) A hydrogen molecule rotating about its center of mass.
(d) An arrow released from a bow.