|| A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod.

1. How far is the center of mass from the 600 g ball?

2. What is the rotational kinetic energy if the structure rotates about its center of mass at 100 rpm?

1. || INT A solid sphere of mass M and radius R rolls at speed v. Find an expression for its kinetic energy in terms of M and v.

2. A 2.0-cm-diameter marble is shot up a $\text{15}°$ ramp at $2.0\text{ m/s}$. How far up the ramp does it travel?

|| CALC An 800 g steel plate has the shape of the isosceles triangle shown in FIGURE P12.51. What are the x- and y-coordinates of the center of mass?

FIGURE P12.51

Hint: Divide the triangle into vertical strips of width $dx$, then relate the mass $dm$ of a strip at position x to the values of x and $dx$.

|| Determine the moment of inertia about the axis of the object shown in FIGURE P12.52.

FIGURE P12.52

|| CALC Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when $d=0$ and when $d=L\text{/}2$.

|| What is the moment of inertia of a 2.0 kg, 20-cm-diameter disk for rotation about an axis (a) through the center, and (b) through the edge of the disk?

|| CALC Calculate the moment of inertia of the rectangular plate in FIGURE P12.55 for rotation about a perpendicular axis through the center.

FIGURE P12.55

1. || INT CALC A disk of mass M and radius R has a hole of radius r centered on the axis. Calculate the moment of inertia of the disk.

2. Confirm that your answer agrees with Table 12.2 when $r=0$ and when $r=R$.

3. A 4.0-cm-diameter disk with a 3.0-cm-diameter hole rolls down a 50-cm-long, $20\text{°}$ ramp. What is its speed at the bottom? What percent is this of the speed of a particle sliding down a frictionless ramp?

|| A 3.0-m-long ladder, as shown in Figure 12.35, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.40. What is the minimum angle the ladder can make with the floor without slipping?

|| BIO A person’s center of mass is easily found by having the person lie on a reaction board. A horizontal, 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman lies on the reaction board with her feet over the pivot. The scale reads 25 kg. What is the distance from the woman’s feet to her center of mass?

|| In FIGURE P12.59, an 80 kg construction worker sits down 2.0 m from the end of a 1450 kg steel beam to eat his lunch. What is the tension in the cable?

FIGURE P12.59

|| Your task in a science contest is to stack four identical uniform bricks, each of length L, so that the top brick is as far to the right as possible without the stack falling over. Is it possible, as FIGURE P12.60 shows, to stack the bricks such that no part of the top brick is over the table? Answer this question by determining the maximum possible value of d.

FIGURE P12.60

|| A 40 kg, 5.0-m-long beam is supported by, but not attached to, the two posts in FIGURE P12.61. A 20 kg boy starts walking along the beam. How close can he get to the right end of the beam without it falling over?

FIGURE P12.61

|| A 120-cm-wide sign hangs from a 5.0 kg, 200-cm-long pole. A cable of negligible mass supports the end of the rod as shown in FIGURE P12.62. What is the maximum mass of the sign if the maximum tension in the cable without breaking is 300 N?

FIGURE P12.62

|| FIGURE P12.63 shows a 15 kg cylinder held at rest on a $\text{20}°$ slope.

1. What is the tension in the cable?

2. What is the magnitude of the static friction force?

FIGURE P12.63

|| INT Blocks of mass $m_1$ and $m_2$ are connected by a massless string that passes over the pulley in FIGURE P12.64. The pulley turns on frictionless bearings. Mass $m_1$ slides on a horizontal, frictionless surface. Mass $m_2$ is released while the blocks are at rest.

1. Assume the pulley is massless. Find the acceleration of $m_1$ and the tension in the string. This is a Chapter 7 review problem.

2. Suppose the pulley has mass $m_{\text{p}}$ and radius R. Find the acceleration of $m_1$ and the tensions in the upper and lower portions of the string. Verify that your answers agree with part a if you set $m_{\text{p}}=0$.

FIGURE P12.64

|| The 2.0 kg, 30-cm-diameter disk in FIGURE P12.65 is spinning at 300 rpm. How much friction force must the brake apply to the rim to bring the disk to a halt in 3.0 s?

FIGURE P12.65

||| INT Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel’s energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm.

1. A motor spins up the flywheel with a constant torque of $50\text{N}\text{m}$. How long does it take the flywheel to reach top speed?

2. How much energy is stored in the flywheel?

3. The flywheel is disconnected from the motor and connected to a machine to which it will deliver energy. Half the energy stored in the flywheel is delivered in 2.0 s. What is the average power delivered to the machine?

4. How much torque does the flywheel exert on the machine?

||| A 30-cm-diameter, 1.2 kg solid turntable rotates on a 1.2-cm-diameter, 450 g shaft at a constant 33 rpm. When you hit the stop switch, a brake pad presses against the shaft and brings the turntable to a halt in 15 seconds. How much friction force does the brake pad apply to the shaft?

|| Your engineering team has been assigned the task of measuring the properties of a new jet-engine turbine. You’ve previously determined that the turbine’s moment of inertia is $2.6\text{kg}{\text{m}}^2$. The next job is to measure the frictional torque of the bearings. Your plan is to run the turbine up to a predetermined rotation speed, cut the power, and time how long it takes the turbine to reduce its rotation speed by $50\%$. Your data are given in the table. Draw an appropriate graph of the data and, from the slope of the best-fit line, determine the frictional torque.

|| A 750 g disk and a 760 g ring, both 15 cm in diameter, are rolling along a horizontal surface at $1.5\text{ m/s}$ when they encounter a $\text{15}°$ slope. How far up the slope does each travel before rolling back down?

|| A hollow sphere is rolling along a horizontal floor at $5.0\text{m}\text{/}\text{s}$ when it comes to a $30\text{°}$ incline. How far up the incline does it roll before reversing direction?

|| The 5.0 kg, 60-cm-diameter disk in FIGURE P12.71 rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released.

1. What is the cylinder’s initial angular acceleration?

2. What is the cylinder’s angular velocity when it is directly below the axle?

FIGURE P12.71

||| A solid spherical marble shot up a frictionless $\text{15}°$ slope rolls 2.50 m to its highest point. If the marble is shot with the same speed up a slightly rough $\text{15}°$ slope, it rolls only 2.30 m. What is the coefficient of rolling friction on the second slope?

|| A long, thin rod of mass M and length L is standing straight up on a table. Its lower end rotates on a frictionless pivot. A very slight push causes the rod to fall over. As it hits the table, what are (a) the angular velocity and (b) the speed of the tip of the rod?

|| The marble rolls down the track shown in FIGURE P12.74 and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off?

FIGURE P12.74

|| The sphere of mass M and radius R in FIGURE P12.75 is rigidly attached to a thin rod of radius r that passes through the sphere at distance $\frac{1}{2}R$ from the center. A string wrapped around the rod pulls with tension T. Find an expression for the sphere’s angular acceleration. The rod’s moment of inertia is negligible.

FIGURE P12.75

|| INT A puck attached to a 50-cm-long string rotates in a horizontal circle on a frictionless table at 40 rpm. The length of the string is gradually reduced to 25 cm as it is slowly pulled through a hole in the table. What is the puck’s angular velocity, in rpm, on the shorter string?

|| A satellite follows the elliptical orbit shown in FIGURE P12.77. The only force on the satellite is the gravitational attraction of the planet. The satellite’s speed at point 1 is $8000\text{m}\text{/}\text{s}$.

1. Does the satellite experience any torque about the center of the planet? Explain.

2. What is the satellite’s speed at point 2?

3. What is the satellite’s speed at point 3?

FIGURE P12.77

|| A 10 g bullet traveling at $400\text{m}\text{/}\text{s}$ strikes a 10 kg, 1.0-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door just after impact?

|| A 200 g, 40-cm-diameter turntable rotates on frictionless bearings at 60 rpm. A 20 g block sits at the center of the turntable. A compressed spring shoots the block radially outward along a frictionless groove in the surface of the turntable. What is the turntable’s rotation angular velocity when the block reaches the outer edge?

|| Sanjay and Ting, each with a mass of 25 kg, are riding opposite each other on the edge of a 150 kg, 3.0-m-diameter playground merry-go-round that’s rotating at 15 rpm. Each walks straight inward and stops 35 cm from the center. What is the new angular velocity, in rpm?

|| A merry-go-round is a common piece of playground equipment. A 3.0-m-diameter merry-go-round with a mass of 250 kg is spinning at 20 rpm. John runs tangent to the merry-go-round at $5.0\text{m}\text{/}\text{s}$, in the same direction that it is turning, and jumps onto the outer edge. John’s mass is 30 kg. What is the merry-go-round’s angular velocity, in rpm, after John jumps on?

||| FIGURE P12.82 shows a cube of mass m sliding without friction at speed $v_0$. It undergoes a perfectly elastic collision with the bottom tip of a rod of length d and mass $M=2m$. The rod is pivoted about a frictionless axle through its center, and initially it hangs straight down and is at rest. What is the cube’s velocity—both speed and direction—after the collision?

FIGURE P12.82

||| During most of its lifetime, a star maintains an equilibrium size in which the inward force of gravity on each atom is balanced by an outward pressure force due to the heat of the nuclear reactions in the core. But after all the hydrogen “fuel” is consumed by nuclear fusion, the pressure force drops and the star undergoes a gravitational collapse until it becomes a neutron star. In a neutron star, the electrons and protons of the atoms are squeezed together by gravity until they fuse into neutrons. Neutron stars spin very rapidly and emit intense pulses of radio and light waves, one pulse per rotation. These “pulsing stars” were discovered in the 1960s and are called pulsars.

1. A star with the mass $(M=2.0\times {10}^{30}\text{kg})$ and size $(R=$ $7.0\times {10}^8\text{m})$ of our sun rotates once every 30 days. After undergoing gravitational collapse, the star forms a pulsar that is observed by astronomers to emit radio pulses every 0.10 s. By treating the neutron star as a solid sphere, deduce its radius.

2. What is the speed of a point on the equator of the neutron star?

   Your answers will be somewhat too large because a star cannot be accurately modeled as a solid sphere. Even so, you will be able to show that a star, whose mass is ${10}^6$ larger than the earth’s, can be compressed by gravitational forces to a size smaller than a typical state in the United States!

|| The earth’s rotation axis, which is tilted $23.5\text{°}$ from the plane of the earth’s orbit, today points to Polaris, the north star. But Polaris has not always been the north star because the earth, like a spinning gyroscope, precesses. That is, a line extending along the earth’s rotation axis traces out a $23.5\text{°}$ cone as the earth precesses with a period of 26,000 years. This occurs because the earth is not a perfect sphere. It has an equatorial bulge, which allows both the moon and the sun to exert a gravitational torque on the earth. Our expression for the precession frequency of a gyroscope can be written $\text{Ω}=\tau /I\omega$. Although we derived this equation for a specific situation, it’s a valid result, differing by at most a constant close to 1, for the precession of any rotating object. What is the average gravitational torque on the earth due to the moon and the sun?

||| CALC Objects that rotate in air or water experience a torque due to drag. With quadratic drag, a drag torque that’s negligible at low rpm quickly becomes significant as the rpm increases. Consider a square bar with cross section $a\times a$ and length L. It is rotating on an axle through its center at angular velocity $\omega$ in a fluid of density $\rho$. Assume that the drag coefficient $C_{\text{d}}$ is constant along the length of the bar. Find an expression for the magnitude of the drag torque on the bar.

Hint: Begin by considering the drag force on a small piece of the bar of length $dr$ at distance r from the axle.

||| The two blocks in FIGURE CP12.86 are connected by a massless rope that passes over a pulley. The pulley is 12 cm in diameter and has a mass of 2.0 kg. As the pulley turns, friction at the axle exerts a torque of magnitude $0.50\text{N}\text{m}$. If the blocks are released from rest, how long does it take the 4.0 kg block to reach the floor?

FIGURE CP12.86

||| A 45 kg figure skater is spinning on the toes of her skates at $1.0\text{rev}\text{/}\text{s}$. Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 kg, 20 cm average diameter, 160 cm tall) plus two rod-like arms (2.5 kg each, 66 cm long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 kg, 20-cm-diameter, 200-cm-tall cylinder. What is her new angular velocity, in rev/s?

||| CALC A rod of length L and mass M has a nonuniform mass distribution. The linear mass density (mass per length) is $\lambda =c{x}^2$, where x is measured from the center of the rod and c is a constant.

1. What are the units of c?

2. Find an expression for c in terms of L and M.

3. Find an expression in terms of L and M for the moment of inertia of the rod for rotation about an axis through the center.

||| In FIGURE CP12.89, a 200 g toy car is placed on a narrow 60-cm-diameter track with wheel grooves that keep the car going in a circle. The 1.0 kg track is free to turn on a frictionless, vertical axis. The spokes have negligible mass. After the car’s switch is turned on, it soon reaches a steady speed of $0.75\text{m}\text{/}\text{s}$ relative to the track. What then is the track’s angular velocity, in rpm?

FIGURE CP12.89

||| A 75 g, 30-cm-long rod hangs vertically on a frictionless, horizontal axle passing through its center. A 10 g ball of clay traveling horizontally at $2.5\text{m}\text{/}\text{s}$ hits and sticks to the very bottom tip of the rod. To what maximum angle, measured from vertical, does the rod (with the attached ball of clay) rotate?

||| BIO The bunchberry flower has the fastest-moving parts ever observed in a plant. Initially, the stamens are held by the petals in a bent position, storing elastic energy like a coiled spring. When the petals release, the tips of the stamen act like medieval catapults, flipping through a $60\text{°}$ angle in just $0.30\text{m}\text{s}$ to launch pollen from anther sacs at their ends. The human eye just sees a burst of pollen; only high-speed photography reveals the details. As FIGURE CP12.91 shows, we can model the stamen tip as a $1.0\text{-mm-long}$, $10\mu \text{g}$ rigid rod with a $10\mu \text{g}$ anther sac at the end. Although oversimplifying, we’ll assume a constant angular acceleration.

1. How large is the “straightening torque”?

2. What is the speed of the anther sac as it releases its pollen?

FIGURE CP12.91