Compound pendulum
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Compound pendulum and its equivalent simple pendulum
The method of solution in the first example would apply to any object of mass m with centre of mass distant r from the axis of rotation.[Diagram goes here - download the original to see it.] At time t the object will have risen [Equation], where [Equation] is a function of time, so the gravitational potential energy of the disc is [Equation] Since the system is frictionless, total energy is conserved, so [Equation] As before, on differentiating both sides [Equation] On dividing by and rearranging we obtain [Equation] which is general the equation of motion for the compound pendulum. If the angle of displacement is small, then hence [Equation] This is the equation of angular simple harmonic motion, and has angular frequency [Equation] If we compare this with the equation for the simple pendulum [Equation] where l is the length of the simple pendulum, we can see that they are very similar If in equation (1) we se [Equation we see that it also takes the for [Equation The quantity [Equation is called the length of the equivalent simple pendulum This means that if we know the mass, m, of the object, the distance, r, of the object's centre of mass from the axis of revolution, and the object's moment of inertia, we can find the object's equivalent length as a simple pendulum. On substitution into the simplified pendulum equation, we obtain its angular frequency, or equivalently, its period. Alternatively, we obtain any one of the terms, I, m, and r given the angular frequency (or period) and two of the others. So we could "work backwards". Example (3) In the first example, find the length of the equivalent simple pendulum. Solution The solution to the first example was [Equation] which gives [Equation] Alternative, recall that the moment of inertia was [Equation] Hence, on substitution into [Equation]
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Contents of
Compound pendulum
1 Compound pendulum
2 Compound pendulum and its equivalent simple pendulum
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