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.

EXERCISE 2.2

1. A baby in a ‘baby bouncer’ is a real-life example of a mass-on-spring oscillator. The baby sits in a sling suspended from a stout rubber cord, and can bounce himself up and down if his feet are just in contact with the ground. Suppose a baby of mass 5.0 kg is suspended from a cord with spring constant 500 N m^–1. Assume g = 10 N kg^–1.

      (a) Calculate the initial (equilibrium) extension of the cord.

      (b) What is the value of angular velocity?

      (c) The baby is pulled down a further distance, 0.10 m, and released. How long after his release does he pass through equilibrium position?

     (d) What is the maximum speed of the baby?

     (e) A simple pendulum has a period of 4.2 s. When it is shortened by 1.0 m the period is only 3.7 s. 

     (f) Calculate the acceleration due to gravity g suggested by the data.

2. A pendulum can only be modelled as a simple harmonic oscillator if the angle over which it oscillates is small. Why is this so?

3. What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s? State the assumptions made.

4. A geologist uses a simple pendulum that has a length of 37.10 cm and a frequency of 0.8190 Hz at a particular location on the Earth. What is the acceleration due to gravity at this location?

5. Find the time taken for a particle moving in S.H.M. from Given that the period of oscillation is 12s.

6. A spring is hanging from a support without any object attached to it and its length is 500 mm. An object of mass 250 g is attached to the end of the spring. The length of the spring is now 850 mm.

     (a) What is the spring constant?  The spring is pulled down 120 mm and then released from rest.

     (b) Describe the motion of the object attached to the end of the spring.

     (c) What is the displacement amplitude?

     (d) What are the natural frequency of oscillation and period of motion?  Another object of mass 250 g is attached to the end of the spring.

     (e) Assuming the spring is in its new equilibrium position, what is the length of the spring?

     (f) If the object is set vibrating, what is the ratio of the periods of oscillation for the two situations?

2.4  KINETIC AND POTENTIAL ENERGY OF AN OSCILLATING SYSTEM

Kinetic energy as the energy of a body in motion, change in velocity will also change it as shown on Fig.2-12. Velocity of an oscillating object at any point is given by equation 2-8;

When the particle is in oscillatory motion, work is done against the force trying to restore it. The energy stored to perform this work is called the potential energy.

Force on the particle;

Work done to restore the position of the particle after being displaced by x is given by;

Note that there is no work done when displacement is zero.

Substitute equation 2-32 into equation 2-33 to get;

2.5  ENERGY CHANGES AND ENERGY CONSERVATION IN AN OSCILLATING SYSTEM

In an oscillation, there is a constant interchange between the kinetic and potential forms and if the system does no work against resistive force its total energy is constant. Fig.2-12 illustrates the variation of potential energy and kinetic energy with displacement x.

Substituting equation 2-3 for sinusoidal displacement into equation 2-34 and equation 2-35 gives;

The total energy of an oscillating system using equations 2.35 and 2.36 is given by;

The equation 2-38 of total energy indicates that this energy is constant and is independent of displacement x. Since the total energy of an oscillating particle is constant, it means that potential energy and kinetic energy vary in such a way that total energy is conserved.

Also substituting equation 2-35 and equation 2-36 into equation 2-37 will give an expression for the total energy of an oscillating system which is independent of time taken.

Fig.2-15 illustrates the variation of energy of an oscillating system with time.

EXAMPLE 2.8

A 0.500-kg cart connected to a light spring for which the force constant is 20.0 N/m oscillates on a horizontal, frictionless air track.

(a) Calculate the total energy of the system and the maximum speed of the cart if the amplitude of the motion is 3.00 cm.

(b) What is the velocity of the cart when the position is 2.00 cm?

(c) Compute the kinetic and potential energies of the system when the position is 2.00 cm.

2.6  SUPERPOSITION OF HARMONICS OF SAME FREQUENCY AND SAME DIRECTION

QUESTIONS

1.Give at least 2 examples of the applications of superposition in real life.

2. Derive the expression for the resultant displacement of two oscillations of the same frequency but acting in opposite directions.

(a) Find the period of its motion. 

(b) Determine the maximum speed of the block. 

(c) What is the maximum acceleration of the block? 

(d) Express the position, speed, and acceleration as functions of time.

3. (a) A 10N weight extends a spring by 5cm. Another 10N weight is added, and the spring extends another 5cm. What is the spring constant of the spring?

(b) A pendulum oscillates with a frequency of 0.5 Hz. What is the length of the pendulum?

4. Christian Huygens (1629–1695), the greatest clockmaker in history, suggested that an international unit of length could be defined as the length of a simple pendulum having a period of exactly 1 s. How much shorter would our length unit be had his suggestion been followed?

5. A simple pendulum is suspended from the ceiling of a stationary elevator, and the period is determined. Describe the changes, if any, in the period when the elevator

(a) accelerates upward,  

(b) accelerates downward, and (c) moves with constant velocity.

6. Imagine that a pendulum is hanging from the ceiling of a car. As the car coasts freely down a hill, is the equilibrium position of the pendulum vertical? Does the period of oscillation differ from that in a stationary car?

7. The diagram below shows a spring of stiffness K, attached to a mass m.

The mass is pulled by a distance a to the left and released. Show that the velocity of the mass can be modeled by;

Where x is the extension in the spring. What important assumption has to be made about the system?

8. What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?