Question

Calculate 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 \(0.0055\, \text{kg} \, \text{m}^2\).

Step-by-step solution

Understand the Problem

We are given a basketball modeled as a hollow sphere with a radius of \( r = 0.12 \) meters and a mass of \( m = 0.57 \) kg. We need to find its moment of inertia.

Formula for Moment of Inertia of a Hollow Sphere

The moment of inertia \( I \) for a hollow sphere is given by the formula: \[ I = \frac{2}{3} m r^2 \] where \( m \) is the mass and \( r \) is the radius of the sphere.

Substitute the Known Values

Insert the given values into the formula: \[ I = \frac{2}{3} \times 0.57 \times (0.12)^2 \]

Calculate the Moment of Inertia

First, calculate the square of the radius: \((0.12)^2 = 0.0144\). Next, multiply the mass and the squared radius by \( \frac{2}{3} \): \[ I = \frac{2}{3} \times 0.57 \times 0.0144 \approx 0.005472 \]\[ I \approx 0.0055 \] kg \( \mathrm{m}^2 \) rounding to four decimal places.

Key concepts

Hollow Sphere

A hollow sphere is a fascinating object in physics, characterized by being empty on the inside. It's a sphere made entirely from its surface with no material filling its interior. An example in everyday life is a basketball, where the material constitutes the outer layer, and the interior is empty.
In physics, analyzing such objects helps us understand how mass distribution with respect to a point affects its rotational characteristics. A hollow sphere is unique because its mass is uniformly distributed across its thin shell, which affects how it behaves when it spins. This characteristic is crucial in calculating its moment of inertia, which tells you about the sphere's resistance to changes in rotational speed.

• Common examples of hollow spheres include bubbles, thin-walled spheres, and some sports balls such as basketballs and soccer balls.
• Hollow spheres are often used in teaching physics because they simplify complex calculations.

Mass and Radius Relationship

In physics, especially in dealing with rotational dynamics, understanding the relationship between mass and radius is critical. The mass of an object refers to the amount of matter it contains and affects how the object responds to external forces.
When dealing with a sphere, either hollow or solid, the radius links the size of the object to its physical properties. The radius defines how far each piece of mass is from the center of the sphere. This distance affects the object's moment of inertia, which itself influences rotational behavior.
In a mathematical sense, the moment of inertia of a hollow sphere depends on both its mass and the square of its radius. This implies that not just the amount of mass matters but also how far the mass extends from the center:

• Inertia increases with more mass because there’s more resistance to changes in motion.
• Inertia also increases with a larger radius, as the mass is farther from the axis of rotation.
• The formula for the moment of inertia involves multiplying the mass and the square of the radius, reflecting how both characteristics work together to define rotational dynamics.

Physics Problem Solving

Solving physics problems successfully often involves several key steps. Whether you're dealing with a straightforward or complex question, applying a systematic approach can aid significantly in finding a solution. Let's break down a method typically used:
First, you should carefully read the problem to understand what is given and what is required. Taking note of all information, such as mass and radius in our example, is essential. Next, relate the problem to a familiar formula or physical concept. Identifying the correct formula for moment of inertia was a vital step here.
Once the formula is identified, substitute the known values into it. This is crucial as it translates the theoretical model into a practical calculation. Finally, carefully perform the calculations, keeping an eye on units and operations like squaring the radius in this problem.

• Reading and understanding the problem is the first step.
• Identify relevant formulas and concepts.
• Substitute given values and carefully perform calculations.
• Review the solution to ensure it makes sense physically and mathematically.

Mathematical Calculations in Physics

Executing mathematical calculations in physics requires attention to detail and a sound understanding of underpinning principles. Physics is grounded in translating real-world phenomena into mathematical language, making accurate calculation essential.
When determining the moment of inertia for a hollow sphere, for instance, it’s not only about applying the formula but understanding why and how each component of the formula is used. Here, we multiply the mass by the square of the radius, integrated together with the constant factor of \( \frac{2}{3} \) from the formula derived from theoretical physics principles.
By multiplying and then computing the squared term, each step requires precision and an understanding of the rationale behind them. This precision helps in reducing errors and ensuring the solution is sensible.
A few tips for ensuring accuracy in physics calculations:

• Always check units at every stage to ensure they are consistent.
• Break down complex operations into smaller, manageable parts.
• Double-check each calculation, especially when they involve squaring or multiples.
• After solving, verify if the result aligns with expected physical properties.