A solid cylinder with a radius of 4.0 cm has the same mass as a solid sphere of radius R. If the cylinder and sphere have the same moment of inertia about their centers, what is the sphere's radius?

Three pairs of balls are connected by very light rods as shown in what we discussed earlier. Rank in order, from smallest to largest, the moments of inertia $I_1, I_2$, and $I_3$ about axes through the centers of the rods.

We can model a small merry-go-round as a uniform circular disk with mass $88 \mathrm{~kg}$ and diameter $1.8 \mathrm{~m}$. How many $22 \mathrm{~kg}$ children need to ride the merry-go-round, standing right at the outer edge, to double the moment of inertia of the system?

A bowling ball is far from uniform. Lightweight bowling balls are made of a relatively low-density core surrounded by a thin shell with much higher density. A $7.0 \mathrm{lb}(3.2 \mathrm{~kg})$ bowling ball has a diameter of $0.216 \mathrm{~m} ; 0.196 \mathrm{~m}$ of this is a $1.6 \mathrm{~kg}$ core, surrounded by a $1.6 \mathrm{~kg}$ shell. This composition gives the ball a higher moment of inertia than it would have if it were made of a uniform material. Given the importance of the angular motion of the ball as it moves down the alley, this has real consequences for the game.
a. Model a real bowling ball as a 0.196-m-diameter core with mass $1.6 \mathrm{~kg}$ plus a thin $1.6 \mathrm{~kg}$ shell with diameter $0.206 \mathrm{~m}$ (the average of the inner and outer diameters). What is the total moment of inertia?
b. How does your answer in part a compare to the moment of inertia of a uniform $3.2 \mathrm{~kg}$ ball with diameter $0.216 \mathrm{~m}$?

A regulation table tennis ball is a thin spherical shell 40 mm in diameter with a mass of 2.7 g. What is its moment of inertia about an axis that passes through its center?

A string is wrapped around a uniform solid cylinder of radius $R$, as shown in the figure. The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass $m$. Note that the positive $y$ direction is downward and counterclockwise torques are positive. Find $\alpha$, the magnitude of the pulley’s angular acceleration as the block descends.

A solid sphere, a solid cylinder, and a hollow cylinder have the same mass and radius and are rolling with the same speed. Which one of the following statements is true? a) The solid sphere has the highest kinetic energy, b) The solid cylinder has the highest kinetic energy, c) The hollow cylinder has the highest kinetic energy, d) All three objects have the same kinetic energy.

A 1500 kg car drives around a flat 200-m-diameter circular track at 25 m/s. What are the magnitude and direction of the net force on the car? What causes this force?

A small grinding wheel has a moment of inertia of $4.0 \times 10 ^ { - 5 } \mathrm { kg } \cdot \mathrm { m } ^ { 2 }$. What net torque must be applied to the wheel for its angular accelaration to be $150 \mathrm { rad } / \mathrm { s } ^ { 2 }$?

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x3, y = 0, x = 1; about x = 2

Find the moment of inertia $I_z$ of the solid of indicated density.
The solid of uniform density inside the paraboloid $z=16-x^2-y^2$ and outside the cylinder $x^2+y^2=9, \quad z \geq 0$

Let $R$ be the region in the first quadrant bounded by the graphs of $y=x^{1 / 5}$ and $y=x^2$. Region $R$ is the base of a solid whose cross sections perpendicular to the $x$-axs are squares. Find the volume of this solid.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. $y=x, y=\sqrt{x} ; \quad \text { about } x=2$

Different hotels on Las Vegas Boulevard ("the strip") in Las Vegas are randomly selected, and their ratings and prices were obtained from Travelocity. Using technology, with x representing the ratings and y representing price, we find that the regression equation has a slope of 130 and a y-intercept of -368. What is the equation of the regression line?