Why kater's pendulum gives accurate value of acceleration due to gravity?
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Why is acceleration due to gravity measured accurately with the help of compound pendulum than simple pendulum?
The time period of a pendulum, simple or compound, is given by;
T = 2 π [ L/ g ]½ , where L = length of the pendulum, T is time period of one oscillation of the pendulum.
There are only two quantities required to be measured for determining the acceleration g due to gravity using a pendulum, namely the length of the pendulum L, and the time period T of the pendulum. The time period T of the pendulum can be measured accurately by measuring the time taken by the pendulum to complete sufficiently large number of oscillations, whether the pendulum is simple or compound. The problem arises in determining the length of the pendulum L. The length of the pendulum is the distance between the point of suspension of the pendulum and the point of oscillation of the pendulum.
In a simple pendulum, the point of suspension is the point, where the inextensible thread put in a split cork and held tight in clamp stand, is just coming out of the cork. This point is not well defined in a simple pendulum.
Similarly the point of oscillation is the centre of gravity of the spherical metallic bob doing the oscillations. It is taken to be the centre of the spherical bob, and the centre is taken r cm below the top of the spherical bob, where r is radius of the mettalic bob. The point of suspension also is not accurately determined in a simple pendulum.
Another source of error in a simple pendulum is that the string is not inextensible and gets extended during oscillations.
All these factors contribute to the uncertainty in the determining the length of the pendulum in a simple pendulum.
In a compound pendulum the point of suspension and point of oscillation are knife edges so are more precisely defined.
In a compound pendulum the point of suspension and the point of oscillation are interchangeable ie if point of oscillation becomes point of suspension and vice versa the time period remains unchanged. So once a near equality in time period is obtained on both sides of the centre of gravity of the metallic bar, determining the length of the pendulum becomes more accurate, and so does determining the acceleration g due to gravity.
The time period of a pendulum, simple or compound, is given by;
T = 2 π [ L/ g ]½ , where L = length of the pendulum, T is time period of one oscillation of the pendulum.
There are only two quantities required to be measured for determining the acceleration g due to gravity using a pendulum, namely the length of the pendulum L, and the time period T of the pendulum. The time period T of the pendulum can be measured accurately by measuring the time taken by the pendulum to complete sufficiently large number of oscillations, whether the pendulum is simple or compound. The problem arises in determining the length of the pendulum L. The length of the pendulum is the distance between the point of suspension of the pendulum and the point of oscillation of the pendulum.
In a simple pendulum, the point of suspension is the point, where the inextensible thread put in a split cork and held tight in clamp stand, is just coming out of the cork. This point is not well defined in a simple pendulum.
Similarly the point of oscillation is the centre of gravity of the spherical metallic bob doing the oscillations. It is taken to be the centre of the spherical bob, and the centre is taken r cm below the top of the spherical bob, where r is radius of the mettalic bob. The point of suspension also is not accurately determined in a simple pendulum.
Another source of error in a simple pendulum is that the string is not inextensible and gets extended during oscillations.
All these factors contribute to the uncertainty in the determining the length of the pendulum in a simple pendulum.
In a compound pendulum the point of suspension and point of oscillation are knife edges so are more precisely defined.
In a compound pendulum the point of suspension and the point of oscillation are interchangeable ie if point of oscillation becomes point of suspension and vice versa the time period remains unchanged. So once a near equality in time period is obtained on both sides of the centre of gravity of the metallic bar, determining the length of the pendulum becomes more accurate, and so does determining the acceleration g due to gravity.
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Answer:
Kater's pendulum was used to measure the acceleration of gravity with greater accuracy than an ordinary pendulum, because it avoids to measure 'l' where 'l' is length is the simple pendulum.
Explanation:
- Kater’s pendulum is a compound pendulum invented by Henry Kater in order to determine the acceleration due to gravity.
- A compound pendulum is the one in which the center of oscillation and center of gravity are separated by some considerable distance.
- Unlike the other pendulums, in the Kater's pendulum the center of gravity and the center of oscillation need not be determined to get more accuracy in the gravity.
- The problem was that there was no way to find the center of oscillation accurately, where Kater's pendulum came up with solution.
- Since the pivot point and the center of oscillation are interchangeable. Any pendulum which is suspended upside down from the center of oscillation, will have the same period of swing. This new oscillation center is the pivot point.
- The distance between the length of the pendulum with the same period and these conjugate points will be equal to each other. This is called the principle of Kater’s pendulum.
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