Asolid, 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 minimum period of the pendulum Tmin is

The period of a pendulum refers to the time required to complete one single revolution or cycle.

Let consider the time period of the disk to that of bob of a pendulum, then, the period of the pendulum can be expressed as:

At minimum, the period of the pendulum needs to obey the condition that:

∴

The above equation becomes:

Taking the differentiation of the above equation with respect to d, we have:

∴

Learn more about the period of a pendulum here:

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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

well clustered together.

points close on a scatter diagram