Question

Consider a cylinder and a hollow cylinder, rotating about an axis going through their centers of mass. If both objects have the same mass and the same radius, which object will have the larger moment of inertia? a) The moment of inertia will be the same for both objects. b) The solid cylinder will have the larger moment of inertia because its mass is uniformly distributed. c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.

Short answer

Answer: The hollow cylinder will have a larger moment of inertia because its mass is located away from the axis of rotation.

Step-by-step solution

Recall the moment of inertia formulas for a solid cylinder and a hollow cylinder.

The formulas for the moment of inertia are given by: Solid cylinder: I_solid = (1/2)MR^2 Hollow cylinder: I_hollow = MR^2 where I represents the moment of inertia, M is the mass of the object, and R is the radius of the object.

Compare the moment of inertia formulas.

We need to compare the formulas for the moment of inertia for a solid cylinder and a hollow cylinder with the same mass and the same radius. Comparing the two formulas, we see that: (1/2)MR^2 < MR^2 This inequality tells us that the moment of inertia for the solid cylinder is less than the moment of inertia for the hollow cylinder when both objects have the same mass and radius.

Choose the correct answer.

Based on the comparison of the moment of inertia formulas for solid and hollow cylinders, we can conclude that the hollow cylinder will have a larger moment of inertia. Therefore, the correct answer is: c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.

Key concepts

Rotational Dynamics

Rotational dynamics is a fundamental concept in physics, specifically in the study of how forces affect the motion of rotating bodies. It concerns the relationship between angular motion, torques, and moment of inertia, which is the rotational equivalent of mass in linear motion.

Understanding the moment of inertia is crucial. It measures an object's resistance to changes in its rotation. For example, the more an object's mass is distributed away from the axis of rotation, the higher its moment of inertia. This is why objects with the same mass but different shapes or mass distributions can have different rotational behaviors.

Two key equations describe the moment of inertia for simple geometric shapes:\begin{itemize}

Solid Cylinder: \( I_{solid} = \frac{1}{2}MR^2 \)

Hollow Cylinder: \( I_{hollow} = MR^2 \)

In these equations, \( M \) is the mass, \( R \) is the radius, and \( I \) represents the moment of inertia. The outcome of these formulas leads to an ingrained concept within rotational dynamics—that the distribution of mass significantly impacts the inertia of a body.

Solid Cylinder

A solid cylinder is a common object studied within physics education, particularly when exploring the principles of rotational motion.

When considering its moment of inertia, the entire mass of the solid cylinder contributes evenly to its rotational inertia. The formula \( I_{solid} = \frac{1}{2}MR^2 \) demonstrates that the moment of inertia for a solid cylinder takes into account that the mass is uniformly distributed throughout the volume of the cylinder.

Consequently, when a force induces the cylinder to rotate about its axis, each part of its mass evenly affects the resistance to this rotational change. Visualizing this evenly spread mass can help students understand why the moment of inertia for a solid cylinder is less than that of a hollow cylinder of equal mass and radius.

Hollow Cylinder

In contrast to the solid cylinder, the hollow cylinder's structure inherently causes its mass to be concentrated at a greater distance from the axis of rotation. This is integral to understanding its higher moment of inertia, as given by the formula \( I_{hollow} = MR^2 \).

The mass distribution in a hollow cylinder results in greater resistance to changes in rotational speed. Hence, when comparing a hollow cylinder with a solid cylinder of the same mass and radius, the hollow cylinder will have a larger moment of inertia. The missing mass at the center of the hollow cylinder means any rotational force has to work against a greater resistance from the mass situated at the outer edge, which contributes to a larger value of \( I \) for the hollow cylinder.

Physics Education

The education of physics principles, especially through exercises and problems like these, helps students develop analytical and problem-solving skills. Effective physics education emphasizes not only the memorization of formulas but also the understanding of underlying concepts.

For instance, the cylinder exercise discussed in this article illustrates how different shapes respond differently to rotational forces, despite having the same mass and radius. Educators should encourage students to visualize physical scenarios and consider practical real-world examples. It reinforces learning when students can relate textbook problems to tangible experiences.

Moreover, interactive and engaging teaching methods should be employed, such as hands-on experiments with physical objects that allow students to observe rotational dynamics in real-time. Simplifying complex ideas and breaking them down into more approachable segments ensures a thorough comprehension, which is the ultimate goal of physics education.