a hollow, spherical shell with mass $2.00$ kg rolls with- out slipping down a $38.0\degree$ slope.
(b) How would the answers to part (a) change if the mass were doubled to $4.00$ kg?

A $2.20$-$\mathrm{kg}$ hoop $1.20$ m in diameter is rolling to the right without slipping on a horizontal floor at a steady $2.60$ rad$/$s.
(b) What is the total kinetic energy of the hoop?

A uniform thin spherical shell has a radius of $2.00 \text{~m}$ and a mass of $300 \text{~kg}$ and its center is located at the origin of a coordinate system. Another uniform thin spherical shell with a radius of $1.00 \text{~m}$ and a mass of $150 \text{~kg}$ is inside the larger shell, with its center at $0.600 \text{~m}$ on the $x$ axis. What is the gravitational force of attraction between the two shells?

A solid uniform cylinder with mass $8.25 \mathrm{~kg}$ and diameter $15.0 \mathrm{~cm}$ is spinning at $220 \mathrm{~rpm}$ on a thin, frictionless axle that passes along the cylinder axis. You design a simple friction brake to stop the cylinder by pressing the brake against the outer rim with a normal force. The coefficient of kinetic friction between the brake and rim is $0.333$. What must the applied normal force be to bring the cylinder to rest after it has turned through $5.25$ revolutions?

A string is wrapped several times around the rim of a small hoop with radius 8.00 cm and mass 0.180 kg. The free end of the string is held in place and the hoop is released from rest. After the hoop has descended 75.0 cm, calculate the angular speed of the rotating hoop.

A hollow spherical shell with a mass of $65\text{ kg}$ rolls without slipping down a slope that makes an angle of $31^{\circ}$ with the horizontal plane. Find the magnitude of the frictional force acting on the spherical shell. Take the free-fall acceleration to be $g=9.80\text{ m/s}^2$.

A disk and a hoop of the same mass and radius are released at the same time at the top of an inclined plane. Calculate the speeds of the disk at the bottom of the inclined plane in Conceptual Checkpoint 10-4 if the height of the incline is $0.82 \mathrm{~m}$.

A $750-$gram grinding wheel $25.0 \mathrm{~cm}$ in diameter is in the shape of a uniform solid disk.When it is in use, it turns at a constant $220 \mathrm{rpm}$ about an axle perpendicular to its face through its center. When the power switch is turned off, you observe that the wheel stops in $45.0 \mathrm{~s}$ with constant angular acceleration due to friction at the axle. What torque does friction exert while this wheel is slowing down?

An Unfair Race This applet shows the results of releasing a frictionless block and a rolling disk with equal masses from the top of identical inclined planes. Which of the following is the best explanation of the results shown in the applet?

A) The disk loses energy to friction as it rolls, but the box is frictionless and so it speeds up more quickly and gets to the bottom first.

B) The potential energy of the disk is converted into translational and rotational kinetic energy, so the translational speed grows more slowly than that of the box, which has no rotational energy.

C) The net forces on the two objects are equal, but the force on the disk gets partially used up in creating the torque necessary to make it roll.

D) The net forces on the two objects are equal, but the force on the disk is not directed parallel to the ramp, and so does not create as great an acceleration down the ramp.

A grindstone in the shape of a solid disk with diameter 0.520 m and a mass of 50.0 kg is rotating at 850 rev/min. You press an ax against the rim with a normal force of 160 N, and the grindstone comes to rest in 7.50 s. Find the coefficient of friction between the ax and the grindstone. You can ignore friction in the bearings.

A yo-yo is made from two uniform disks, each with mass m and radius R, connected by a light axle of radius b. A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds. Find the linear acceleration and angular acceleration of the yo-yo and the tension in the string.

What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface?
(b) a uniform sphere;

A $2.00$-kg textbook rests on a frictionless, hori- zontal surface. A cord attached to the book passes over a pulley whose diameter is $0.150$ m, to a hanging book with mass $3.00 \mathrm{~kg}$. The system is released from rest, and the books are observed to move $1.20$ m in $0.800$ s.
(b) What is the moment of inertia of the pulley about its rotation axis?

What fraction of a solid disk's kinetic energy is rotational if it's rolling without slipping?