AP Physics C: Mechanics Practice Quiz: Rotational Inertia

Correct Answer: B

Rotational inertia (I) is the rotational analog of mass. Just as mass measures an object's resistance to a change in linear velocity (linear acceleration), rotational inertia measures a rigid system's resistance to a change in rotational velocity (angular acceleration). This is the fundamental description of rotational inertia.

Correct Answer: B

The parallel axis theorem states $I'=I_{cm}+Md^{2}$. Since M (mass) and $d^2$ (distance squared) are always non-negative, the term $Md^2$ is always greater than or equal to zero. The minimum value of $I'$ occurs when $d=0$, which means the axis passes through the center of mass. Therefore, $I_{cm}$ is the minimum rotational inertia for any set of parallel axes.

Correct Answer: C

The equation $I=\int r^{2}dm$ calculates rotational inertia by summing the contributions of all infinitesimal mass elements ($dm$) in the solid. The term 'r' specifically represents the perpendicular distance of each mass element $dm$ from the chosen axis of rotation.

Correct Answer: C

This is a direct application of the parallel axis theorem, $I'=I_{cm}+Md^{2}$. Here, the new axis is parallel to the axis through the center of mass, and the distance 'd' between these two axes is the radius R. Therefore, the new rotational inertia is $I' = I_{cm} + MR^2$.

Correct Answer: C

The name 'parallel axis theorem' itself states the primary condition for its use. The theorem relates the rotational inertia about an axis through the center of mass ($I_{cm}$) to the rotational inertia about another axis that is parallel to it.

Correct Answer: D

Using the parallel axis theorem, $I'=I_{cm}+Md^{2}$. We are given $I_{cm} = 4$ kg·m², M = 3 kg, and d = 2 m. Plugging these values in: $I' = 4 + (3)(2^2) = 4 + (3)(4) = 4 + 12 = 16$ kg·m².

Correct Answer: B

Sphere A's rotational inertia is $I_{cm}$. Sphere B is rotated about an axis that does not pass through its center of mass. According to the parallel axis theorem, $I_B = I_{cm} + Md^2$, where d is the radius of the sphere. Since $Md^2$ is a positive value, the rotational inertia of sphere B will be greater than that of sphere A.

Correct Answer: C

The integral $I=\int r^{2}dm$ sums up the product of each mass element ($dm$) and the square of its distance ($r^2$) from the axis. This shows that mass elements farther from the axis of rotation (larger 'r') contribute significantly more to the total rotational inertia than mass elements closer to the axis. Therefore, the distribution of mass is the key factor.

Correct Answer: A

The parallel axis theorem, $I'=I_{cm}+Md^{2}$, governs this situation. The rotational inertia $I'$ depends on the square of the distance 'd' from the center of mass. As 'd' increases, the term $Md^2$ increases, and therefore the total rotational inertia $I'$ increases.

Correct Answer: C

By definition within the parallel axis theorem, $I_{cm}$ is the rotational inertia of the rigid system calculated with respect to an axis that is parallel to the new axis and passes through the system's center of mass.

Correct Answer: A

We are given that $I' = 2I_{cm}$. Substituting this into the parallel axis theorem gives: $2I_{cm} = I_{cm} + Md^2$. Subtracting $I_{cm}$ from both sides gives $I_{cm} = Md^2$. To solve for d, we rearrange the equation: $d^2 = I_{cm}/M$, and then take the square root of both sides: $d = \sqrt{I_{cm}/M}$.

Correct Answer: B

Rotational inertia is not an intrinsic property like mass; its value depends on the choice of the axis of rotation. Both the integral form ($I=\int r^{2}dm$) and the parallel axis theorem ($I'=I_{cm}+Md^{2}$) show that the calculation is dependent on the axis chosen.

Correct Answer: C

The purpose of the parallel axis theorem ($I'=I_{cm}+Md^{2}$) is to provide a straightforward way to find the rotational inertia ($I'$) about an axis that is parallel to an axis passing through the center of mass, for which the rotational inertia ($I_{cm}$) is already known or can be easily found.