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?

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