Hollow Cylinder

Definition

A hollow cylinder is a three-dimensional geometric shape that is cylindrical in nature, with a hollow or empty space running through the center. This shape is commonly used in various engineering and scientific applications, particularly in the context of rotational dynamics and Newton's Second Law for Rotation.

5 Must Know Facts For Your Next Test

1. The moment of inertia of a hollow cylinder is determined by its mass and the distribution of that mass around the central axis of rotation.
2. The moment of inertia of a hollow cylinder is typically greater than that of a solid cylinder with the same mass and dimensions, due to the hollow nature of the shape.
3. Torque acting on a hollow cylinder will cause it to experience angular acceleration, which is related to the moment of inertia and the applied torque through Newton's Second Law for Rotation.
4. The hollow cylinder shape is often used in engineering applications, such as in the design of mechanical shafts, bearings, and other rotating components, due to its favorable strength-to-weight ratio.
5. The analysis of the rotational dynamics of a hollow cylinder is an important concept in the study of rigid body motion and the application of Newton's Second Law for Rotation.

Review Questions

• Explain how the moment of inertia of a hollow cylinder differs from that of a solid cylinder with the same mass and dimensions.
  • The moment of inertia of a hollow cylinder is typically greater than that of a solid cylinder with the same mass and dimensions. This is because the mass of the hollow cylinder is distributed further away from the central axis of rotation, which increases the object's resistance to changes in its rotational motion. The hollow nature of the cylinder allows for the mass to be concentrated closer to the outer edges, resulting in a higher moment of inertia compared to a solid cylinder of the same mass and size.
• Describe the relationship between the torque applied to a hollow cylinder and the resulting angular acceleration, as per Newton's Second Law for Rotation.
  • According to Newton's Second Law for Rotation, the angular acceleration experienced by a hollow cylinder is directly proportional to the torque applied to it and inversely proportional to its moment of inertia. Specifically, the equation $\tau = I\alpha$ relates the torque $\tau$, the moment of inertia $I$, and the angular acceleration $\alpha$. For a given torque, the angular acceleration of a hollow cylinder will be less than that of a solid cylinder with the same mass and dimensions, due to the higher moment of inertia of the hollow shape.
• Explain why the hollow cylinder shape is often preferred in engineering applications over a solid cylinder design.
  • The hollow cylinder shape is often preferred in engineering applications over a solid cylinder design due to its favorable strength-to-weight ratio. By concentrating the mass of the cylinder towards the outer edges, the hollow shape allows for a reduction in the overall weight of the component while maintaining a similar level of structural integrity and resistance to rotational forces. This makes the hollow cylinder design well-suited for applications where weight and rotational dynamics are important considerations, such as in the design of mechanical shafts, bearings, and other rotating components. The higher moment of inertia of the hollow cylinder also allows for better control and stability in rotational systems.

Related terms

Moment of Inertia:

The measure of an object's resistance to changes in its rotational motion, which depends on the object's mass distribution.

Torque:

The rotational equivalent of force, which causes an object to rotate about an axis or pivot point.

Angular Acceleration:

The rate of change of an object's angular velocity, which is the measure of how quickly an object is rotating.