Worksheet: Moment of Inertia

In this worksheet, we will practice calculating the angular mass, called the moment of inertia, of rotating objects of various regular shapes.

Q1:

Which of the following correctly shows the SI unit for the moment of inertia?

  • Akg2⋅m2
  • Bkg/m2
  • Ckg2⋅m
  • Dkg⋅m2
  • E()kgm

Q2:

Which of the following formulas correctly relates the moment of inertia of an object, 𝐼, to the mass of the object, 𝑚, and the distance of the object from the axis around which it rotates, 𝑑?

  • A𝐼=𝑚𝑑
  • B𝐼=𝑚𝑑
  • C𝐼=𝑚𝑑
  • D𝐼=𝑚𝑑
  • E𝐼=𝑚𝑑

Q3:

A thin hoop has a radius of 1.25 m and a mass of 750 g. The hoop can rotate around an axis that intersects the hoop at the points A and B, as shown by the blue arrow in the diagram. The hoop can also rotate around an axis that intersects the midpoint of the line from A to B and does not intersect the hoop, as shown by the orange arrow. What is the difference in the moment of inertia of the hoop around these two axes?

Q4:

Earth has a mass of 5.97×10 kg and a radius of 6,370 km.

If Earth is assumed to be a solid sphere of uniform density, what is its moment of inertia?

  • A3.23×10 kg⋅m2
  • B3.04×10 kg⋅m2
  • C2.42×10 kg⋅m2
  • D9.69×10 kg⋅m2
  • E1.94×10 kg⋅m2

If Earth was a disk, what would its moment of inertia be?

  • A2.42×10 kg⋅m2
  • B5.41×10 kg⋅m2
  • C4.84×10 kg⋅m2
  • D1.21×10 kg⋅m2
  • E1.90×10 kg⋅m2

Q5:

A point object with a mass of 15 µg can move around a circular track that has a radius of 25 cm. What is the moment of inertia of the object around the centre of the track?

  • A3.8×10 kg⋅m2
  • B1.7×10 kg⋅m2
  • C9.4×10 kg⋅m2
  • D9.4×10 kg⋅m2
  • E3.8×10 kg⋅m2

Q6:

Consider two cuboids. Each can be rotated around its center of mass, as shown in the diagram. The cuboids have the same mass and the same moment of inertia as each other. Which of the following is a true statement comparing the lengths of the cuboids, 𝐿 and 𝐿, to their widths, 𝑊 and 𝑊?

  • A𝐿𝐿=𝑊𝑊
  • B𝐿𝐿=𝑊𝑊
  • C(𝐿𝐿)=(𝑊𝑊)
  • D(𝐿𝐿)=(𝑊𝑊)
  • E𝐿𝐿=𝑊𝑊

Q7:

The objects shown in the diagram both rotate around the same axis and have the same moment of inertia. The radius of both objects, 𝑟, is 25 cm. What is the ratio of the mass of the thin hoop to the mass of the disk?

Q8:

The objects shown in the diagram rotate around the same axis and have the same moment of inertia and the same radius. What is the ratio of the mass of the hollow sphere to the mass of the solid sphere? Answer to two significant figures.

Q9:

The objects shown in the diagram rotate around the same axis and have the same moment of inertia. The radius of the disk, 𝑟, equals the outer radius of the ring, 𝑟. The inner radius of the ring, 𝑟, is 90% of the ring’s outer radius. What is the ratio of the mass of the disk to the mass of the ring? Answer to two significant figures.

Q10:

The objects shown in the diagram both rotate around the same axis and have the same moment of inertia. The radius 𝑟 of both objects is the same. What is the ratio of the mass of the disk to the mass of the hoop?

Q11:

The objects shown in the diagram both rotate around the same axis and have the same moment of inertia. The radius 𝑟 of both objects is 50 cm. The thin hoop has negligible width and the thick hoop has a width 𝑤=15cm. What is the ratio of the mass of the thin hoop to the mass of the thick hoop?

Q12:

A rod that has a mass of 275 g has a length 𝐿=25.0cm and a cross-sectional diameter 𝑑=3.00mm. The rod can be rotated around one of its ends (case I), around the midpoint of its length (case II), and around the center of its cross-sectional face (case III), as shown in the diagram. In cases I and II, consider the rod to have negligible radius. In case III, consider the rod as a solid cylinder.

What is the ratio of the moment of inertia of the rod in case I to that in case II?

What is the ratio of the moment of inertia of the rod in case II to that in case III?

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