A wagon wheel is constructed as shown in Fig. E9.35. The...

A wagon wheel is constructed as shown in Fig. E9.35. The radius of the wheel is $0.300 \mathrm{~m},$ and the rim has mass $1.40 \mathrm{~kg}$. Each of the eight spokes that lie along a diameter and are $0.300 \mathrm{~m}$ long has mass $0.280 \mathrm{~kg}$. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel? (Use Table $9.2 .$

A wagon wheel is constructed as shown in Fig. E9.35. The radius of the wheel is $0.300 \mathrm{~m},$ and the rim has mass $1.40 \mathrm{~kg}$. Each of the eight spokes that lie along a diameter and are $0.300 \mathrm{~m}$ long has mass $0.280 \mathrm{~kg}$. What is the moment of inertia of the wheel about
an axis through its center and perpendicular to the plane of the wheel? (Use Table $9.2 .$

Key Concepts

Moment of Inertia

Moment of inertia quantifies how a mass is distributed with respect to an axis of rotation and determines the resistance of an object to changes in its rotational motion. It is calculated by integrating or summing the mass elements multiplied by the square of their distances from the rotation axis.

Composite Rigid Body Analysis

When dealing with composite objects, the total moment of inertia is found by calculating the individual moments of inertia for each component about the same axis and then summing them. This approach allows complex geometries to be broken down into simpler parts whose moments of inertia are known or easier to compute.

Thin Ring (Rim) Inertia

For an object where the mass is concentrated along a circular path, such as a thin ring or rim, the moment of inertia about an axis perpendicular to its plane and through its center is given by the product of the mass and the square of the radius. This formula arises from the uniform distribution of mass at a constant distance from the center.

Uniform Rod Inertia about an End

A uniform rod rotating about an axis perpendicular to its length at one end has a moment of inertia that can be derived from integrating its mass distribution. The standard result is one-third of the mass times the square of its length, and this formula is useful when the rod’s mass is distributed linearly from the pivot point.

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