Questions: Scale drawings of three objects, A, B, and C are shown below. Each object has the same mass and uniform thickness, with the mass distributed uniformly within the shaded region. Which one has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P?

Solution Steps

We need to determine which of the three objects (A, B, or C) has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P. All objects have the same mass and uniform thickness, with mass distributed uniformly within the shaded region.

The moment of inertia (I) depends on the mass distribution relative to the axis of rotation. For a given mass, the farther the mass is from the axis, the greater the moment of inertia.

• Object A: A small disk with point P at its center. The moment of inertia for a disk about its center is \( I_A = \frac{1}{2}MR^2 \).
• Object B: A ring with point P at its center. The moment of inertia for a ring about its center is \( I_B = MR^2 \).
• Object C: A large disk with point P at its center. The moment of inertia for a disk about its center is \( I_C = \frac{1}{2}MR^2 \).

• For Object A: \( I_A = \frac{1}{2}MR_A^2 \)
• For Object B: \( I_B = MR_B^2 \)
• For Object C: \( I_C = \frac{1}{2}MR_C^2 \)

Given that \( R_B > R_C > R_A \), and considering the coefficients:

• \( I_B \) will be the largest because it has the largest radius and the coefficient is 1.
• \( I_C \) will be larger than \( I_A \) because \( R_C > R_A \) and both have the same coefficient.

Final Answer