Question

A basketball has a radius of \(0.12 \mathrm{~m}\) and a mass of \(0.57 \mathrm{~kg}\). Assuming the ball to be a hollow sphere, what is its moment of inertia?

Short answer

The moment of inertia is approximately 0.00547 kg⋅m².

Step-by-step solution

Understand the Moment of Inertia Formula for Hollow Sphere

The moment of inertia (I) of a hollow sphere about an axis through its center is given by the formula: \[ I = \frac{2}{3} m r^2 \]. Here, \( m \) is the mass of the sphere, and \( r \) is the radius of the sphere.

Substitute the Given Values into the Formula

We are given the mass \( m = 0.57 \; \text{kg} \) and the radius \( r = 0.12 \; \text{m} \). Substitute these values into the formula: \[ I = \frac{2}{3} \times 0.57 \; \text{kg} \times (0.12 \; \text{m})^2 \].

Calculate the Radius Squared

First, calculate the square of the radius: \( r^2 = (0.12 \; \text{m})^2 = 0.0144 \; \text{m}^2 \).

Calculate the Mass and Radius Squared Product

Multiply the mass by the square of the radius: \( 0.57 \; \text{kg} \times 0.0144 \; \text{m}^2 = 0.008208 \; \text{kg} \cdot \text{m}^2 \).

Calculate the Moment of Inertia

Now multiply the result by \( \frac{2}{3} \) to find the moment of inertia: \[ I = \frac{2}{3} \times 0.008208 \; \text{kg} \cdot \text{m}^2 = 0.005472 \; \text{kg} \cdot \text{m}^2 \].

Key concepts

Hollow Sphere

A hollow sphere is a shell that is devoid of any substance inside, which makes it different from a solid sphere. When dealing with physics problems involving a hollow sphere, it's essential to understand that all the mass is concentrated on the outer shell. This unique configuration affects how the sphere rotates about an axis passing through its center. Unlike solid spheres, where mass distribution is uniform throughout the volume, the mass in a hollow sphere is spread over the surface. Key points to remember about hollow spheres:

• All the mass is on the exterior surface.
• The formula for calculating moment of inertia changes from that of a solid sphere.
• The hollow nature implies a different distribution of mass, affecting its rotational inertia.

In physics, grasping these differences is crucial for correctly solving problems involving the motion and rotatory dynamics of hollow objects like basketballs, globes, or any spherical containers.

Mass and Radius

The mass and radius are vital parameters when calculating the moment of inertia for any spherical object. These two factors determine not just the moment of inertia but also offer insights into how easily an object can rotate.Mass and radius are fundamental:

• Mass (\( m \)): It represents how much "stuff" is contained in the object. Heavier spheres have a greater moment of inertia.
• Radius (\( r \)): This is the distance from the center to the surface of the sphere. The larger the radius, the greater the moment of inertia.

Considering the equation \(I = \frac{2}{3} m r^2\), both mass and radius significantly influence the result. For instance, in our basketball example, a small radius and modest mass result in a relatively small moment of inertia. In essence, these two properties are central to understanding an object's reluctance to change its rotational state.

Physics Problem Solving

Physics problem solving often centers around formula application, conceptual understanding, and clear logical steps. When faced with a problem like calculating the moment of inertia for a hollow sphere, follow these guiding strategies:Problem-solving strategies:

• Identify the problem type: Recognize whether the object is a hollow sphere, solid sphere, disc, etc. Different scenarios require different formulas.
• Understand the formula: For each type of object, there's a specific formula. For hollow spheres, \(I = \frac{2}{3} m r^2\) is what you need. Grasp why this formula applies by considering the mass distribution.
• Substitution of known values: Insert the given values for mass and radius into the formula carefully to avoid mistakes.
• Perform calculations step by step: Break down each mathematical operation into digestible parts, solving part by part to minimize errors.

These fundamentals make solving physics questions more manageable and help solidify a strong basis for tackling complex problems in mechanics and rotational dynamics.