Determination the value of g acceleration due to gravity by means of a compound pendulum.

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Name of the Experiment:
To determine the value of g, acceleration due to gravity, by means of a Compound Pendulum

Objectives of the Experiment:
• To study the motion of a compound pendulum,
• To study simple harmonic motion,
• To determine the acceleration due to gravity using the theory, results, and analysis of this experiment.
Theory:

A simple pendulum consists of a small body called a “bob” (usually a sphere) attached to
the end of a string the length of which is great compared with the dimensions of the bob and the
mass of which is negligible in comparison with that of the bob. Under these conditions the mass
of the bob may be regarded as concentrated at its center of gravity, and the length of the
pendulum is the distance of this point from the axis of suspension. When the dimensions of the
suspended body are not negligible in comparison with the distance from the axis of suspension to
the center of gravity, the pendulum is called a compound, or physical, pendulum. A rigid body
mounted upon a horizontal axis so as to vibrate under the force of gravity is a compound
pendulum. Compound pendulum is a rigid body of any shape free to turn about a horizontal axis.
3. A graph is drawn with the distance d of the various holes from the end A along the X-axis and the period T of the pendulum at these holes along the Y-axis. The graph has two branches, which are symmetrical about G. To determine the length of the equivalent simple pendulum corresponding to any period, a straight line is drawn parallel to the X- axis from a given period T on the Y- axis, cutting the graph at four points A, B, C, D. The distances AC and BD, determined from the graph, are equal to the corresponding length l. The average length l = (AC+BD)/2 and l/T2 are calculated. In a similar way, l/T2 is calculated for different periods by drawing lines parallel to the X-axis from the corresponding values of T along the Y- axis. L/T2 should be constant over all periods T, so the average over all suspension points is taken. Finally, the acceleration due to gravity is calculated from the equation g= 4π2(l/T2).
4. Tmin is where the tangent EF to the two branches of the graph crosses the Y-axis. At Tmin, the distance EF = l = 2kG can be determined, which gives us kG, the radius of gyration of the pendulum about its Centre of mass, and one more value of g, from g= 4π2(2kG/Tmin2) .
5. kG can also be determined as follows. A line is drawn parallel to the Y -axis from the point G corresponding to the Centre of mass on the X-axis, crossing the line ABCD at P. The distances AP = PD = AD/2 = h and BP = PC = BC/2 = h′ are obtained from the graph. The radius of gyration kG about the Centre of mass of the bar is then determined by equation (4). The average value of kG over the different measured periods T is taken, and the moment of inertia of the bar about a perpendicular axis through its Centre of mass is calculated using the equation IG=MkG2.

Apparatus:
A Compound Pendulum (metallic rectangular bar with hole in each 5cm distance).
A small metallic wedge
A holder in fixed point
Stop watch

Experimental Procedures:
1. The compound bar pendulum AB is suspended by passing a knife edge through the first hole at the end A. The pendulum is pulled aside through a small angle and released, whereupon it oscillates in a vertical plane with small amplitude. The time for 10 oscillations is measured. From this the period T of oscillation of the pendulum is determined.
2. In a similar manner, periods of oscillation are determined by suspending the pendulum through the remaining holes on the same side of the Centre of mass G of the bar. The bar is then inverted and periods of oscillation are determined by suspending the pendulum through all the holes on the opposite side of G. The distances d of the top edges of different holes from the end A of the bar are measured for each hole. The position of the Centre of mass of the bar is found by balancing the bar horizontally on a knife edge. The mass M of the pendulum is determined by weighing the bar with an accurate scale or balance.


Calculation:
From the graph,
Length L=(AC+BD)/2 cm =cm and
Time period, T = Sec
Therefore, g =(4π^2 L)/T^2 cm/sec2 = cm/sec2
%of error calculation = |(Experimental value-Actual value)/(Actual Value)|×100%
=