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**Question 14.1** 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.

**Question 14.2** 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.

**Question 14.3** 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) ?

**Question 14.4** 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)

(f) 1 + ωt + ω2t2

**Question 14.5** 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.

**Question 14.6** Which of the following relationships between the acceleration a and the displacement
x of a particle involve simple harmonic motion?

(a) a = 0.7x

(b) a = –200x2

(c) a = –10x

(d) a = 100x3

**Question 14.7** 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.

**Question 14.8** 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 ?

**Question 14.9** A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table
as shown in Fig. 14.28. 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.
Fig.14.28
Determine
(i) the frequency of oscillations, (ii) maximum acceleration of the mass,
and (iii) the maximum speed of the mass.

**Question 14.10** 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?

**Question 14.11** Figures 14.29 correspond to two circular motions. The radius of the circle, the
period of revolution, the initial position, and the sense of revolution (i.e. clockwise
or anti-clockwise) are indicated on each figure. Obtain the corresponding simple harmonic motions of the x-projection of the radius
vector of the revolving particle P, in each case.

**Question 14.12** Plot the corresponding reference circle for each of the following simple harmonic
motions. Indicate the initial (t =0) position of the particle, the radius of the circle,
and the angular speed of the rotating particle. For simplicity, the sense of rotation
may be fixed to be anticlockwise in every case: (x is in cm and t is in s).

(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

**Question 14.13** Figure 14.30
(a) shows a spring of force constant k clamped rigidly at one end and
a mass m attached to its free end. A force F applied at the free end stretches the
spring. Figure 14.30 (b) shows the same spring with both ends free and attached to
a mass m at either end. Each end of the spring in Fig. 14.30(b) is stretched by the
same force F.
(a) What is the maximum extension of the spring in the two cases ?
(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the
period of oscillation in each case ?

**Question 14.14** The piston in the cylinder head of a locomotive has a stroke (twice the amplitude)
of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency
of 200 rad/min, what is its maximum speed ?

**Question 14.15** The acceleration due to gravity on the surface of moon is 1.7 m s–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 m s–2)

**Question 14.16** Answer the following questions :

(a) Time period of a particle in SHM depends on the force constant k and mass m
of the particle:
T
m
k
= 2π . A simple pendulum executes SHM approximately. Why then is
the time period of a pendulum independent of the mass of the pendulum?

(b) The motion of a simple pendulum is approximately simple harmonic for small
angle oscillations. For larger angles of oscillation, a more involved analysis
shows that T is greater than 2π
l
g
. Think of a qualitative argument to
appreciate this result.

(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 cabin
that is freely falling under gravity ?

**Question 14.17** 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 ?

**Question 14.18** A cylindrical piece of cork of density of base area A and height h floats in a liquid of
density ρl. The cork is depressed slightly and then released. Show that the cork
oscillates up and down simple harmonically with a period
T
h
g 1
= 2π
ρ
ρ
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

**Question 14.19** 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.
Additional Exercises

**Question 14.20** An air chamber of volume V has a neck area of cross section a into which a ball of
mass m 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].

**Question 14.21** 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.

**Question 14.22** 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.

**Question 14.23 **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).

**Question 14.24** A body describes simple harmonic motion with an 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.

**Question 14.25** 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.]

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- Chapter 3: Motion in a Straight Line
**[Ques wise Ans]** - Chapter 1 Physical World
- Chapter 2 Units and Measurement
- Chapter 3 Motion in a Straight Line
- Chapter 4 Motion in a Plane
- Chapter 5 Laws of Motion
- Chapter 7 Systems of Particles and Rotational Motion
- Chapter 6 Work Energy and Power
- Chapter 8 Gravitation
- Chapter 9 Mechanical Properties of Solids
- Chapter 10 Mechanical Properties of Fluids
- Chapter 11 Thermal Properties of Matter
- Chapter 12 Thermodynamics
- Chapter 13 Kinetic Theory
- Chapter 14 Oscillations
- Chapter 15 Waves

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