Chapter 9

A flexible straight wire \(75.0 \mathrm{~cm}\) long is bent into the arc of a circle of radius \(2.50 \mathrm{~m}\). What angle (in radians and degrees) will this arc subtend at the center of the circle?

(a) What angle in radians is subtended by an arc \(1.50 \mathrm{~m}\) in length on the circumference of a circle of radius \(2.50 \mathrm{~m} ?\) What is this angle in degrees? (b) An arc \(14.0 \mathrm{~cm}\) in length on the circumference of a circle subtends an angle of \(128^{\circ}\). What is the radius of the circle? (c) The angle between two radii of a circle with radius \(1.50 \mathrm{~m}\) is 0.700 rad. What length of are is intercepted on the circumference of the circle by the two radii?

(a) Calculate the angular velocity (in rad/s) of the second, minute, and hour hands on a wall clock. (b) What is the period of each of these hands?

The once-popular LP (long-play) records were 12 in. in diameter and turned at a constant \(33 \frac{1}{3} \mathrm{rpm} .\) Find (a) the angular speed of the LP in rad \(/ \mathrm{s}\) and \((\mathrm{b})\) its period in seconds.

If a wheel \(212 \mathrm{~cm}\) in diameter takes \(2.25 \mathrm{~s}\) for each revolution, find its (a) period and (b) angular speed in rad/s.

A curve ball is a type of pitch in which the baseball spins on its axis as it heads for home plate. If a curve ball is thrown at \(35.8 \mathrm{~m} / \mathrm{s}\) (80 mph) with a spin rate of 30 rev \(/\) s, how many revolutions does it complete before reaching home plate? Assume that home plate is \(18.3 \mathrm{~m}(60 \mathrm{ft})\) from the pitching mound and that the baseball travels at a constant velocity.

A laser beam aimed from the earth is swept across the face of the moon. (a) If the beam is rotated at an angular velocity of \(1.50 \times 10^{-3} \mathrm{rad} / \mathrm{s},\) at what speed does the laser light move across the moon's surface? (See Appendix E for the moon's orbital radius.) (b) If the diameter of the laser spot on the moon is \(6.00 \mathrm{~km},\) what is the angle of divergence of the laser beam?

At \(t=0\), a cooling fan running at \(200 \mathrm{rad} / \mathrm{s}\) is turned off and then slows down at a rate of \(20 \mathrm{rad} / \mathrm{s}^{2}\). Simultaneously (at \(t=0\) ), a second cooling fan is turned on and begins to spin from rest with an acceleration of \(60 \mathrm{rad} / \mathrm{s}^{2}\). (a) Find the time at which both fans have the same angular speed. (b) What is the angular speed of the fans at this time?

A turntable that spins at a constant 78.0 rpm takes \(3.50 \mathrm{~s}\) to reach this angular speed after it is turned on. Find (a) its angular acceleration (in rad \(/ \mathrm{s}^{2}\) ), assuming it to be constant, and (b) the number of degrees it turns through while speeding up.

A circular saw blade \(0.200 \mathrm{~m}\) in diameter starts from rest. In 6.00 s, it reaches an angular velocity of \(140 \mathrm{rad} / \mathrm{s}\) with constant angular acceleration. Find the angular acceleration and the angle through which the blade has turned in this time.

A wheel turns with a constant angular acceleration of \(0.640 \mathrm{rad} / \mathrm{s}^{2}\). (a) How much time does it take for the wheel to reach an angular velocity of \(8.00 \mathrm{rad} / \mathrm{s},\) starting from rest? (b) Through how many revolutions does the wheel turn in this interval?

An electric fan is turned off, and its angular velocity decreases uniformly from 500.0 rev \(/\) min to 200.0 rev \(/ \min\) in 4.00 s. (a) Find the angular acceleration in rev \(/ \mathrm{s}^{2}\) and the number of revolutions made by the motor in the \(4.00 \mathrm{~s}\) interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

A flywheel in a motor is spinning at 500.0 rpm when a power failure suddenly occurs. The flywheel has mass \(40.0 \mathrm{~kg}\) and diameter \(75.0 \mathrm{~cm} .\) The power is off for \(30.0 \mathrm{~s},\) and during this time the flywheel slows down uniformly due to friction in its axle bearings. During the time the power is off, the flywheel makes 200.0 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on? (b) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on, and how many revolutions would the wheel have made during this time?

A flywheel having constant angular acceleration requires \(4.00 \mathrm{~s}\) to rotate through 162 rad. Its angular velocity at the end of this time is \(108 \mathrm{rad} / \mathrm{s}\). Find (a) the angular velocity at the beginning of the 4.00 s interval; (b) the angular acceleration of the flywheel.

A potter's wheel is spinning with an initial angular velocity of \(12 \mathrm{rad} / \mathrm{s} .\) It rotates through an angle of \(60 \mathrm{rad}\) in the process of coming to rest. (a) What is the angular acceleration of the wheel? (b) How long does it take for it to come to rest?

a car is traveling at a constant speed on the highway. Its tires have a diameter of \(61.0 \mathrm{~cm}\) and are rolling without sliding or slipping. If the angular speed of the tires is \(50.0 \mathrm{rad} / \mathrm{s},\) what is the speed of the car, in SI units?

(a) A cylinder \(0.150 \mathrm{~m}\) in diameter rotates in a lathe at \(620 \mathrm{rpm}\). What is the tangential speed of the surface of the cylinder? (b) The proper tangential speed for machining cast iron is about \(0.600 \mathrm{~m} / \mathrm{s}\). At how many revolutions per minute should a piece of stock \(0.0800 \mathrm{~m}\) in diameter be rotated in a lathe to produce this tangential speed?

A wheel rotates with a constant angular velocity of \(6.00 \mathrm{rad} / \mathrm{s}\) (a) Compute the radial acceleration of a point \(0.500 \mathrm{~m}\) from the axis, using the relation \(a_{\mathrm{rad}}=\omega^{2} r .(\mathrm{b})\) Find the tangential speed of the point, and compute its radial acceleration from the relation \(a_{\mathrm{rad}}=v^{2} / r\)

Find the required angular speed (in rpm) of an ultracentrifuge for the radial acceleration of a point \(2.50 \mathrm{~cm}\) from the axis to equal \(400,000 \mathrm{~g}\)

II A flywheel with a radius of \(0.300 \mathrm{~m}\) starts from rest and accelerates with a constant angular acceleration of \(0.600 \mathrm{rad} / \mathrm{s}^{2}\). Compute the magnitude of the tangential acceleration, the radial acceleration, and the resultant acceleration of a point on its rim (a) at the start, (b) after it has turned through \(60.0^{\circ},\) and \((\mathrm{c})\) after it has turned through \(120.0^{\circ} .\)

A car is traveling at a speed of \(101 \mathrm{~km} / \mathrm{h}\) on the highway and has a small stone stuck between the treads of one of its tires. The tires have diameter \(61.0 \mathrm{~cm}\) and are rolling without sliding or slipping. What are (a) the maximum and (b) the minimum speeds of the stone as observed by a pedestrian standing on the side of the highway?

Electric toothbrushes can be effective in removing dental plaque. One model consists of a head \(1.1 \mathrm{~cm}\) in diameter that rotates back and forth through a \(70.0^{\circ}\) angle 7600 times per minute. The rim of the head contains a thin row of bristles. (See Figure \(9.27 .)\) (a) What is the average angular speed in each direction of the rotating head, in \(\mathrm{rad} / \mathrm{s} ?\) (b) What is the average linear speed in each direction of the bristles against the teeth? (c) Using your own observations, what is the approximate speed of the bristles against your teeth when you brush by hand with an ordinary toothbrush?

The spin cycles of a washing machine have two angular speeds, 423 rev \(/ \mathrm{min}\) and \(640 \mathrm{rev} / \mathrm{min} .\) The internal diameter of the drum is \(0.470 \mathrm{~m}\). (a) What is the ratio of the maximum radial force on the laundry for the higher angular speed to that for the lower speed? (b) What is the ratio of the maximum tangential speed of the laundry for the higher angular speed to that for the lower speed? (c) Find the laundry's maximum tangential speed and the maximum radial acceleration, in terms of \(g\).

A slender metal rod has a mass \(M\) and length \(L\). The rod is first rotated about a perpendicular axis through its center with an angular velocity \(\omega\) (Figure \(9.28 \mathrm{a}\) ). It is then rotated about a perpendicular axis through its end at the same angular velocity (Figure \(9.28 \mathrm{~b}\) ). Find the ratio of the kinetic energy of the first case to that of the second case.

A thin uniform bar has two small balls glued to its ends. The bar is \(2.00 \mathrm{~m}\) long and has mass \(4.00 \mathrm{~kg},\) while the balls each have mass \(0.500 \mathrm{~kg}\) and can be treated as point masses. Find the moment of inertia of this combination about each of the following axes: (a) an axis perpendicular to the bar through its center; (b) an axis perpendicular to the bar through one of the balls; (c) an axis parallel to the bar through both balls.

Four small \(0.200 \mathrm{~kg}\) spheres, each of which you can regard as a point mass, are arranged in a square \(0.400 \mathrm{~m}\) on a side and connected by light rods. (See Figure \(9.29 .)\) Find the moment of inertia of the system about an axis (a) through the center of the square, perpendicular to its plane at point \(O ;\) (b) along the line \(A B ;\) and (c) along the line \(C D\).

Suppose you are given a steel bar and you cut it in half. How does the moment of inertia of one of the two halves compare to that of the original bar? Assume rotation about a perpendicular axis through one end of the bars.

A wagon wheel is constructed as shown in Figure \(9.31 .\) The radius of the wheel is \(0.300 \mathrm{~m}\), and the rim has a mass of \(1.40 \mathrm{~kg}\). Each of the wheel's eight spokes, which come out from the center and are 0.300 m long, has a mass of \(0.280 \mathrm{~kg}\). What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel?

The flywheel of a gasoline engine is required to give up 500 \(\mathrm{J}\) of kinetic energy while its angular velocity decreases from 650 rev \(/\) min to 520 rev \(/\) min. What moment of inertia is required?

An airplane propeller is \(2.08 \mathrm{~m}\) in length (from tip to tip) with mass \(117 \mathrm{~kg}\) and is rotating at \(2400 \mathrm{rpm}\) (rev/min) about an axis through its center. You can model the propeller as a slender rod. (a) What is its rotational kinetic energy? (b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to \(75.0 \%\) of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

It has been suggested that we should use our power plants to generate energy in the off-hours (such as late at night) and store it for use during the day. One idea put forward is to store the energy in large flywheels. Suppose we want to build such a flywheel in the shape of a hollow cylinder of inner radius \(0.500 \mathrm{~m}\) and outer radius \(1.50 \mathrm{~m},\) using concrete of density \(2.20 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). (a) If, for stability, such a heavy flywheel is limited to 1.75 seconds for each revolution and has negligible friction at its axle, what must be its length to store \(2.5 \mathrm{MJ}\) of energy in its rotational motion? (b) Suppose that by strengthening the frame you could safely double the flywheel's rate of spin. What length of flywheel would you need in that case? (Solve this part without reworking the entire problem!)

A solid copper disk has a radius of \(0.2 \mathrm{~m},\) a thickness of \(0.015 \mathrm{~m}\), and a mass of \(17 \mathrm{~kg}\). (a) What is the moment of inertia of the disk about a perpendicular axis through its center? (b) If the copper disk were melted down and re-formed into a solid sphere, what would its moment of inertia be?

We can roughly model a gymnastic tumbler as a uniform solid cylinder of mass \(75 \mathrm{~kg}\) and diameter \(1.0 \mathrm{~m}\). If this tumbler rolls forward at 0.50 rev \(/ \mathrm{s}\), (a) how much total kinetic energy does he have and (b) what percent of his total kinetic energy is rotational?

A bicycle racer is going downhill at \(11.0 \mathrm{~m} / \mathrm{s}\) when, to his horror, one of his \(2.25 \mathrm{~kg}\) wheels comes off when he is \(75.0 \mathrm{~m}\) above the foot

A \(2.20 \mathrm{~kg}\) hoop \(1.20 \mathrm{~m}\) in diameter is rolling to the right without slipping on a horizontal floor at a steady \(3.00 \mathrm{rad} / \mathrm{s}\). (a) How fast is its center moving? (b) What is the total kinetic energy of the hoop?

A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping along a horizontal surface at a speed \(v\). They then encounter a hill that rises at an angle \(\theta\) above the horizontal. To what height \(h\) does each sphere roll before coming to rest?

A size-5 soccer ball of diameter \(22.6 \mathrm{~cm}\) and mass \(426 \mathrm{~g}\) rolls up a hill without slipping, reaching a maximum height of \(5.00 \mathrm{~m}\) above the base of the hill. We can model this ball as a thin-walled, hollow sphere. (a) At what rate was it rotating at the base of the hill? (b) How much rotational kinetic energy did it then have?

A solid uniform marble and a block of ice, each with the same mass, start from rest at the same height \(H\) above the bottom of a hill and move down it. The marble rolls without slipping, but the ice slides without friction. (a) Find the speed of each of these objects when it reaches the bottom of the hill. (b) Which object is moving faster at the bottom, the ice or the marble? (c) Which object has more kinetic energy at the bottom, the ice or the marble?

What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) a uniform solid cylinder; (b) a uniform sphere; (c) a thin-walled, hollow sphere; (d) a hollow cylinder with outer radius \(R\) and inner radius \(R / 2\).

A \(150.0 \mathrm{~kg}\) cart rides down a set of tracks on four solid steel wheels, each with radius \(20.0 \mathrm{~cm}\) and mass \(45.0 \mathrm{~kg} .\) The tracks slope downward at an angle of \(20^{\circ}\) to the horizontal. If the cart is released from rest a distance of \(16.0 \mathrm{~m}\) from the bottom of the track (measured along the slope), how fast will it be moving when it reaches the bottom? Assume that the wheels roll without slipping, and that there is no energy loss due to friction.

A uniform marble rolls down a symmetric bowl, starting from rest at the top of the left side. The top of each side is a distance \(h\) above the bottom of the bowl. (a) How far up the right side of the bowl will the marble go if the interior surface of the bowl is rough so that the marble rolls without slipping? (b) How high would the marble go if the bowl's surface were frictionless? (c) For which case is the marble moving faster when it reaches the bottom of the bowl?

A \(7300 \mathrm{~N}\) elevator is to be given an acceleration of \(0.150 \mathrm{~g}\) by connecting it to a cable of negligible weight wrapped around a turning cylindrical shaft. If the shaft's diameter can be no larger than 16.0 \(\mathrm{cm}\) due to space limitations, what must be its minimum angular acceleration to provide the required acceleration of the elevator?

A \(392 \mathrm{~N}\) wheel comes off a moving truck and rolls without slipping along a highway. At the bottom of a hill it is rotating at \(25.0 \mathrm{rad} / \mathrm{s} .\) The radius of the wheel is \(0.600 \mathrm{~m},\) and its moment of inertia about its rotation axis is \(0.800 \mathrm{MR}^{2}\). Friction does work on the wheel as it rolls up the hill to a stop, a height \(h\) above the bottom of the hill; this work has absolute value 3500 J. Calculate \(h\).

The odometer (mileage gauge) of a car tells you the number of miles you have driven, but it doesn't count the miles directly. Instead, it counts the number of revolutions of your car's wheels and converts this quantity to mileage, assuming a standard-size tire and that your tires do not slip on the pavement. (a) A typical midsize car has tires 24 in. in diameter. How many revolutions of the wheels must the odometer count in order to show a mileage of 0.10 mile? (b) What will the odometer read when the tires have made 5000 revolutions? (c) Suppose you put oversize 28 -in.-diameter tires on your car. How many miles will you really have driven when your odometer reads 500 miles?

Your car's speedometer works in much the same way as its odometer (see the previous problem), except that it converts the angular speed of the wheels to a linear speed of the car, assuming standard-size tires and no slipping on the pavement. (a) If your car has standard 24 -in.-diameter tires, how fast are your wheels turning when you are driving at a freeway speed of \(55 \mathrm{mph} ?\) (b) How fast are you going when your wheels are turning at 500 rpm? (c) If you put on undersize 20 -in.-diameter tires, what will the speedometer read when you are actually traveling at \(50 \mathrm{mph} ?\)

A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. Whenever the bus was stopped at a station, the wheel was brought up to speed with the use of an electric motor that could then be attached to the electric power lines. The flywheel was a solid cylinder with a mass of \(1000 \mathrm{~kg}\) and a diameter of \(1.80 \mathrm{~m} ;\) its top angular speed was \(3000 \mathrm{rev} / \mathrm{min} .\) At this angular speed, what was the kinetic energy of the flywheel?

A \(55 \mathrm{~kg}\) woman is riding a 24 kg bike at a speed of \(7 \mathrm{~m} / \mathrm{s}\). The wheels of the bike can be treated as thin rings, each of mass \(3 \mathrm{~kg}\) and radius \(0.33 \mathrm{~m}\). What percentage of the total kinetic energy of the woman-bike system is carried in the rotational kinetic energy of the wheels?

A vacuum cleaner belt is looped over a shaft of radius \(0.45 \mathrm{~cm}\) and a wheel of radius \(2.00 \mathrm{~cm}\). The motor turns the shaft at \(60.0 \mathrm{rev} / \mathrm{s}\) and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed (see Figure 9.36 ). Assume that the belt doesn't slip on either the shaft or the wheel. (a) What is the speed of a point on the belt? (b) What is the angular velocity of the wheel, in \(\mathrm{rad} / \mathrm{s} ?\)

A thin uniform rod \(50.0 \mathrm{~cm}\) long with mass \(0.320 \mathrm{~kg}\) is bent at its center into a \(\mathrm{V}\) shape, with a \(70.0^{\circ}\) angle at its vertex. Find the \(\mathrm{mo}\) ment of inertia of this \(V\) -shaped object about an axis perpendicular to the plane of the \(\mathrm{V}\) at its vertex.

In redesigning a piece of equipment, you need to replace a solid spherical steel part with a similar steel part that has half the radius. How does the moment of inertia of the new part compare to that of the old? Express your answer as a ratio \(I_{\text {new }} / I_{\text {old }}\)