Question

If the hollow spheres of Question 10.4 were replaced by solid spheres of the same mass and radius, the ratio of the moments of inertia for the two cases would a) increase. b) decrease. c) stay the same. d) be zero.

Short answer

Answer: (b) decrease

Step-by-step solution

Moment of Inertia of a Hollow Sphere

The moment of inertia for a hollow sphere with mass M and radius R can be written as: \(I_{hollow} = \frac{2}{3}MR^2\).

Moment of Inertia of a Solid Sphere

The moment of inertia for a solid sphere with mass M and radius R can be written as: \(I_{solid} = \frac{2}{5}MR^2\).

Comparing the Moments of Inertia

Now, let's compare the moments of inertia for both cases to determine the ratio. Divide the moment of inertia of the solid sphere by the moment of inertia of the hollow sphere: \(I_{ratio} = \frac{I_{solid}}{I_{hollow}} = \frac{\frac{2}{5}MR^2}{\frac{2}{3}MR^2}\) Notice that the mass M and the radius R are the same in both spheres, so they will cancel each other out: \(I_{ratio} = \frac{\frac{2}{5}}{\frac{2}{3}} = \frac{2}{5} \cdot \frac{3}{2} = \frac{6}{10} = \frac{3}{5}\)

Determine the Answer

Since the ratio of the moments of inertia is \(\frac{3}{5}\), we can compare it to the original moment of inertias: - If the ratio \(\frac{3}{5} > 1\), the moment of inertia would increase. - If the ratio \(\frac{3}{5} < 1\), the moment of inertia would decrease. - If the ratio \(\frac{3}{5} = 1\), the moment of inertia would stay the same. Since \(\frac{3}{5} < 1\), the moment of inertia would decrease when changing a hollow sphere to a solid sphere of the same mass and radius. Therefore, the correct answer is (b) decrease.

Key concepts

Moment of Inertia for Solid Sphere

Understanding the moment of inertia for solid spheres is crucial when tackling the physics problems involving rotational motion. The moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a uniform solid sphere, where the mass is distributed evenly throughout the object, the moment of inertia is calculated using the formula:
\[I_{solid} = \frac{2}{5}MR^2\].
In this formula, \(M\) stands for the mass of the sphere, and \(R\) represents the radius. The constant \(\frac{2}{5}\) arises from the integration of the mass distribution throughout the sphere. It's essential to understand that for objects of the same mass and size, the moment of inertia depends on how the mass is distributed in relation to the axis of rotation—a key concept when comparing different shapes or density distributions, such as contrasting a solid sphere with a hollow one.

Moment of Inertia for Hollow Sphere

The moment of inertia for a hollow sphere, such as a thin-walled ball, is different from that of a solid one because the mass is concentrated further away from the center. This geometry results in a different formula: \[I_{hollow} = \frac{2}{3}MR^2\].
The variable \(M\) is the mass of the hollow sphere, while \(R\) is its radius. The fact that the constant is \(\frac{2}{3}\) rather than \(\frac{2}{5}\) highlights the increased resistance to rotational acceleration compared to a solid sphere. When dealing with hollow spheres, remember that having more mass positioned far from the axis of rotation leads to a higher moment of inertia.

Comparing Moments of Inertia

Comparing the moments of inertia between different objects or shapes is a foundational concept in physics that sheds light on how various bodies behave under rotational forces. As seen in the example given, when comparing the moment of inertia of a solid sphere and a hollow sphere with identical mass and radius, the formulas imply innate differences due to their structures. To compare them, we take the ratio of their moments of inertia, as shown in the solution steps. The calculation reveals that for the same mass and radius, a solid sphere has a lower moment of inertia than a hollow sphere: \[I_{solid} < I_{hollow}\].
Which translates into the solid sphere being easier to spin or rotate. Understanding this relationship helps in predicting rotational behavior in various physical systems, such as planetary dynamics, industrial machinery, or even sports equipment.

Physics Problem Solving

Physics problem solving involves a systematic approach to understanding the principles at play and applying mathematical tools to find a solution. In the given exercise, the steps to find the solution included identifying the correct formulas, substituting known values, and performing algebraic manipulation to compare the moments of inertia. A strong foundation in concepts like moments of inertia, combined with methodical problem-solving skills, enables students to confidently tackle complex scenarios in physics. Remember, always double-check formulas, cancel out like terms, and interpret your results in the context of the physical situation for effective problem solving.