Lesson: Moment of Inertia of Solids

Mathematics

In this lesson, we will learn how to find the moment of inertia and radius of gyration of a solid and how to use the parallel axis theorem to find the moment of inertia of a composite solid at different axes.

Worksheet: 24 Questions

Q1:

Matthew has a model train that uses a circular cone as a flywheel. The cone has radius π‘Ÿ and height β„Ž.

Find its moment of inertia about its axis of symmetry.

What is its radius of gyration about this axis?

Matthew would like to replace the cone with a circular cylinder of the same mass. He does not want this change to affect the performance of his train, so he needs a cylinder with the same moment of inertia as the cone. Find an expression for the radius 𝑅 of the cylinder that he needs to buy.

Q2:

Suppose 𝑛 particles are attached to a light circular ring with radius π‘Ÿ. If the total mass of the particles is 𝑀, find the moment of inertia of the resulting system about an axis passing through the center of the ring which is perpendicular to the plane of the ring.

Q3:

Jennifer is building a model race car. She wants the car to be as responsive as possible, so she is looking for a drive wheel with the smallest moment of inertia. The figure shows the configuration of the drive wheel, π·π‘Š, in relation to the engine’s output shaft, 𝑂, and the two drive shafts, π·π‘†οŠ§ and π·π‘†οŠ¨.

She has two options for the drive wheel. The specifications for each drive wheel are detailed in the table.

What is the moment of inertia of drive wheel A about a perpendicular axis passing through the center of the disk? Give your answer in gram square millimeters.

What is the moment of inertia of drive wheel B about a perpendicular axis passing through the center of the disk? Give your answer in gram square millimeters.

Which drive wheel should she choose to maximize the responsiveness of the model car?

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