Unit 8: Moment of Inertia

Practice Problems

Note: problem difficulty is ranked using a star system.

  • (*) One-star problems are fundamental to the unit, and can be done relatively quickly. Use these problems to introduce yourself to the material.

  • (**) Two-star problems are more difficult, and require an understanding of one or two key concepts. Use these problems to test your understanding of the material.

  • (***) Three-star problems are the most difficult, and require some creative thinking in addition to a deep familiarity with multiple key concepts. Use these problems to challenge yourself; if you can complete one of these, you’re on your way to mastering the material.

*Q8.1) Two cylinders are released from rest atop a ramp at the same instant and let roll to the bottom. One cylinder is hollow and made of steel while the other is solid and made of wood. If both cylinders have the same mass and the same radius, which of the following describes what would happen:

A) The wooden cylinder will reach the bottom first

B) The steel cylinder will reach the bottom first

C) They will both reach the bottom at the same time

*Q8.2) A child stands up and walks towards the center of a merry-go-round as it rotates. What happens to the moment of inertia of the system?

*Q8.3) What would happen to the moment of inertia of a propeller if you added more blades to it?

*Q8.4) What would happen to the moment of inertia of the solar system if another planet were put into orbit around the Sun?

**Q8.5) Consider an object made of four identical 2kg point masses positioned at the corners of a square of sidelength 1m as shown below. Calculate the object’s moment of inertia as it rotates around the axes shown in red.

a)

b)

c)

**Q8.6) Consider an object made of the same four identical point masses as before, but this time positioned equidistant from one another along a line of length 1m as shown below. Calculate the object’s moment of inertia as it rotates around the axes shown in red.

a)

b)

c)

***Q8.7) Calculate the moment of inertia of a methane molecule rotating about an axis that passes through the central carbon atom and one of the hydrogen atoms. You may treat the hydrogen atoms as identical point masses of mass 1.00784 amu (atomic mass units), and you can use 108.7 pm (picometers) as the C-H bondlength. Provide your answer in units of [ amu pm² ]. Hint: the four faces of a tetrahedron are equilateral triangles.

Disclaimer: this is a very hard problem. If you manage to solve it, I would love to see your solution.