i) the acceleration due to gravity at the place and
ii) the radius of gyration of the bar pendulum about its centre of gravity and hence
iii) calculate the moment of inertia about an axis through its centre of gravity perpendicular to its length.
In the fig. O is the centre of suspension, G is the centre of gravity and O' is the centre of oscillation, then OO' is the length l of an equivalent simple pendulum. Acceleration due to gravity at the place is given by,
Where T is the period of oscillations of the pendulum about O.
If and the radius of gyration is given by,
Also, if t is the minimum period,
Moment of inertia I of the pendulum about the axis through its centre of gravity,
where M is the mass of the pendulum.
The compound bar pendulum AB is suspended by passing the knife edge through first hole from the end A. A long knitting needle is kept vertically infront of the pendulum to mark its equilibrium position. The pendulum is pulled aside through the small angle and released. The pendulum oscillates in a vertical plane with a small amplitude. The time for 10 oscillations is calculated. From this the period(T) of the oscillation of the pendulum is determined.
In a similar manner periods of oscillations are determined by suspending the pendulum through all the holes on the side of the centre of gravity G of the bar.(The bar is then inverted and the period of oscillations are determined by suspending the pendulum through all the holes on the other side of centre of gravity G). The distances d of top edges of the different holes from the end of the bar measured for each case.
The distance of the centre of gravity of the bar from the end A is noted by balancing horizontally on a knife edge. The mass of the pendulum is also determined.
A graph is drawn with distance (d) of the various holes from the end A along the X-axis and the period (T) of the pendulum about these holes along the Y-axis. The graph has exactly two similar parts 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 the point on the Y- axis cutting the graph at four points ABCD. The distances AC and BD are determined from the graph. The length l is calculated as l=AC+BD/2. Then l / T 2 is calculated. In the similar way , l / T 2 is calculated for different periods by drawing line parallel to the X-axis from the corresponding values of T along the y- axis. The average value l / T 2 is found out . The acceleration due to gravity is calculated from the equation,
Then minimum period Tmin= t is noted by drawing a tangent EF at the curved part of the graph.
The distance EF is measured as 2k.Acceleration due to gravity is also calculated from the equation,
From the graph the distance EF is measured as 2k. From this K, the radius of gyration of the pendulum about its centre of gravity is calculated. The radius of gyration k also be determined as follows.
A line is drawn parallel to the Y -axis from the point G corresponding to the centre of gravity on the X-axis meeting the line ABCD at P. The distances AP = PD = AD/2 = h1 and BP = PC = BC/2 = h2 are calculated. The radius of gyration k about the centre of gravity of the bar is determined from the equation,
The average value of k is determined. The moment of inertia of the bar about a perpendicular axis through its centre of gravity is calculated using the equation,