Chapter 10

Two solid steel balls, one small and one large, are on an inclined plane. The large ball has a diameter twice as large as that of the small ball. Starting from rest, the two balls roll without slipping down the incline until their centers of mass are \(1 \mathrm{~m}\) below their starting positions. What is the speed of the large ball \(\left(v_{\mathrm{L}}\right)\) relative to that of the small ball \(\left(v_{\mathrm{S}}\right)\) after rolling \(1 \mathrm{~m} ?\) a) \(v_{\mathrm{L}}=4 v_{\mathrm{S}}\) d) \(v_{\mathrm{L}}=0.5 v_{\mathrm{S}}\) b) \(v_{\mathrm{L}}=2 v_{\mathrm{S}}\) e) \(v_{\mathrm{L}}=0.25 v_{\mathrm{S}}\) c) \(v_{\mathrm{L}}=v_{\mathrm{S}}\)

Consider a cylinder and a hollow cylinder, rotating about an axis going through their centers of mass. If both objects have the same mass and the same radius, which object will have the larger moment of inertia? a) The moment of inertia will be the same for both objects. b) The solid cylinder will have the larger moment of inertia because its mass is uniformly distributed. c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.

A basketball of mass \(610 \mathrm{~g}\) and circumference \(76 \mathrm{~cm}\) is rolling without slipping across a gymnasium floor. Treating the ball as a hollow sphere, what fraction of its total kinetic energy is associated with its rotational motion? a) 0.14 b) 0.19 c) 0.29 d) 0.40 e) 0.67

A solid sphere rolls without slipping down an incline, starting from rest. At the same time, a box starts from rest at the same altitude and slides down the same incline, with negligible friction. Which object arrives at the bottom first? a) The solid sphere arrives first. b) The box arrives first. c) Both arrive at the same time. d) It is impossible to determine.

A cylinder is rolling without slipping down a plane, which is inclined by an angle \(\theta\) relative to the horizontal. What is the work done by the friction force while the cylinder travels a distance \(s\) along the plane \(\left(\mu_{s}\right.\) is the coefficient of static friction between the plane and the cylinder)? a) \(+\mu_{s} m g s \sin \theta\) b) \(-\mu_{s} m g s \sin \theta\) c) \(+m g s \sin \theta\) d) \(-m g s \sin \theta\) e) No work done.

A ball attached to the end of a string is swung in a vertical circle. The angular momentum of the ball at the top of the circular path is a) greater than the angular momentum at the bottom of the circular path. b) less than the angular momentum at the bottom of the circular path. c) the same as the angular momentum at the bottom of the circular path.

You are unwinding a large spool of cable. As you pull on the cable with a constant tension, what happens to the angular acceleration and angular velocity of the spool, assuming that the radius at which you are extracting the cable remains constant and there is no friction force? a) Both increase as the spool unwinds. b) Both decrease as the spool unwinds. c) Angular acceleration increases, and angular velocity decreases. d) Angular acceleration decreases, and angular velocity increases. e) It is impossible to tell.

A disk of clay is rotating with angular velocity \(\omega .\) A blob of clay is stuck to the outer rim of the disk, and it has a mass \(\frac{1}{10}\) of that of the disk. If the blob detaches and flies off tangent to the outer rim of the disk, what is the angular velocity of the disk after the blob separates? a) \(\frac{5}{6} \omega\) b) \(\frac{10}{11} \omega\) c) \(\omega\) d) \(\frac{11}{10} \omega\) e) \(\frac{6}{5} \omega\)

An ice skater spins with her arms extended and then pulls her arms in and spins faster. Which statement is correct? a) Her kinetic energy of rotation does not change because, by conservation of angular momentum, the fraction by which her angular velocity increases is the same as the fraction by which her rotational inertia decreases. b) Her kinetic energy of rotation increases because of the work she does to pull her arms in. c) Her kinetic energy of rotation decreases because of the decrease in her rotational inertia; she loses energy because she gradually gets tired.

An ice skater rotating on frictionless ice brings her hands into her body so that she rotates faster. Which, if any, of the conservation laws hold? a) conservation of mechanical energy and conservation of angular momentum b) conservation of mechanical energy only c) conservation of angular momentum only d) neither conservation of mechanical energy nor conservation of angular momentum

If the iron core of a collapsing star initially spins with a rotational frequency of \(f_{0}=3.2 \mathrm{~s}^{-1},\) and if the core's radius decreases during the collapse by a factor of \(22.7,\) what is the rotational frequency of the iron core at the end of the collapse? a) \(10.4 \mathrm{kHz}\) b) \(1.66 \mathrm{kHz}\) c) \(65.3 \mathrm{kHz}\) d) \(0.46 \mathrm{kHz}\) e) \(5.2 \mathrm{kHz}\)

A uniform solid sphere of radius \(R, \operatorname{mass} M,\) and moment of inertia \(I=\frac{2}{5} M R^{2}\) is rolling without slipping along a horizontal surface. Its total kinetic energy is the sum of the energies associated with translation of the center of mass and rotation about the center of mass. Find the fraction of the sphere's total kinetic energy that is attributable to rotation.

In another race, a solid sphere and a thin ring roll without slipping from rest down a ramp that makes angle \(\theta\) with the horizontal. Find the ratio of their accelerations, \(a_{\text {ring }} / a_{\text {sphere }}\)

A uniform solid sphere of mass \(m\) and radius \(r\) is placed on a ramp inclined at an angle \(\theta\) to the horizontal. The coefficient of static friction between sphere and ramp is \(\mu_{s} .\) Find the maximum value of \(\theta\) for which the sphere will roll without slipping, starting from rest, in terms of the other quantities.

A round body of mass \(M\), radius \(R,\) and moment of inertia \(I\) about its center of mass is struck a sharp horizontal blow along a line at height \(h\) above its center (with \(0 \leq h \leq R,\) of course). The body rolls away without slipping immediately after being struck. Calculate the ratio \(I /\left(M R^{2}\right)\) for this body.

A projectile of mass \(m\) is launched from the origin at speed \(v_{0}\) at angle \(\theta_{0}\) above the horizontal. Air resistance is negligible. a) Calculate the angular momentum of the projectile about the origin. b) Calculate the rate of change of this angular momentum. c) Calculate the torque acting on the projectile, about the origin, during its flight.

A solid sphere of radius \(R\) and mass \(M\) is placed at a height \(h_{0}\) on an inclined plane of slope \(\theta\). When released, it rolls without slipping to the bottom of the incline. Next, a cylinder of same mass and radius is released on the same incline. From what height \(h\) should it be released in order to have the same speed as the sphere at the bottom?

It is harder to move a door if you lean against it (along the plane of the door) toward the hinge than if you lean against the door perpendicular to its plane. Why is this so?

A figure skater draws her arms in during a final spin. Since angular momentum is conserved, her angular velocity will increase. Is her rotational kinetic energy conserved during this process? If not, where does the extra energy come from or go to?

Does a particle traveling in a straight line have an angular momentum? Explain.

A cylinder with mass \(M\) and radius \(R\) is rolling without slipping through a distance \(s\) along an inclined plane that makes an angle \(\theta\) with respect to the horizontal. Calculate the work done by (a) gravity, (b) the normal force, and (c) the frictional force.

Using the conservation of mechanical energy, calculate the final speed and the acceleration of a cylindrical object of mass \(M\) and radius \(R\) after it rolls a distance \(s\) without slipping along an inclined plane of angle \(\theta\) with respect to the horizontal

A couple is a set of two forces of equal magnitude and opposite directions, whose lines of action are parallel but not identical. Prove that the net torque of a couple of forces is independent of the pivot point about which the torque is calculated and of the points along their lines of action where the two forces are applied.

Why does a figure skater pull in her arms while increasing her angular velocity in a tight spin?

To turn a motorcycle to the right, you do not turn the handlebars to the right, but instead slightly to the left. Explain, as precisely as you can, how this counter-steering turns the motorcycle in the desired direction. (Hint: The wheels of a motorcycle in motion have a great deal of angular momentum.)

A light rope passes over a light, frictionless pulley. One end is fastened to a bunch of bananas of mass \(M,\) and \(a\) monkey of the same mass clings to the other end. The monkey climbs the rope in an attempt to reach the bananas. The radius of the pulley is \(R\). a) Treating the monkey, bananas, rope, and pulley as a system, evaluate the net torque about the pulley axis. b) Using the result of part (a) determine the total angular momentum about the pulley axis as a function of time

A uniform solid cylinder of mass \(M=5.00 \mathrm{~kg}\) is rolling without slipping along a horizontal surface. The velocity of its center of mass is \(30.0 \mathrm{~m} / \mathrm{s}\). Calculate its energy.

Determine the moment of inertia for three children weighing \(60.0 \mathrm{lb}, 45.0 \mathrm{lb}\) and \(80.0 \mathrm{lb}\) sitting at different points on the edge of a rotating merry-go-round, which has a radius of \(12.0 \mathrm{ft}\).

A \(24-\mathrm{cm}\) -long pen is tossed up in the air, reaching a maximum height of \(1.2 \mathrm{~m}\) above its release point. On the way up, the pen makes 1.8 revolutions. Treating the pen as a thin uniform rod, calculate the ratio between the rotational kinetic energy and the translational kinetic energy at the instant the pen is released. Assume that the rotational speed does not change during the toss.

A solid ball and a hollow ball, each with a mass of \(1.00 \mathrm{~kg}\) and radius of \(0.100 \mathrm{~m}\), start from rest and roll down a ramp of length \(3.00 \mathrm{~m}\) at an incline of \(35.0^{\circ} .\) An ice cube of the same mass slides without friction down the same ramp. a) Which ball will reach the bottom first? Explain! b) Does the ice cube travel faster or slower than the solid ball at the base of the incline? Explain your reasoning. c) What is the speed of the solid ball at the bottom of the incline?

The Crab pulsar \(\left(m \approx 2 \cdot 10^{30} \mathrm{~kg}, R=5 \mathrm{~km}\right)\) is a neutron star located in the Crab Nebula. The rotation rate of the Crab pulsar is currently about 30 rotations per second, or \(60 \pi \mathrm{rad} / \mathrm{s} .\) The rotation rate of the pulsar, however, is decreasing; each year, the rotation period increases by \(10^{-5}\) s. Justify the following statement: The loss in rotational energy of the pulsar is equivalent to 100,000 times the power output of the Sun. (The total power radiated by the Sun is about \(\left.4 \cdot 10^{26} \mathrm{~W} .\right)\)

A uniform solid sphere of mass \(M\) and radius \(R\) is rolling without sliding along a level plane with a speed \(v=3.00 \mathrm{~m} / \mathrm{s}\) when it encounters a ramp that is at an angle \(\theta=23.0^{\circ}\) above the horizontal. Find the maximum distance that the sphere travels up the ramp in each case: a) The ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height. b) The ramp provides enough friction to prevent the sphere from sliding, so both the linear and rotational motion stop (instantaneously).

A force, \(\vec{F}=(2 \hat{x}+3 \hat{y}) \mathrm{N},\) is applied to an object at a point whose position vector with respect to the pivot point is \(\vec{r}=(4 \hat{x}+4 \hat{y}+4 \hat{z}) \mathrm{m} .\) Calculate the torque created by the force about that pivot point.

A thin uniform rod (length \(=1.00 \mathrm{~m},\) mass \(=2.00 \mathrm{~kg})\) is pivoted about a horizontal frictionless pin through one of its ends. The moment of inertia of the rod through this axis is \(\frac{1}{3} m L^{2} .\) The rod is released when it is \(60.0^{\circ}\) below the horizontal. What is the angular acceleration of the rod at the instant it is released?

An object made of two disk-shaped sections, \(\mathrm{A}\) and \(\mathrm{B}\), as shown in the figure, is rotating about an axis through the center of disk A. The masses and the radii of disks \(A\) and \(B\). respectively are, \(2.00 \mathrm{~kg}\) and \(0.200 \mathrm{~kg}\) and \(25.0 \mathrm{~cm}\) and \(2.50 \mathrm{~cm}\). a) Calculate the moment of inertia of the object. b) If the axial torque due to friction is \(0.200 \mathrm{~N} \mathrm{~m}\), how long will it take for the object to come to a stop if it is rotating with an initial angular velocity of \(-2 \pi \mathrm{rad} / \mathrm{s} ?\)

You are the technical consultant for an action-adventure film in which a stunt calls for the hero to drop off a 20.0 -m-tall building and land on the ground safely at a final vertical speed of \(4.00 \mathrm{~m} / \mathrm{s}\). At the edge of the building's roof, there is a \(100 .-\mathrm{kg}\) drum that is wound with a sufficiently long rope (of negligible mass), has a radius of \(0.500 \mathrm{~m}\), and is free to rotate about its cylindrical axis with a moment of inertia \(I_{0}\). The script calls for the 50.0 -kg stuntman to tie the rope around his waist and walk off the roof. a) Determine an expression for the stuntman's linear acceleration in terms of his mass \(m\), the drum's radius \(r\) and moment of inertia \(I_{0}\). b) Determine the required value of the stuntman's acceleration if he is to land safely at a speed of \(4.00 \mathrm{~m} / \mathrm{s},\) and use this value to calculate the moment of inertia of the drum about its axis. c) What is the angular acceleration of the drum? d) How many revolutions does the drum make during the fall?

In a tire-throwing competition, a man holding a \(23.5-\mathrm{kg}\) car tire quickly swings the tire through three full turns and releases it, much like a discus thrower. The tire starts from rest and is then accelerated in a circular path. The orbital radius \(r\) for the tire's center of mass is \(1.10 \mathrm{~m},\) and the path is horizontal to the ground. The figure shows a top view of the tire's circular path, and the dot at the center marks the rotation axis. The man applies a constant torque of \(20.0 \mathrm{~N} \mathrm{~m}\) to accelerate the tire at a constant angular acceleration. Assume that all of the tire's mass is at a radius \(R=0.35 \mathrm{~m}\) from its center. a) What is the time, \(t_{\text {throw }}\) required for the tire to complete three full revolutions? b) What is the final linear speed of the tire's center of mass (after three full revolutions)? c) If, instead of assuming that all of the mass of the tire is at a distance \(0.35 \mathrm{~m}\) from its center, you treat the tire as a hollow disk of inner radius \(0.30 \mathrm{~m}\) and outer radius \(0.40 \mathrm{~m}\), how does this change your answers to parts (a) and (b)?

A uniform rod of mass \(M=250.0 \mathrm{~g}\) and length \(L=50.0 \mathrm{~cm}\) stands vertically on a horizontal table. It is released from rest to fall. a) What forces are acting on the rod? b) Calculate the angular speed of the rod, the vertical acceleration of the moving end of the rod, and the normal force exerted by the table on the rod as it makes an angle \(\theta=45.0^{\circ}\) with respect to the vertical. c) If the rod falls onto the table without slipping, find the linear acceleration of the end point of the rod when it hits the table and compare it with \(g\).

A wheel with \(c=\frac{4}{9}\), a mass of \(40.0 \mathrm{~kg}\), and a rim radius of \(30.0 \mathrm{~cm}\) is mounted vertically on a horizontal axis. A 2.00 -kg mass is suspended from the wheel by a rope wound around the rim. Find the angular acceleration of the wheel when the mass is released.

The flywheel of an old steam engine is a solid homogeneous metal disk of mass \(M=120 . \mathrm{kg}\) and radius \(R=80.0 \mathrm{~cm} .\) The engine rotates the wheel at \(500 .\) rpm. In an emergency, to bring the engine to a stop, the flywheel is disengaged from the engine and a brake pad is applied at the edge to provide a radially inward force \(F=100 .\) N. If the coefficient of kinetic friction between the pad and the flywheel is \(\mu_{\mathrm{k}}=0.200,\) how many revolutions does the flywheel make before coming to rest? How long does it take for the flywheel to come to rest? Calculate the work done by the torque during this time.

The turbine and associated rotating parts of a jet engine have a total moment of inertia of \(25 \mathrm{~kg} \mathrm{~m}^{2}\). The turbine is accelerated uniformly from rest to an angular speed of \(150 \mathrm{rad} / \mathrm{s}\) in a time of \(25 \mathrm{~s}\). Find a) the angular acceleration, b) the net torque required, c) the angle turned through in \(25 \mathrm{~s}\) d) the work done by the net torque, and e) the kinetic energy of the turbine at the end of the \(25 \mathrm{~s}\).

A sphere of radius \(R\) and mass \(M\) sits on a horizontal tabletop. A horizontally directed impulse with magnitude \(J\) is delivered to a spot on the ball a vertical distance \(h\) above the tabletop. a) Determine the angular and translational velocity of the sphere just after the impulse is delivered. b) Determine the distance \(h_{0}\) at which the delivered impulse causes the ball to immediately roll without slipping.

A circular platform of radius \(R_{p}=4.00 \mathrm{~m}\) and mass \(M_{\mathrm{p}}=400 .\) kg rotates on frictionless air bearings about its vertical axis at 6.00 rpm. An 80.0 -kg man standing at the very center of the platform starts walking \((\) at \(t=0)\) radially outward at a speed of \(0.500 \mathrm{~m} / \mathrm{s}\) with respect to the platform. Approximating the man by a vertical cylinder of radius \(R_{\mathrm{m}}=0.200 \mathrm{~m}\) determine an equation (specific expression) for the angular velocity of the platform as a function of time. What is the angular velocity when the man reaches the edge of the platform?

A 25.0 -kg boy stands \(2.00 \mathrm{~m}\) from the center of a frictionless playground merry-go-round, which has a moment of inertia of \(200 . \mathrm{kg} \mathrm{m}^{2} .\) The boy begins to run in a circular path with a speed of \(0.600 \mathrm{~m} / \mathrm{s}\) relative to the ground. a) Calculate the angular velocity of the merry-go-round. b) Calculate the speed of the boy relative to the surface of the merry-go- round.

-10.62 The Earth has an angular speed of \(7.272 \cdot 10^{-5} \mathrm{rad} / \mathrm{s}\) in its rotation. Find the new angular speed if an asteroid \(\left(m=1.00 \cdot 10^{22} \mathrm{~kg}\right)\) hits the Earth while traveling at a speed of \(1.40 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\) (assume the asteroid is a point mass compared to the radius of the Earth) in each of the following cases: a) The asteroid hits the Earth dead center. b) The asteroid hits the Earth nearly tangentially in the direction of Earth's rotation. c) The asteroid hits the Earth nearly tangentially in the direction opposite to Earth's rotation.

Most stars maintain an equilibrium size by balancing two forces - an inward gravitational force and an outward force resulting from the star's nuclear reactions. When the star's fuel is spent, there is no counterbalance to the gravitational force. Whatever material is remaining collapses in on itself. Stars about the same size as the Sun become white dwarfs, which glow from leftover heat. Stars that have about three times the mass of the Sun compact into neutron stars. And a star with a mass greater than three times the Sun's mass collapses into a single point, called a black hole. In most cases, protons and electrons are fused together to form neutrons-this is the reason for the name neutron star. Neutron stars rotate very fast because of the conservation of angular momentum. Imagine a star of mass \(5.00 \cdot 10^{30} \mathrm{~kg}\) and radius \(9.50 \cdot 10^{8} \mathrm{~m}\) that rotates once in 30.0 days. Suppose this star undergoes gravitational collapse to form a neutron star of radius \(10.0 \mathrm{~km} .\) Determine its rotation period.

In experiments at the Princeton Plasma Physics Laboratory, a plasma of hydrogen atoms is heated to over 500 million degrees Celsius (about 25 times hotter than the center of the Sun) and confined for tens of milliseconds by powerful magnetic fields \((100,000\) times greater than the Earth's magnetic field). For each experimental run, a huge amount of energy is required over a fraction of a second, which translates into a power requirement that would cause a blackout if electricity from the normal grid were to be used to power the experiment. Instead, kinetic energy is stored in a colossal flywheel, which is a spinning solid cylinder with a radius of \(3.00 \mathrm{~m}\) and mass of \(1.18 \cdot 10^{6} \mathrm{~kg}\). Electrical energy from the power grid starts the flywheel spinning, and it takes 10.0 min to reach an angular speed of \(1.95 \mathrm{rad} / \mathrm{s}\). Once the flywheel reaches this angular speed, all of its energy can be drawn off very quickly to power an experimental run. What is the mechanical energy stored in the flywheel when it spins at \(1.95 \mathrm{rad} / \mathrm{s}\) ? What is the average torque required to accelerate the flywheel from rest to \(1.95 \mathrm{rad} / \mathrm{s}\) in \(10.0 \mathrm{~min} ?\)

A 2.00 -kg thin hoop with a 50.0 -cm radius rolls down a \(30.0^{\circ}\) slope without slipping. If the hoop starts from rest at the top of the slope, what is its translational velocity after it rolls \(10.0 \mathrm{~m}\) along the slope?

An oxygen molecule \(\left(\mathrm{O}_{2}\right)\) rotates in the \(x y\) -plane about the \(z\) -axis. The axis of rotation passes through the center of the molecule, perpendicular to its length. The mass of each oxygen atom is \(2.66 \cdot 10^{-26} \mathrm{~kg},\) and the average separation between the two atoms is \(d=1.21 \cdot 10^{-10} \mathrm{~m}\) a) Calculate the moment of inertia of the molecule about the \(z\) -axis. b) If the angular speed of the molecule about the \(z\) -axis is \(4.60 \cdot 10^{12} \mathrm{rad} / \mathrm{s},\) what is its rotational kinetic energy?

A 0.050 -kg bead slides on a wire bent into a circle of radius \(0.40 \mathrm{~m}\). You pluck the bead with a force tangent to the circle. What force is needed to give the bead an angular acceleration of \(6.0 \mathrm{rad} / \mathrm{s}^{2} ?\)