Problem 68 - University Physics with Modern Physics

Question

A uniform disk has radius \(R$$_0\) and mass \(M$$_0\). Its moment of inertia for an axis perpendicular to the plane of the disk at the disk's center is \\( \frac{1}{2} \\)\(M_0R_0^2\). You have been asked to halve the disk's moment of inertia by cutting out a circular piece at the center of the disk. In terms of \(R_0\), what should be the radius of the circular piece that you remove?

Step-by-Step Solution

Answer

The radius of the removed piece should be \( \frac{R_0}{\sqrt{2}} \).

Step 1: Understand the Problem

We have a disk with a known moment of inertia and we need to remove a central circular piece to reduce its moment of inertia by half. The goal is to find the radius of this central piece removed.

Step 2: Current Moment of Inertia

The initial moment of inertia of the entire disk is \( \frac{1}{2} M_0 R_0^2 \), where \( R_0 \) is the radius of the entire disk and \( M_0 \) is its mass.

Step 3: Halve the Moment of Inertia

We want the moment of inertia to be half of its original value. Hence, the desired moment of inertia is \( \frac{1}{4} M_0 R_0^2 \).

Step 4: Determine Moment of Inertia of Removed Piece

Assume the removed piece has radius \( r \). Its mass \( m \) is proportionally \( \frac{\pi r^2}{\pi R_0^2} M_0 = \frac{r^2}{R_0^2} M_0 \). The moment of inertia for the removed piece is \( \frac{1}{2} m r^2 = \frac{1}{2} \frac{r^2}{R_0^2} M_0 r^2 = \frac{1}{2} M_0 \frac{r^4}{R_0^2} \).

Step 5: Set Up Equation for Reduced Moment of Inertia

The moment of inertia after removing the piece should satisfy: \( \frac{1}{2} M_0 R_0^2 - \frac{1}{2} M_0 \frac{r^4}{R_0^2} = \frac{1}{4} M_0 R_0^2 \).

Step 6: Solve for \( r \)

Rewrite the equation: \( \frac{1}{2} M_0 R_0^2 - \frac{1}{2} \frac{r^4}{R_0^2} M_0 = \frac{1}{4} M_0 R_0^2 \). Simplify to get \( \frac{1}{4} M_0 R_0^2 = \frac{1}{2} M_0 \frac{r^4}{R_0^2} \). Divide both sides by \( \frac{M_0}{2} \): \( \frac{R_0^2}{4} = \frac{r^4}{R_0^2} \). Multiply through by \( R_0^2 \) to obtain \( r^4 = \frac{R_0^4}{4} \). Take the fourth root to find \( r = \frac{R_0}{\sqrt{2}} \).

Key Concepts

Understanding a Uniform Disk

A uniform disk is a flat, circular object with even mass distribution across its entire area. This is an important feature because it simplifies calculations related to physical properties like the moment of inertia.
- The uniform nature means that every small section of the disk has the same density.- This symmetrical mass distribution around the center is key to many physics problems.In our problem, the disk's radius is denoted by \( R_0 \) and the mass is \( M_0 \). These properties allow us to calculate the original moment of inertia easily.

Understanding Mass Distribution

Mass distribution refers to how mass is spread out in an object. For a disk, since it is uniform, the mass is evenly distributed. This uniform mass density affects how much torque is needed to rotate the disk.
The formula to determine the mass of a piece cut out from the disk depends on the area it covers: - The mass of the original disk is proportional to the area \( \pi R_0^2 \).- When a circular piece with radius \( r \) is removed, its mass \( m \) is calculated as \( \frac{r^2}{R_0^2} M_0 \). This evenly spread mass makes computation of changes in the disk's physical properties, such as moment of inertia, more straightforward.

Impact of the Axis of Rotation

The moment of inertia depends heavily on the axis of rotation. For our disk, the axis in question is perpendicular to the disk's plane, passing through its center. This setup is crucial:- The moment of inertia here is calculated because the mass is rotating around this specific axis.- The initial moment of inertia of a whole disk with this axis is \( \frac{1}{2} M_0 R_0^2 \). Removing a section of the disk changes the distribution of mass around the axis, which affects the new moment of inertia. Therefore, altering the disk's structure, like cutting out a central piece, directly impacts how easily the disk can rotate about the same axis.