Question

write the equation of motion of a simple pendulum and use it to obtain the condition under which its motion is simple harmonic in nature?

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Answer 1

Answer:

F = -kx

Explanation:

The general equation for simple harmonic motion along the x-axis results from a straightforward application of Newton's second law to a particle of mass m acted on by a force: F = -kx, where x is the displacement from equilibrium and k is called the spring constant.

or

Calculate the period of oscillations according to the formula above: T = 2π * √(L/g) = 2π * √(2/9.80665) = 2.837 s . Find the frequency as the reciprocal of the period: f = 1/T = 0.352 Hz

or

The Equation of Motion

A simple pendulum consists of a ball (point-mass) m hanging from a (massless) string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. By applying Newton's secont law for rotational systems, the equation of motion for the pendulum may be obtained

τ=Iα⇒ −mg sinθ L=mL2d2θdt2 and rearranged as

d2θdt2+gLsinθ=0

If the amplitude of angular displacement is small enough, so the small angle approximation ($\sin\theta\approx\theta$) holds true, then the equation of motion reduces to the equation of simple harmonic motion

d2θdt2+gLθ=0

The simple harmonic solution is

θ(t)=θocos(ωt),where θo

is the initial angular displacement, and

ω=√g/L

the natural frequency of the motion. The period of this sytem (time for one oscillation) is

T=2πω=2π√Lg.

Small Angular Displacements Produce Simple Harmonic Motion

The period of a pendulum does not depend on the mass of the ball, but only on the length of the string. Two pendula with different masses but the same length will have the same period. Two pendula with different lengths will different periods; the pendulum with the longer string will have the longer period.

How many complete oscillations do the blue and brown pendula complete in the time for one complete oscillation of the longer (black) pendulum?

From this information and the definition of the period for a simple pendulum, what is the ratio of lengths for the three pendula?

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. This simple approximation is illustrated in the animation at left. All three pendulums cycle through one complete oscillation in the same amount of time, regardless of the initial angle.

The Real (Nonlinear) Simple Pendulum

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 form

d2θdt2+gLsinθ=0

This 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