Question

A solid, uniform disk of mass M and radius a may be rotated about an axis parallel to the disk axis, at variable distances from the center of the disk.
What is Tmin, the minimum period of the pendulum?

Answer 1

Answer:

the function varies linearly with the radius of the disk, so the smallest period is zero for a radius of zero centimeters

Explanation:

This system performs a simple harmonic movement where the angular velocity is given by

         w = √ k / I

Where k is the constant recovered from the axis of rotation and I is the moment of inertia of the disk

The expression for the moment of inertia is

           I = 1/2 m r²

Angular velocity, frequency and period are related

         w = 2π f = 2π / T

Substituting

             2π / T = √ k / I

             T = 2π √ I / k

             T = 2π √ (½ m r² / k)

             T = (2π √m / 2k)   r

 

We can see that the function varies linearly with the radius of the disk, so the smallest period is zero for a radius of zero centimeters