Physics

Unit 2: SIMPLE HARMONIC MOTION

Topic Area: OSCILLATIONS AND WAVES

Sub-Topic Area: ENERGY CHANGES IN SIMPLE HARMONIC MOTION

Key unit competence: By the end of the unit I should be able to analyze energy changes in simple harmonic motion.

Unit Objectives:

By the end of this unit I will be able to;

   ◊ Determine the periodic time of an oscillating mass by practically and by calculation accurately.

   ◊ Derive and apply the equation of simple harmonic motion correctly

   ◊ Determine the periodic time of the simple pendulum correctly.

2.0 INTRODUCTION

You are familiar with many examples of repeated motion in your daily life. If an object returns to its original position a number of times, we call its motion repetitive. Typical examples of repetitive motion of the human body are heartbeat and breathing. Many objects move in a repetitive way, such as a swing, a rocking chair and a clock pendulum. Probably the first understanding of repetitive motion grew out of the observations of motion of the sun and phases of the moon.

Strings undergoing repetitive motion are the physical basis of all string musical instruments. What are the common properties of these diverse examples of repetitive motion?

In this unit we will discuss the physical characteristics of repetitive motion and develop techniques that can be used to analyze this motion quantitatively.

Opening question

Clearly analyze the images of Fig. 2-1 given below and explain what you think will happen in each case when the mass is displaced.

2.1 KINEMATICS OF SIMPLE HARMONIC MOTION

One common characteristic of the motions of the heartbeat, clock pendulum, violin string and the rotating phonograph turntable is that each motion has a well defined time interval for each complete cycle of its motion. Any motion that repeats itself with equal time intervals is called periodic motion. Its period is the time required for one cycle of the motion. 

Simple harmonic motion (SHM) is a type of motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

SHM is an oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position. This means that Simple harmonic motion occurs when

     • the force F acting on an object is directly proportional to the displacement x from a fixed point and is always towards this point.

• This means that acceleration is directly proportional to displacement from a fixed point and it is always directed towards this point.

Definition of terms

Time Period or Periodic Time T: It is the time taken for the particle to complete one oscillation, that is, the time taken for the particle to move from its starting position and return to its original position and is generally denoted by the symbol T.

Frequency f means how many oscillations occur in one second. Since the time period is the time taken for one oscillation, the frequency is expressed by; (f is frequency in one oscillation and T is the time period)

The frequency is measured in s ^–1. This unit is known as the hertz (Hz) in honour of the physicist Heinrich Hertz.

Amplitude A is the maximum displacement of the particle from its resting position or mean position.

Fig.2-3 shows the displacement-time graph of a periodic motion of a particle. From this graph, displacement can be represented as;

Angular velocity (w): Angular velocity is the rate of change of angular displacement. It is measured in (rad/s). This is related to periodic time according to equation (2-4).

Linear velocity : Linear velocity is the rate of change of linear displacement. It is measured in (m/s).

Linear acceleration a of a particle is the rate of change of linear velocity of that particle with time. It is measured in m/s².

2.2 EQUATION OF SIMPLE HARMONIC MOTION

The equation of simple harmonic motion is derived based on the conditions necessary for periodic motion to be simple harmonic.

Linear velocity can be related to displacement as shown below:

But from trigonometric identities;

                                                   EXAMPLE 2.1

A particle moving with SHM has velocities 4 cm/s and 3 cm/s at distances
 3 cm and 4 cm respectively from equilibrium position. Find

(a) the amplitude of oscillation

(b) the period

(c) velocity of the particle as it passes through the equilibrium position.

QUESTIONS

(i) Measure the gradient, m of your graph.

(j) Calculate the intercept c on the vertical axis.

(k) Calculate the constant a of the rule from c = log a.

(l) Calculate the period of a cantilever from T =  aLm

(m) Calculate the value of T from log T = m log L + log a  for value of
 L = 70.0 cm.

Compare and comment on the results in procedures (l) and (m).

EXAMPLE 2-3

The displacement of an object undergoing simple harmonic motion is given by the equation x(t) = 3.00 sin . Where x is in meters, t is in seconds

and the argument of the sine function is in radians.

(a) What is the amplitude of motion?

(b) What is the frequency of oscillation?

(c) What are the position, velocity and acceleration of the object at t = 0?

2.3 SIMPLE HARMONIC OSCILLATORS

A simple harmonic oscillator is a physical system in which a particle oscillates above and below a mean position at one or more characteristic frequencies. Such systems often arise when a contrary force results from displacement from a force-neutral position and gets stronger in proportion to the amount of displacement. Below are some of the physical oscillators;

2.3.1 Simple Pendulum

A simple pendulum consists of a small bob of mass m suspended from a fixed support through a light, inextensible string of length L as shown on Fig.2-5. This system can stay in equilibrium if the string is vertical. This is called the mean position or the equilibrium position. If the particle is pulled aside and released, it oscillates in a circular arc with the center at the point of suspension ‘O’.

Equation 2-18 represents the periodic time of a simple pendulum. Thus, the following are the factors affecting the periodic time of the simple pendulum;

       • Length of string

       • Angle from which pendulum is dropped

       • Acceleration due to gravity

       • Air resistance

EXAMPLE 2.4

A small piece of lead of mass 40 g is attached to the end of a light string of length 50 cm and it is allowed to hang freely. The lead is displaced to 0.5 cm above its rest position, and released.

(a) Calculate the period of the resulting motion, assuming it is simple harmonic.

(b) Calculate the maximum speed of the lead piece. (Take g = 9.81 ms–2)

EXAMPLE 2.5

What happens to the period of a simple pendulum if the pendulum’s length is doubled? What happens to the period if the mass of the suspended bob is doubled?

2.3.2 Mass suspended from a Coiled Spring

The extension of the spiral spring which obeys Hook’s law is directly proportional to the extending tension. A mass m is attached to the end of the spring which exerts a downward tension mg on it and stretches it by e as shown in Fig.2-7 below;

EXAMPLE 2.7

A light spiral spring is loaded with a mass of 50 g and it extends by 10 cm. Calculate the period of small vertical oscillations.

2.3.3Liquid in a U-tube

Consider a U-shaped tube filled with a liquid. If the liquid on one side of a U-tube is depressed by blowing gently down that side, the level of the liquid will oscillate for a short time about the respective positions O and C before finally coming to rest.