Question

Why is mass not a factor in the equation for the period of a simple pendulum?

Answer 1

Let the length of the pendulum be L and its mass be M.

So, moment of inertia of the point mass = I= M*L^2

Also, torque=rF sin , where r is position of the point where force is applied, F is force and is angle between r and F.

So, torque on pendulum due to gravity = LMg sin (here r=L and F=Mg where g is gravitational acceleration)

For small angles, sin=

So, torque = LMg

Now, for simple harmonic motion, torque = I*^2*, where I is moment of inertia, is anguar frequency and is angular position.

So,LMg = I*^2*

=>LMg = M*L^2*^2*

=>^2 = g/L

=>=(g/L)^0.5

Now, time period = 2/ = 2(L/g)^0.5

So, time period for simple pendulum depends only on gravitational acceleration g and length L. It does not depend on mass M.