Many kinds of

oscillatory motion
are sinusoidal in
time, or nearly so,
and are referred to
as simple harmonic
motion.

Oscillations & Waves

03 foda,k iy ;rx.

CONTENTS

03-1 Simple Harmonic Motion –

Spring Oscillations
03-2 Definition of S.H.M.
03-3 Characteristic equation of the
Simple Harmonic
simple harmonic motion
03-4 Simple harmonic motion as a Motion
projection of a uniform (ir, wkqj¾;S p,s;h)
circular motion
03-5 Equation of Displacement
03-6 Displacement – time graph
corresponding to simple
harmonic motion
03-7 Small oscillations of a simple
pendulum

Physics
FOR G.C.E. ADVANCED LEVEL EXAMINATION
2021 THEORY
H. MARIE F. SILVA
M any objects vibrate or oscillate—an object on the end of a spring, a tuning
fork, the balance wheel of an old watch, a pendulum, a plastic ruler held
firmly over the edge of a table and gently struck, the strings of a guitar or piano.
Spiders detect prey by the vibrations of their webs; cars oscillate up and down when
they hit a bump; buildings and bridges vibrate when heavy trucks pass or the wind is
fierce.
Vibrations and wave motion are intimately related. Waves—whether ocean waves,
waves on a string, earthquake waves, or sound waves in air—have as their source a

03-1 Simple Harmonic Motion – Spring

vibration. Indeed, when a wave travels through a medium, the medium oscillates
(such as air for sound waves).

Oscillations
When an object vibrates or oscillates back and forth, over the same path, each
oscillation taking the same amount of time, the motion is periodic. The simplest
form of periodic motion is represented by an object oscillating on the end of a
uniform coil spring.

We assume that the mass of the spring can be ignored, and that the spring is
mounted horizontally, as shown in Fig.03–1a, so that the object of mass m slides
without friction on the horizontal surface. Any spring has a natural length at which
it exerts no force on the mass m. The position of the mass at this point is called the
equilibrium position. If the mass is moved either to the left, which compresses the
spring, or to the right, which stretches it, the spring exerts a force on the mass that
acts in the direction of returning the mass to the equilibrium position; hence it is
called a restoring force. We consider the common situation where we can assume
Figure 03-1
the restoring force F is directly proportional to the displacement x the spring has
been stretched (Fig.03–1b) or compressed (Fig.03–1c) from the equilibrium
position:

F = -kx [force exerted by spring] (03-1)

Note that the equilibrium position has been chosen at x = 0 and the minus sign in
Eq. 03–1 indicates that the restoring force is always in the direction opposite to the
to the displacement x. For example, if we choose the positive direction to the right
in Fig.03–1, x is positive when the spring is stretched (Fig.03–1b), but the direction
of the restoring force is to the left (negative direction). If the spring is compressed,
x is negative (to the left) but the force F acts toward the right (Fig.03–1c).

The proportionality constant k in Eq. 03–1 is called the spring constant for that
particular spring, or its spring stiffness constant (units = N/m). To stretch the spring
a distance x, an (external) force must be exerted on the free end of the spring with a
magnitude at least equal to
2
MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
 F= +kx [external force on spring]

Note that the force F in Eq.03–1 is not a constant, but varies with position.
Therefore, the acceleration of the mass m is not constant.

Let us examine what happens when our uniform spring is initially compressed a
distance x = -A as shown in Fig. 03–2a, and then our object of mass m is released
on the frictionless surface. The spring exerts a force on the mass that accelerates it
toward the equilibrium position. Because the mass has inertia, it passes the
equilibrium position with considerable speed. Indeed, as the mass reaches the
equilibrium position, the force on it decreases to zero, but its speed at this point is a
maximum, vmax (Fig.03–2b). As the mass moves farther to the right, the force on it
acts to slow it down, and it stops for an instant at x = A (Fig. 03–2c). It then begins
moving back in the opposite direction, accelerating until it passes the equilibrium
point (Fig. 03–2d), and then slows down until it reaches zero speed at the original
starting point, x = -A (Fig.03–2e). It then repeats the motion, moving back and
forth symmetrically between x = A and x = -A.

EXERCISE A A mass is oscillating on a frictionless surface at the end of a

horizontal spring. Where, if anywhere, is the acceleration of the mass zero (see Fig.
03–2)? Figure 03-2
(a) At x = -A (d) At both x = -A and x = +A
(b) At x = 0 (e) nowhere.
(c) At x = +A

3
MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
4
MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
5
MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
6
MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
7
MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
Simple Harmonic Motion

1. An object undergoing simple harmonic motion takes 0.25 s to travel from one point of zero velocity to the
next such point. The distance between those points is 36 cm. Calculate the

(a) period,

(b) frequency, and

(c) amplitude of the motion.

2. A 0.12 kg body undergoes simple harmonic motion of amplitude 8.5 cm and period 0.20 s.

(a) What is the magnitude of the maximum force acting on it?

(b) If the oscillations are produced by a spring, what is the spring constant?

3. What is the maximum acceleration of a platform that oscillates at amplitude 2.20 cm and frequency 6.60
Hz?

4. An automobile can be considered to be mounted on four identical springs as far as vertical oscillations are
concerned. The springs of a certain car are adjusted so that the oscillations have a frequency of 3.00 Hz.

(a) What is the spring constant of each spring if the mass of the car is 1450kg and the mass is evenly
distributed over the springs?

(b) What will be the oscillation frequency if five passengers, averaging 73.0 kg each, ride in the car with
an even distribution of mass?

5. In an electric shaver, the blade moves back and forth over a distance of 2.0 mm in simple harmonic motion,
with frequency 120 Hz. Find

(a) the amplitude,

(b) the maximum blade speed, and

(c) the magnitude of the maximum blade acceleration.

6. A particle with a mass of 1.00 kg is oscillating with simple harmonic motion with a period of 1.00 s and a
maximum speed of 103 m/s. Calculate

(a) the angular frequency and

(b) the maximum displacement of the particle.

7. An oscillating block–spring system takes 0.75s to begin repeating its motion. Find

(a) the period,

(b) the frequency in hertz, and

(c) the angular frequency in radians per second.

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MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
8. In Fig. 15-31, two identical springs of spring constant 7580
N/m are attached to a block of mass 0.245 kg.What is the
frequency of oscillation on the frictionless floor?

9. An oscillator consists of a block of mass 0.500 kg connected to

a spring.When set into oscillation with amplitude 35.0 cm,the oscillator repeats its motion every 0.500 s.
Find the

(a) period,

(b) frequency,

(c) angular frequency,

(d) spring constant,

(e) maximum speed, and

(f) magnitude of the maximum force on the block from the spring.

Simple Pendulum

1. A pendulum has a period of 1.85s on Earth. What is its period on Mars, where the acceleration of gravity is
about 0.37 that on Earth?

2. A pendulum makes 28 oscillations in exactly 50s. What is its

(a) period and

(b) frequency?

3. What is the period of a simple pendulum 47cm long

(a) on the Earth, and

(b) when it is in a freely falling elevator?

Energy in Simple Harmonic Motion

1. When the displacement in SHM is one-half the amplitude xm, what fraction of the total energy is

(a) kinetic energy and

(b) potential energy?

(c) At what displacement, in terms of the amplitude, is the energy of the system half kinetic energy and
half potential energy?

2. Find the mechanical energy of a block–spring system with a spring constant of 1.3 N/cm and an amplitude
of 2.4 cm.

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MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.
3. An oscillating block–spring system has a mechanical energy of 1.00J, an amplitude of 10.0cm, and a
maximum speed of 1.20m/s. Find

(a) the spring constant,

(b) the mass of the block, and

(c) the frequency of oscillation.

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MRS. H. MARIE F. SILVA
BSc, Dip. in Edu., MSc. In Phy. Edu.