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A composite body consists of a solid cylinder and a solid sphere, both of mass M

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Question: A composite body consists of a solid cylinder and a solid sphere, both of mass M and radius R. What is the total moment of inertia about the same axis?

Options:

  1. (7/10) MR^2
  2. (9/10) MR^2
  3. (11/10) MR^2
  4. (13/10) MR^2

Correct Answer: (7/10) MR^2

Solution: 

The total moment of inertia is I_cylinder + I_sphere = (1/2 MR^2) + (2/5 MR^2) = (7/10) MR^2.

A composite body consists of a solid cylinder and a solid sphere, both of mass M

Practice Questions

Q1
A composite body consists of a solid cylinder and a solid sphere, both of mass M and radius R. What is the total moment of inertia about the same axis?
  1. (7/10) MR^2
  2. (9/10) MR^2
  3. (11/10) MR^2
  4. (13/10) MR^2

Questions & Step-by-Step Solutions

A composite body consists of a solid cylinder and a solid sphere, both of mass M and radius R. What is the total moment of inertia about the same axis?
  • Step 1: Identify the shapes involved in the composite body. We have a solid cylinder and a solid sphere.
  • Step 2: Note the mass (M) and radius (R) of both the cylinder and the sphere.
  • Step 3: Find the moment of inertia formula for the solid cylinder. It is I_cylinder = (1/2) * M * R^2.
  • Step 4: Find the moment of inertia formula for the solid sphere. It is I_sphere = (2/5) * M * R^2.
  • Step 5: Add the moments of inertia of the cylinder and the sphere together: I_total = I_cylinder + I_sphere.
  • Step 6: Substitute the formulas into the equation: I_total = (1/2) * M * R^2 + (2/5) * M * R^2.
  • Step 7: To add the fractions, find a common denominator. The common denominator for 2 and 5 is 10.
  • Step 8: Rewrite the fractions: (1/2) = (5/10) and (2/5) = (4/10).
  • Step 9: Now add the fractions: I_total = (5/10) * M * R^2 + (4/10) * M * R^2 = (9/10) * M * R^2.
  • Step 10: Therefore, the total moment of inertia about the same axis is (9/10) * M * R^2.
  • Moment of Inertia – The moment of inertia is a measure of an object's resistance to rotational motion about a given axis, depending on the mass distribution relative to that axis.
  • Composite Bodies – When calculating the moment of inertia for composite bodies, the total moment of inertia is the sum of the individual moments of inertia of each component.
  • Parallel Axis Theorem – This theorem may be relevant if the axis of rotation is not through the center of mass, but in this case, it is not needed as both shapes are rotating about the same axis.
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