Question

DOES THE TIME PERIOD DEPEND ON THE AMPLITUDE OF THE PENDULUM?

Answer 1

Answer:

No ..................

Answer 2

Answer:

The period of a pendulum does not depend on the mass of the ball, but only on the length of the string. ... With the assumption of small angles, the frequency and period of the pendulum are independent of the initial angular displacement amplitude. A pendulum will have the same period regardless of its initial angle.When the angular displacement amplitude of the pendulum is large enough that the small angle approximation no longer holds, then the equation of motion must remain in its nonlinear formd2θdt2+gLsinθ=0d2θdt2+gLsin⁡θ=0This differential equation does not have a closed form solution, but instead must be solved numerically using a computer. Mathematica numerically solves this differential equation very easily with the built in function NDSolve[ ].

The small angle approximation is valid for initial angular displacements of about 20° or less. If the initial angle is smaller than this amount, then the simple harmonic approximation is sufficient. But, if the angle is larger, then the differences between the small angle approximation and the exact solution quickly become apparent.

In the animation below left, the initial angle is small. The dark blue pendulum is the small angle approximation, and the light blue pendulum (initially hidden behind) is the exact solution. For a small initial angle, it takes a rather large number of oscillations before the difference between the small angle approximation (dark blue) and the exact solution (light blue) begin to noticeable diverge.