Worksheet: Moment of Inertia

In this worksheet, we will practice calculating the moment of inertia of a system given the rotational motions of its components.

Q1:

A uniform rod of mass 1.0 kg and length 2.0 m is free to rotate about one of its ends, as shown in the accompanying diagram. If the rod is released from rest at an angle of 60 above the horizontal, what is the speed of the tip of the rod as it passes the horizontal position?

Q2:

A skater is spinning on the spot. The skater has a mass of 60.0 kg and can be approximated as a uniform cylinder of radius 0.110 m.

Calculate the moment of inertia of the skater.

The skater extends her arms, and in this configuration her body can be approximated as a cylinder of mass 52.5 kg and radius 0.110 m. From the surface of this cylinder her arms project in opposite directions and perpendicularly to the cylinder’s surface. Each arm can be modelled as 0.900 m long cylinder of mass 3.75 kg. Calculate the moment of inertia of the skater in this configuration.

Q3:

A solid sphere of radius 10 cm is allowed to rotate freely about an axis. The sphere is given a sharp blow so that its center of mass starts from the position shown in the following figure with speed 15 cm/s. What is the maximum angle that the diameter makes with the vertical?

  • A 1 2
  • B 8 . 9
  • C 7 . 9
  • D 1 0
  • E 1 1

Q4:

A rod and a sphere are combined to form a system. The rod’s length 𝐿 is 0.50 m and its mass is 2.0 kg. The sphere’s radius 𝑅 is 20.0 cm and its mass is 1.0 kg. The system can rotate either about the point 𝐴, at the opposite end of the rod to the sphere, or about the point 𝐵, where the rod and the sphere connect, as shown in the diagram.

Find the moment of inertia of the system about the point 𝐴.

Find the moment of inertia of the system about the point 𝐵.

Q5:

The major and minor radii of a thin elliptical disk of uniform density are 2.00 m and 1.00 m, respectively. The disk has a mass of 1.00 kg. Determine the moment of inertia about an axis that is perpendicular with the disk’s surface and intersects the center of the disk.

Q6:

A three-rod assembly forms an equilateral triangle. The mass of each rod is 1.0 kg and the length of each rod is 1.0 m. Determine the moment of inertia about an axis that is perpendicular to the surface of the triangle intersecting the center of the triangle.

Q7:

How do shear stresses and normal stresses differ?

  • ANormal stresses act at the surfaces of materials, whereas shear stresses act within materials.
  • BShear stresses are the absolute stresses, whereas normal stresses have been divided by the total stress.
  • CNormal stresses act within materials, whereas shear stresses act at the surfaces of materials.
  • DThey have different units.
  • EShear stresses act to slide parallel planes of material apart without changing separation, whereas normal stresses act to increase or decrease the separation between parallel planes of material.

Q8:

What is the dimension of mass moment of inertia?

  • A 𝑀 𝑇 𝐿
  • B 𝑀 𝐿
  • C 𝐿
  • D 𝑀 𝐿
  • E 𝑀 𝐿

Q9:

The box shown in the accompanying diagram extends from 𝑥=𝑒 to 𝑥=(𝑒+𝑎) along the 𝑥-axis, from 𝑦=𝑓 to 𝑦=(𝑓+𝑐) along the 𝑦-axis, and from 𝑧=𝑑 to 𝑧=(𝑑+𝑏) and along the 𝑧-axis. The box has a uniform density 𝜌 kilograms per cubic meter. What is the mass moment inertia of the box about the 𝑧-axis?

  • A ( 𝑑 + 𝑏 ) 𝑑 𝑎 𝑏 𝜌
  • B ( 𝑑 + 𝑏 ) 𝑑 𝑎 𝑐 𝜌 9
  • C ( 𝑑 + 𝑏 ) 𝑑 𝑎 𝑐 𝜌 2
  • D ( ( 𝑑 + 𝑏 ) 𝑑 ) 𝑎 𝑐 𝜌 3
  • E ( 𝑑 + 𝑏 ) 𝑑 𝑎 𝑐 𝜌

Q10:

The box shown in the accompanying diagram extends from 𝑥=𝑒 to 𝑥=(𝑒+𝑎) along the 𝑥-axis, from 𝑦=𝑓 to 𝑦=(𝑓+𝑐) along the 𝑦-axis, and from 𝑧=𝑑 to 𝑧=(𝑑+𝑏) and along the 𝑧-axis. The box has a uniform density 𝜌 kg/m3. What is the mass moment inertia of the box about the 𝑥-axis?

  • A ( 𝑓 + 𝑑 ) 𝑓 3 𝑎 𝑏 𝜌
  • B ( 𝑒 + 𝑐 ) 𝑒 3 𝑎 𝑏 𝜌
  • C 𝑎 𝑏 𝑐 𝜌
  • D 𝑏 𝑑 𝜌
  • E ( 𝑓 + 𝑐 ) 𝑓 3 𝑎 𝑏 𝜌

Q11:

The box shown in the accompanying diagram extends from 𝑥=𝑒 to 𝑥=(𝑒+𝑎) along the 𝑥-axis, from 𝑦=𝑓 to 𝑦=(𝑓+𝑐) along the 𝑦-axis, and from 𝑧=𝑑 to 𝑧=(𝑑+𝑏) and along the 𝑧-axis. The box has a uniform density 𝜌 kg/m3. What is the mass moment inertia of the box about the 𝑦-axis?

  • A ( 𝑒 + 𝑎 ) 𝑎 3 𝜌 𝑏 𝑐
  • B ( 𝑒 + 𝑎 ) 𝑎 3 𝜌 𝑎 𝑐
  • C ( 𝑒 + 𝑎 ) 𝑎 3 𝜌 𝑏 𝑐
  • D ( 𝑒 + 𝑎 ) 𝑎 𝜌 𝑏 𝑐
  • E ( 𝑐 + 𝑎 ) 𝑎 3 𝜌 𝑏 𝑐

Q12:

Which of the following is consistent with the parallel axis theorem for the mass moment of inertia?

  1. The mass moment of inertia of an object about an axis through the center of mass is smaller than that about any other axis in the same direction.
  2. The mass moments of inertia of an object about any parallel axes are identical.
  3. The mass moment of inertia of an object about an axis (𝑎) is equal to the moment about an axis (𝑏) through the center of mass plus the mass times the square of the distance from (𝑎) to (𝑏).
  4. The mass moment of inertia about parallel axes is proportional to the square of the separation between axes.
  • AIII and IV only
  • BII only
  • CI and III only
  • DIII only
  • EIV only

Q13:

Which of the following formulas correctly expresses the relationship between the mass moment of inertia 𝐼 and the radius of gyration 𝑘 for an object of mass 𝑚?

  • A 𝐼 = 𝑚 𝑘
  • B 𝐼 = 𝑚 𝑘
  • C 𝐼 = 𝑚 𝑘
  • D 𝐼 = 𝑚 𝑘
  • E 𝐼 = 𝑚 𝑘

Q14:

Which of the following formulas correctly expresses the mass moment of inertia 𝐼 for an object of mass density 𝜌, mass 𝑚, and volume 𝑉? The quantity 𝑙 is the distance of a volume element d𝑉 from the axis of rotation.

  • A 𝐼 = 𝑚 𝑉 𝜌 𝑉 d
  • B 𝐼 = 𝑚 𝜌 𝑉 d
  • C 𝐼 = 𝑙 𝜌 𝑚 𝑉 d
  • D 𝐼 = 𝑚 𝜌 𝑉 d
  • E 𝐼 = 𝑙 𝜌 𝑉 d

Q15:

A merry-go-round can be modeled as a solid disk of mass 500 kg and a radius of 2.0 m. A child of mass 25 kg stands at a point on the merry-go-round that is 1.0 m away from its axis of rotation, as shown in the diagram. Find the moment of inertia of the system consisting of the merry-go-round and the child. Give your answer to four significant figures.

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