Problem 44 - College Physics

Question

A compound disk of outside diameter 140.0 \(\mathrm{cm}\) is made up of a uniform solid disk of radius 50.0 \(\mathrm{cm}\) and area density 3.00 \(\mathrm{g} / \mathrm{cm}^{2}\) surrounded by a concentric ring of inner radius \(50.0 \mathrm{cm},\) outer radius \(70.0 \mathrm{cm},\) and area density 2.00 \(\mathrm{g} / \mathrm{cm}^{2} .\) Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

Step-by-Step Solution

Answer

The total moment of inertia is \(8.6435 \times 10^6 \text{ g cm}^2\).

Step 1: Understand the Problem

The compound disk consists of two parts: a solid disk and a concentric ring, each with different dimensions and mass distributions. We need to find the moment of inertia of this combination about the central axis.

Step 2: Moment of Inertia Formula for Solid Disk

For a solid disk with radius \(R_1\) and area density \(\sigma_1\), the moment of inertia about the central axis is given by: \[I_{ ext{solid disk}} = \frac{1}{2} M_1 R_1^2\]where \(M_1 = \sigma_1 \times \pi R_1^2\) is the mass of the solid disk.

Step 3: Calculate Mass and Inertia of Solid Disk

For the solid disk, \(R_1 = 50.0 \text{ cm}\) and \(\sigma_1 = 3.00 \text{ g/cm}^2\). Calculate the mass:\[M_1 = 3.00 \times \pi \times (50.0)^2 = 23562.5 \text{ g}\]Then, calculate the moment of inertia:\[I_{ ext{solid disk}} = \frac{1}{2} \times 23562.5 \times (50.0)^2 = 2.9478 \times 10^6 \text{ g cm}^2\]

Step 4: Moment of Inertia Formula for Concentric Ring

The inertia for the ring is calculated by subtracting the inertia of the inner disk from the outer disk. For a ring with inner radius \(R_1\), outer radius \(R_2\), and area density \(\sigma_2\), the moment of inertia is:\[I_{ ext{ring}} = \pi \sigma_2 (R_2^4 - R_1^4)\]where \(R_1 = 50.0 \text{ cm}\) and \(R_2 = 70.0 \text{ cm}\).

Step 5: Calculate Mass and Inertia of Concentric Ring

Calculate the mass of the ring:\[M_2 = 2.00 \times \pi \times ((70.0)^2 - (50.0)^2) = 7539.8 \text{ g}\]Calculate the moment of inertia:\[I_{ ext{ring}} = \pi \times 2.00 \times \left((70.0)^4 - (50.0)^4\right) = 5.6957 \times 10^6 \text{ g cm}^2\]

Step 6: Combine Moments of Inertia

Add the moments of inertia from the solid disk and the concentric ring to find the total moment of inertia:\[I_{ ext{total}} = I_{ ext{solid disk}} + I_{ ext{ring}} = 2.9478 \times 10^6 + 5.6957 \times 10^6\]Thus, the total moment of inertia is:\[I_{ ext{total}} = 8.6435 \times 10^6 \text{ g cm}^2\].

Key Concepts

Solid Disk Moment of Inertia

Understanding the concept of moment of inertia for a solid disk is crucial for analyzing compound objects. The moment of inertia is a measure of an object's tendency to resist angular acceleration. In simpler terms, it tells us how difficult it is to spin an object around an axis. For a uniform solid disk, the moment of inertia depends on its mass and radius. It can be calculated using the formula:
\[I_{\text{solid disk}} = \frac{1}{2} M_1 R_1^2\]
Here, the mass \(M_1\) is derived from the area density \(\sigma_1\) and the disk's radius \(R_1\) as \(M_1 = \sigma_1 \times \pi R_1^2\). This equation highlights how the distribution of mass impacts rotational motion.

Concentric Ring Calculations

Calculating the moment of inertia for a concentric ring involves more steps than for a solid disk. A concentric ring is essentially a circular band. To find its moment of inertia, you need to consider both the outer and inner radii. This is because the ring is formed by subtracting the inner disk from the larger outer disk. The formula for the ring's moment of inertia is:
\[I_{\text{ring}} = \pi \sigma_2 (R_2^4 - R_1^4)\]
Here, \(\sigma_2\) is the area density, and \(R_1\) and \(R_2\) are the inner and outer radii, respectively. This formula shows how inertia depends on the size of the ring and the mass distribution across its area.

Mass Distribution in Compound Objects

Compound objects, like the disk in this scenario, are made up of multiple parts with varying mass distributions. When calculating the total moment of inertia, it's important to consider the separate contributions from each part. For this compound disk:

• The solid disk's mass is concentrated towards the center, giving it a specific moment of inertia.
• The concentric ring's mass is spread between two radii, requiring a different calculation.

Each component's contribution is assessed individually, using its own specifications and added together to find the total moment of inertia.

Radius and Density Relationship

The radius and density of an object play significant roles in determining both mass and moment of inertia. In our exercise:

• Radius impacts the moment arm, which influences how much torque is needed to rotate an object. Larger radii increase the moment of inertia.
• Density determines how mass is spread over the object's area. It informs the mass calculation, which then feeds into the moment of inertia formula.

The balance between radius and density is essential in physics problems, as it affects an object's response to rotational forces. Analyzing this relationship helps in understanding the rotational dynamics of compound shapes like our disk example.