Rotational Inertia (College Board AP® Physics 1: Algebra-Based): Exam Questions

Four objects, each of mass $M$, are placed at the corners of a square with side length $L$ and arranged such that the axis of rotation lies in the plane of the objects, as shown in the figure.

Which of the following expressions correctly represents the rotational inertia of this configuration?

Two pulleys of equal mass and radius are mounted on a frictionless horizontal axis through their centers, as shown in the figure. One pulley is a uniform solid disk, and the other is a hoop.

How does the rotational inertia of each pulley compare, and why?

A thin uniform rod of length $L$ lies along the x-axis, as shown in the figure.

Which of the following correctly describes the axes about which the rod has the smallest and greatest rotational inertia?

Four objects, each of mass $m$, are arranged at the corners of a rectangle that rotates about an axis perpendicular to the midpoint of one of the sides of the rectangle and out of the page, as shown in the figure.

Which of the following expressions correctly represents the total rotational inertia of the system?

Two objects are connected by a massless rod to form a dumbbell.

Which of the following diagrams shows the dumbbell with the greatest rotational inertia about an axis perpendicular to the rod and passing through its center?

A uniform disk of mass $M$ and radius $R$ is mounted to a frictionless axle, as shown in the figure. A thin uniform rod of mass $\frac{1}{4}M$ and length $2R$ is rigidly attached to the disk so that they rotate together. A block of mass $\frac{2}{3}M$ is attached to the end of the rod.

The rotational inertia of the disk about its center is $\frac{1}{2}M{R}^2$. The rotational inertia of a rod about one end is $\frac{1}{3}m{L}^2$, where $m$ is the mass of the rod and $L$ is the length.

Which of the following expressions correctly represents the total rotational inertia of the disk, rod, and block?

Two solid spheres of unequal mass and radius are connected by a massless rod of length $L$. The rotation of the two-sphere system is considered about four parallel axes, A, B, C, and D, as shown in the figure. The sphere on axis A has mass $3M$, and the sphere on axis D has mass $M$. The rotational inertias of the two-sphere system about the axes are $I_A$, $I_B$, $I_C$, and $I_D$, respectively.

Which of the following correctly ranks $I_A$, $I_B$, $I_C$, and $I_D$ from greatest to least?

Two identical solid hemispheres, each of mass $m$ and radius $r$ are joined together and allowed to rotate about axis AB, as shown in the figure. The rotational inertia of a solid sphere of mass $M$ and radius $R$ about its center is $\frac{2}{5}M{R}^2$.

Which of the following expressions correctly represents the rotational inertia of this system about axis AB?

A thin circular disc of mass $M$ and radius $R$ has rotational inertia $I=\frac{1}{2}M{R}^2$ about an axis passing through its center and perpendicular to its plane. The same disc has rotational inertia $\frac{I}{2}$ about an axis passing through its center and parallel to its plane.

Which of the following expressions gives the rotational inertia of the disc about an axis tangential to its edge and parallel to its plane?

Two uniform solid spheres of mass $M$ and radius $R$ are connected by a cylindrical rod of mass $\frac{1}{2}M$, length $6R$, and radius $\frac{1}{5}R$. The system is made to rotate about an axis through the system's center of mass, as shown in the figure. The rotational inertia of a solid sphere about its center is $\frac{2}{5}M{R}^2$. The rotational inertia of a cylinder of length $L$ about a central axis perpendicular to its length is $\frac{1}{4}M{R}^2+\hspace{0.17em}\frac{1}{12}M{L}^2$.

Which of the following is most nearly the proportion the cylindrical rod contributes to the total rotational inertia of the system about its center of mass?