For procedures, see your mechanical laboratory manual.

Title: Kater's Pendulum


Aim

To find 'g' the acceleration due to gravity.

Theory

A Kater's pendulum is a reversible free-swinging pendulum invented by British physicist and army captain Henry Kater in 1817 for use as a gravimeter instrument to measure the local acceleration of gravity.

An important application of the pendulum is the determination of the value of the acceleration due to gravity. By adding a second knife-edge pivot and two adjustable masses to the physical pendulum described in the Physical Pendulum demo, the value of g can be determined to 0.2% precision.

An improvement in the precision of the measurement of g was developed in 1817 by
Kater. He realized that by using a compound pendulum and suspending it from each end in turn the requirement to measure the distance from the center of mass to the pivot could be removed. He made a very accurate measurement of g in London, a value that was used to define the meter for many years.

Using a simple pendulum, the value of g can be determined by measuring the length L and the period T. The value of T can be obtained with considerable precision by simply timing a large number of swings, but comparable precision in the length of the pendulum is not so easy. For example, it's hard to estimate where exactly the center of the mass is.

To overcome this difficulty we can turn a physical pendulum into a so-called reversible (Kater's) pendulum. Two knife-edge pivot points and two adjustable masses are positioned on the rod so that the period of swing is the same from either edge.


In Fig. 1, we consider the force of gravity to be acting at G. If hi is the distance to G from the suspension point Oi at the knife-edge Ki, the equation of motion of the pendulum is
where Ii is the moment of inertia of the pendulum about the suspension point Oi, and I can be 1 or 2. Comparing to the equation of motion for a simple pendulum
we see that the two equations of motion are the same if we take
It is convenient to define the radius of gyration of a compound pendulum such that if all its mass M were at a distance from Oi, the moment of inertia about Oi would be Ii, which we do by writing
Inserting this definition into equation (1) shows that
If IG is the moment of inertia of the pendulum about its center of mass G, we can also define the radius of gyration about the center of mass by writing
The parallel axis theorem gives us
so that, using (2), we have
The period of the pendulum from either suspension point is then
Squaring (3), one can show that
and in turn,
which allows us to calculate g,

Applications

Pendulums are used to regulate pendulum clocks and are used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geophysical surveys, and even as a standard of length.

References

  1. Kater's pendulum, Advanced mechanics virtual lab, Physical sciences, Amrita Vishwa Vidyapeetham University.
  2. Kater's pendulum, wikipedia.org.
  3. Reversible (Kater's) Pendulum, Harvard Natural Sciences Lecture Demonstrations, Havard University.
  4. Kater’s Pendulum, 2nd-year laboratory script, Department of Physics and Astronomy, The University of Sheffield.
  5. AM-2 The Kater Reversible Pendulum, Lock Haven University.
  6. Kater's pendulum, oxfordreference.com.
  7. Precision Kater Pendulum by Randall D. Peters.