Simple Harmonic Motion - Part II

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2. Simple pendulum and its uses
A simple pendulum is an example of simple harmonic motion. A simple pendulum is made up of a bob made out of a metal ball, attached to a thread that is supported rigidly at one end.  

The bob of the pendulum is free to swing. When the pendulum is at rest, it is in its mean position at A. The mean position is also the equilibrium position. At this position, all the forces are balanced and no net force is acting on the pendulum. Thus initially at A, the pendulum is stationary. Now if the bob of the pendulum is taken to one side, at position B and released, the bob will undergo back and forth motion. The motion will trace an arc of a circle whose radius is equal to the length of the pendulum. The bob will move towards position C through A. If there is no frictional force, then the bob will move in the back and forth manner continuously, between the two extreme positions B and C.

Some parameters associated with a simple pendulum :  
1. Length of the pendulum : The length of the thread from which the pendulum is hung, up to the centre of the bob is called the length of the pendulum. This is denoted by L. It has to be borne in mind that the length of the thread plus the diameter of the bob does not constitute L. In fact L = length of the thread + radius of the bob.
2. Oscillations of the pendulum : One complete to and fro motion is known as the oscillation of the pendulum. One oscillation is the motion taken for the bob to go from position B (initial position) to A (mean position) to C (the other extreme position) and back to B. One oscillation can also be defined as the motion taken for the bob to go from A to B to C (through A) and back to position A.
3. Time period of the pendulum :  The time taken for one oscillation by the pendulum is called the time period of the pendulum. Time period is also called period of the pendulum. This is denoted by T.
4. Amplitude of the pendulum : The maximum displacement of the bob from its mean or equilibrium position is known as the amplitude of the pendulum. The straight-line  length AB or AC is the amplitude of the pendulum. It has to be borne in mind that the amplitude is the straight-line distance between points A and B or C and not the length of the arc AB or AC.

Since a simple pendulum is a good example of simple harmonic motion, the study of the time period of the pendulum is important. Let us now look at how the various parameters enumerated above affect the time period of the pendulum.

Take a simple pendulum and measure the time period by varying the following parameters :
1. Vary the length L of the pendulum.
2. Vary the mass of the bob of the pendulum.
3. Vary the material of the bob.
4. Vary the amplitude of the pendulum.

To measure the time period, take the bob of the pendulum to one side and release the bob gently. Let the pendulum swing for a short while. Start a stopwatch, when the bob reaches at one extreme position of its swing. Count a number of oscillations that the pendulum takes. Stop the stopwatch when the bob completes a certain number of oscillations. Lets say that you have measured time taken for the bob to complete 10 oscillations. The time period will be the time measured in seconds divided by 10. This number in seconds will give you the period of the pendulum in seconds. This procedure is necessary because it may not be possible to calculate time taken for just one oscillation. This method gives an averaged and more accurate value of the time period.  

From the above experiment, you will conclude that the time period of the pendulum  

  • Depends on the length of the pendulum L. In fact T2 is proportional to L.

  • Does not depend on the mass of the bob

  • Does not depend on the material of the bob

  • Does not depend on the amplitude of the pendulum

Thus  T2    L  

L = constant x T2

     L
    =  constant
    T2  

It has been calculated that the constant of proportionality is equal to acceleration due to gravity g divided by 4p2. The left hand side has units of meters per second square or m/s2. The units and the value of the constant is 0.248m/ s2.
                     g
Constant =  
                    42

               L                  g
Thus         = 
              T2                     42

                L x 42
T2  =   
                   g

T =    (L x 42 / g )

T =  2  (L / g)

Energy changes in the pendulum in the pendulum : Since the bob swings from one end to the other, the kinetic and potential energy of the system changes continuously. This has been studied in the chapter of Work, Energy and Power.

Uses of a simple pendulum : Simple pendulums are used to keep time in clocks in olden days. A simple pendulum is also used to calculate acceleration due to gravity g at various places on the earth and above the earth. The oscillations of the pendulum can be maintained continuously if there is no friction of the bob and the thread with its surroundings. But if there is a change in the period, then there is friction. The frictional force of the medium surrounding the pendulum can be calculated with the help of experiments with the simple pendulum.

 History of the pendulum

The discovery of oscillatory motion of a pendulum and its properties are attributed to the famous astronomer Galileo. The story goes that as a small child he used to be taken to a cathedral in Pisa, Italy, by his parents. The cathedral is next to the leaning tower of Pisa. Child Galileo was once bored listening to sermons in the cathedral. There were gas lamps hanging from the ceiling of the cathedral. Due to the breeze, the lamps were swinging from the ceiling. Galileo noticed that the swinging was periodic in nature by measuring time from the pulse or heart


Galileo

beats on his wrist. He saw that the swinging lamp came to its one extreme position after a fixed number of beats on the heart beats on his wrist. Galileo later discovered the detailed the nature of simple pendulums and their properties.

The swinging lamp (shown below) is still present in the cathedral at Pisa. It is a mute reminder of history of science.

 

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