How to find the acceleration due to gravity and moment of inertia using Compound pendulum


compound bar pendulum setup along with its graph
    The Compound Pendulum is an essential experiment in the  first year B.Sc. Physics practical course as part of the TANSCHE program. It builds on the concept of the simple pendulum and helps students explore more complex ideas related to rotational motion and moment of inertia. By examining how a rigid body swings around a horizontal axis, students can gain practical experience with important mechanics concepts.
    This guide will walk you through the step-by-step process of conducting the Compound Pendulum experiment, making sure you understand everything from the setup to the final calculations. Whether your goal is to measure the acceleration due to gravity or investigate the moment of inertia, this experiment provides valuable insights that connect theoretical knowledge with hands-on practice.

Compound pendulum

Aim:

        To determine the acceleration due to gravity and radius of gyration of the compound bar pendulum about its centre of gravity.

Apparatus required:

        Compound bar pendulum, stop clock, metre scale etc

Formula:

        Acceleration due to gravity is,

                                

    Radius of gyration of the compound bar pendulum about its centre of gravity is,

                            

Here,

             g is the acceleration due to gravity m/s2
            l is the equivalent length of the simple pendulum m
            T is the time period of oscillation for length s
            K is the radius of gyration of the bar about its centre of gravity m
            P, Q is the points corresponding to minimum time periods s

Procedure:

The pendulum is hung by inserting the knife edge through the first hole close to end A. We then measure how far the knife edge is from end A. The pendulum is set to oscillate with a small range. We have to note the time for 10 oscillations. Doing this twice to find the average time period T. Next, we repeat the process by hanging the bar through all the holes on one side of its center of gravity. After that, flip the bar and do the experiment again, but this time we only measure from end A. Record all the measurements in a table.
    A graph is plotted with the time period on the y-axis and the distance from end A on the x-axis. The graph shows two identical sections that mirror each other around the bar's center. To find the length of a simple pendulum that matches any period T, we draw a horizontal line from the y-axis at that period, which intersects the curve at points A, B, D, and E. We then measure the distances AD and BE from the graph. The length of the equivalent simple pendulum for time period T is calculated using the formula l = (AD + BE)/2. We also calculate l/T², repeating this for various time periods, and find that l/T² remains constant.
    Finally, we draw a common tangent PQ to the two curves at their lowest points and measure the distance PQ. The radius of gyration of the bar around its center of gravity is then found using K = PQ/2.

Tabular Column 1: To find the time period

S.No

Distance of the centre of the hole from A end 

cm

Time for 10 oscillations s

Time for one oscillation 

s

Trial 1

Trial 2

Mean








Tabular Column 2: To find the l/T2

Time period T

s

T2

s2

Equivalent length of simple pendulum cm

l/T²

AD

BE

l = (AD + BE) / 2



















Result:

    Using compound bar pendulum,

        (i) Acceleration due to gravity (g) = ______ m/s2

        (ii) Radius of gyration of the compound bar pendulum about its centre of gravity (K) = _____ m

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