Objective To find the acceleration due to gravity by means of a simple pendulum.

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G4 Acceleration if free fall by means of a simple pendulum

Objective

To find the acceleration due to gravity by means of a simple pendulum.

Theory

Simple harmonic motion is the repetitive back-and-forth movement through a central, or equilibrium, position in which the maximum displacement on one side is equal to the maximum displacement on the other. The acceleration of a particle is directly proportional to its displacement from the equilibrium position but always opposite to its displacement from the equilibrium position. ( a = -)

A pendulum can display the simple harmonic motion. A simple pendulum consists merely of a bob (weight) suspended from a fixed point by a rod or string of length (L). The mass of the rod or string is assumed to be so much less than the suspended mass that it can be ignored. If the suspended mass is displaced to the left or right, while the rod or string is kept taut, and then released, the mass will swing freely back and forth under the gravity’s influence.

Restoring force F = - mg

             F = ma

     - mg = ma

             a = -g  (provided that  is small)

  a = -gθ = -g(x/l)     (∵ θ radius=arc length)    

             a=-  (where  g/l)

Thus, the periodic time T of a simple pendulum of length l is given by      

  T == 2π        

The period of oscillation of a simple pendulum depends on the length and is directly proportional to the square root of the length.

Moreover, we know that the period of oscillation is independent of the amplitude when the amplitude is within small limits (length remaining constant). This means the amplitude can vary within small limits, but the period of oscillation will be the same. The period of oscillation is also independent of the nature of the bob, i.e., the size, shape and material of the bob can vary, but the period of oscillation will remain the same.

Apparatus

Pendulum bob (e.g. a metal sphere with a hook attached, or with a hole bored through its centre), cotton thread, stop-watch, meter rule, retort stand and clamp, G-clamp, protractor, pieces of paper, small improvised vice, vernier caliper (available upon request), and triple beam balance (available upon request).

Method

About two-meter thread was tied to a pendulum bob and the cotton thread was suspended rigidly from the jaws of an improvised vice, such as two small wooden blocks held in a clamp. Alternatively two coins, two halves of a cork split lengthwise, or the jaws of a pair of pliers served equally of suspension when gripped in a clamp.

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(figure showing the set up with two small wooden blocks held in clamp)

A piece of paper with a vertical mark on it was placed behind the pendulum so that when the latter was at rest it hid the vertical mark from an observer standing in front of the pendulum.

        

The length L of the cotton thread was measured from the point of attachment to the center of mass of the bob by meter rule. (This was called effective length of the pendulum.)

The pendulum bob was set to swing through an arc of about 10

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