will the periods be different for large and small oscillations?

Question

for general equation of motion for a pendulum. will the periods be different for large and small oscillations? if so for which will it be longer? explain

Answer 1

Unlike the case of and ideal mass and spring system, a pendulum in a gravity field is not really a harmonic oscillator. That means, the position, velocity and acceleration of the pendulum are not exactly described by sinusoids of a single, pure frequency.

For small oscillations of a pendulum, the approximate relation sin(theta) ~= theta is used to simplify the equation of motion that describes the system. When this approximation is used, the pendulum appears to be a harmonic oscillator.

The approximation becomes less accurate as the amplitude (i.e.angular swing) of the pendulum becomes larger. In the exact solution, the period becomes longer as the amplitude increases.

You can readily work out the exact equation of motion of a simple, idealized pendulum to be: g sin(theta) = L d^2(theta)/dt^2

Answer 2

There are two scenarios. One is where the velocity increases after the end of one oscilation . The other the velocity repeats after each oscilation.
In the case of the pendulum the swing of the pendulum peters out due to friction but the period does not change.

Answer 3

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