A compound disk of outside diameter 140.0 cm is made up of a uniform solid disk of radius 50.0 cm and area density 3.00 g/cm² surrounded by a concentric ring of inner radius 50.0 cm, outer radius 70.0 cm, and area density 2.00 g/cm². Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

Verified step by step guidance

1. Step 1: Understand the problem. The moment of inertia of a compound disk is calculated by summing the moments of inertia of its individual components. Here, the disk consists of two parts: a solid disk and a concentric ring. Both parts have different area densities and dimensions, so their contributions to the total moment of inertia must be calculated separately.
2. Step 2: Recall the formula for the moment of inertia of a solid disk about an axis perpendicular to its plane and passing through its center. The formula is: $I=\frac{1}{2}M{R}^2$, where $M$ is the mass and $R$ is the radius of the disk. To find the mass, use the area density and the area of the disk: $M=\mathrm{ho}A$, where $\mathrm{ho}$ is the area density and $A$ is the area.
3. Step 3: Calculate the moment of inertia of the solid disk. First, find the area of the disk using the formula for the area of a circle: $A=\mathrm{\backslash (\backslash text\{\pi \}\backslash )}{R}^2$. Then, multiply the area by the area density to find the mass. Substitute the mass and radius into the moment of inertia formula for the solid disk.
4. Step 4: Recall the formula for the moment of inertia of a ring about an axis perpendicular to its plane and passing through its center. The formula is: $I=M\frac{(}{R^2}2$, where $M$ is the mass, $R$ is the outer radius, and $r$ is the inner radius. To find the mass, use the area density and the area of the ring: $M=\mathrm{ho}\left(A\right)$, where the area of the ring is the difference between the areas of the outer and inner circles.
5. Step 5: Calculate the moment of inertia of the ring. First, find the area of the ring by subtracting the area of the inner circle from the area of the outer circle. Multiply the area by the area density to find the mass. Substitute the mass, inner radius, and outer radius into the moment of inertia formula for the ring. Finally, sum the moments of inertia of the solid disk and the ring to find the total moment of inertia of the compound disk.

Key Concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It depends on the mass distribution relative to that axis; the further the mass is from the axis, the greater the moment of inertia. For composite objects, the total moment of inertia is the sum of the moments of inertia of its individual components.

Area Density

Area density, often expressed in units like g/cm², refers to the mass per unit area of a surface. It is crucial for calculating the mass of irregular shapes or composite objects, as it allows us to determine the total mass by integrating the area density over the object's surface area.

Composite Shapes

Composite shapes are objects made up of multiple simpler geometric forms. To find properties like moment of inertia for these shapes, one can analyze each component separately and then combine their contributions. This approach often involves using parallel axis theorem and integrating over the defined areas of each component.