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?

\(\bullet\) (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 arc is intercepted on the circumference of the circle by the two radii?

\(\bullet\) (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?

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

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

Find the angular velocity, in \(\mathrm{rad} / \mathrm{s},\) of \((\mathrm{a})\) the earth due to its daily spin on its axis, (b) the earth due to its yearly motion around the sun, and (c) our moon due to its monthly motion around the earth. Consult Appendix E as needed.

\(\bullet\) An airplane propeller is rotating at 1900 \(\mathrm{rpm.}\) (a) Compute the propeller's angular velocity in rad/s. (b) How many seconds does it take for the propeller to turn through \(35^{\circ} ?\) (c) If the propeller were turning at \(18 \mathrm{rad} / \mathrm{s},\) at how many rpm would it be turning? (d) What is the period (in seconds) of this propeller?

A A wall clock on Planet \(X\) has two hands that are aligned at midnight and turn in the same direction at uniform rates, one at 0.0425 \(\mathrm{rad} / \mathrm{s}\) and the other at 0.0163 \(\mathrm{rad} / \mathrm{s}\) . At how many seconds after midnight are these hands (a) first aligned and (b) next aligned?

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

\(\cdot\) When the power is turned off on a turntable spinning at 78.0 rpm, you find that it takes 10.5 revolutions for it to stop while slowing down at a uniform rate. (a) What is the angular acceleration (in rad \(s^{2} )\) of this turntable? (b) How long does it take to stop after the power is turned off?

DVDs. The angular speed of digital video discs (DVDs) varies with whether the inner or outer part of the disc is being read. (CDs function in the same way.) Over a 133 min playing time, the angular speed varies from 570 rpm to 1600 rpm. Assuming it to be constant, what is the angular acceleration (in rad/s' \(s\) ) of such a DVD?

A circular saw blade 0.200 \(\mathrm{m}\) in diameter starts from rest. In 6.00 \(\mathrm{s}\) , 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.

\(\cdot\) A wheel turns with a constant angular acceleration of 0.640 \(\mathrm{rad} / \mathrm{s}^{2} .\) (a) How much time does it take 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/s' and the number of revolutions made by the motor in the 4.00 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 \(\mathrm{rpm}\) 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 slown 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 A flywheel having constant angular acceleration requires 4.00 s to rotate through 162 rad. Its angular velocity at the end of this time is 108 rad/s. Find (a) the angular velocity at the beginning of the 4.00 s interval; (b) the angular acceleration of the flywheel.

Emilie's potter's wheel rotates with a constant 2.25 \(\mathrm{rad} / \mathrm{s}^{2}\) angular acceleration. After 4.00 \(\mathrm{s}\) , the wheel has rotated through an angle of 60.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00 s interval?

\(\cdot\) A car is traveling at a speed of 63 \(\mathrm{mi} / \mathrm{h}\) on a freeway. If its tires have diameter 24.0 in and are rolling without sliding or slipping, what is their angular velocity?

\(\bullet\) (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?

\(\cdot \mathrm{A}\) 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 \(a_{\mathrm{rad}}=\omega^{2} r .\) (b) Find the tangential speed of the point, and compute its radial acceleration from the relation \(a_{\text { rad }}=v^{2} / r\) .

\(\bullet\) Ultracentrifuge. 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 \(400,000 g .\)

\(\bullet\) 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 \(120.0^{\circ} .\)

\(\bullet\) Electric drill. According to the shop manual, when drilling a 12.7 -mm-diameter hole in wood, plastic, or aluminum, a drill should have a speed of 1250 rev/min. For a 12.7 - -mm-diameter drill bit turning at a constant 1250 \(\mathrm{rev} / \mathrm{min}\) , find (a) the maximum linear speed of any part of the bit and (b) the maximum radial acceleration of any part of the bit.

\(\bullet\) Dental hygiene. Electric toothbrushes can be effective in removing dental plaque. One model consists of a head 1.1 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 9.25.) (a) What is the average angular speed in each direction of the rotating head, in rad/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?

\(\bullet\) The spin cycles of a washing machine have two angular speeds, 423 \(\mathrm{rev} / \mathrm{min}\) 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 \(g\) .

\(\bullet\) A twirler's baton is made of a slender metal cylinder of mass \(M\) and length \(L .\) Each end has a rubber cap of mass \(m,\) and you can accurately treat each cap as a particle in this problem. Find the total moment of inertia of the baton about the usual twirling axis (perpendicular to the baton through its center).

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.

Compound objects. Moment of inertia is a scalar. There- fore, if several objects are connected together, the moment of inertia of this compound object is simply the scalar (algebraic) sum of the moments, of inertia of each of the component. objects. Use this principle to answer each of the following questions about the moment of inertia of compound objects: (a) A thin uniform 2.50 \(\mathrm{kg}\) bar 1.50 \(\mathrm{m}\) long has a small 1.25 \(\mathrm{kg}\) mass glued to each end. What is the moment of inertia of this object about an axis perpendicular to the bar through its center? (b) What is the moment of inertia of the object in part (a) about an axis perpendicular to the bar at one end? (c) \(A 725\) g metal wire is bent into the shape of a hoop 60.0 \(\mathrm{cm}\) in diameter. Six wire spokes, each of mass 112 g, are added from the center of the hoop to the rim. What is the moment of inertia of this object about an axis perpendicular to it through its center?

You need to design an industrial turntable that is 60.0 cm in diameter and has a kinetic energy of 0.250 J when turning at 45.0 rpm v (a) What must be the moment of inertia of the turntable about the rotation axis? (b) If your workshop makes this turntable in the shape of a uniform solid disk, what must be its mass?

A grinding wheel in the shape of a solid disk is 0.200 m in diameter and has a mass of 3.00 kg. The wheel is rotating at 2200 rpm about an axis through its center. (a) What is its kinetic energy? (b) How far would it have to drop in free fall to acquire the same amount of kinetic energy?

An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm v 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?

Storing energy in flywheels. 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 fly wheels. 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 second 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 compound disk of outside diameter 140.0 \(\mathrm{cm}\) is made up of a uniform solid disk of radius 50.0 \(\mathrm{cm}\) and area density 3.00 \(\mathrm{g} / \mathrm{cm}^{2}\) surrounded by a concentric ring of inner radius \(50.0 \mathrm{cm},\) outer radius \(70.0 \mathrm{cm},\) and area density 2.00 \(\mathrm{g} / \mathrm{cm}^{2} .\) Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

\(\bullet\) A solid uniform sphere and a uniform spherical shell, both having the same mass and radius, roll without slipping down a hill that rises at an angle \(\theta\) above the horizontal. Both spheres start from rest at the same vertical height \(h\) . (a) How fast is each sphere moving when it reaches the bottom of the hill? (b) Which sphere will reach the bottom first, the hollow one or the solid one?

\(\bullet\) 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?

\(\bullet\) 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 string is wrapped several times around the rim of a small hoop with a radius of 0.0800 \(\mathrm{m}\) and a mass of 0.180 \(\mathrm{kg}\) . If the free end of the string is held in place and the hoop is released from rest (see Figure 9.30), calculate the angular speed of the rotating hoop after it has descended 0.750 \(\mathrm{m} .\)

\(\bullet\) 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. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half has no friction because it is coated with oil. (a) How far up the smooth side will the marble go, measured vertically from the bottom? (b) How high would the marble go if both sides were as rough as the left side? (c) How do you account for the fact that the marble goes higher with friction on the right side than without friction?

3\. An apparatus for launching a small boat consists of a 150.0 kg cart that 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 at an angle of \(7.50^{\circ}\) to the horizontal, and the boat's mass is 750.0 kg. If the boat is released from rest a distance of 16.0 \(\mathrm{m}\) from the water (measured along the slope), how fast will it be mov- ing when it reaches the water? Assume the wheels roll without slipping, and that there is no energy loss due to friction.

\(\bullet\) 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 \(\mathrm{A} 392\) -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 rad/s. The radius of the wheel is \(0.600 \mathrm{m},\) and its moment of inertia about its rotation axis is 0.800\(M R^{2} .\) 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 \(h .\)

Odometer. 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 inches 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 \(5,000\) revolutions? (c) Suppose you put oversize 28 -inch-diameter tires on your car. How many miles will you really have driven when your odometer reads 500 miles?

\(\bullet\) When a toy car is rapidly scooted across the floor, it stores energy in a flywheel. The car has mass \(0.180 \mathrm{kg},\) and its fly- wheel has moment of inertia \(4.00 \times 10^{-5} \mathrm{kg} \cdot \mathrm{m}^{2} .\) The car is 15.0 \(\mathrm{cm}\) long. An advertisement claims that the car can travel at a scale speed of up to 700 \(\mathrm{km} / \mathrm{h}(440 \mathrm{mi} / \mathrm{h}) .\) The scale speed is the speed of the toy car multiplied by the ratio of the length of an actual car to the length of the toy. Assume a length of 3.0 \(\mathrm{m}\) for a real car. (a) For a scale speed of \(700 \mathrm{km} / \mathrm{h},\) what is the actual translational speed of the car? (b) If all the kinetic energy that is initially in the flywheel is converted to the translational kinetic energy of the toy, how much energy is originally stored in the flywheel? (c) What ini- tial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

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?

\(\bullet\) 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 9.32 ). 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 rad/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 moment 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 part of mass \(M\) with a hollow spherical shell of the same size. If both parts must spin at the same rate about an axis through their center, and the new part must have the same kinetic energy as the old one, what must be the mass of the new part in terms of \(M ?\)

A solid uniform spherical stone starts moving from rest at the top of a hill. At the bottom of the hill the ground curves upward, launching the stone vertically a distance \(H\) below its start. How high will the stone go (a) if there is no friction on the hill and (b) if there is enough friction on the hill for the stone to roll without slipping? (c) Why do you get two different answers even though the stone starts with the same gravitational potential energy in both cases?

\(\bullet\) The kinetic energy of walking. If a person of mass \(M\) simply moved forward with speed \(V,\) his kinetic energy would be \(\frac{1}{2} M V^{2} .\) However, in addition to possessing a forward motion, various parts of his body (such as the arms and legs) undergo rotation. Therefore, his total kinetic energy is the sum of the energy from his forward motion plus the rotational kinetic energy of his arms and legs. The purpose of this problem is to see how much this rotational motion contributes to the person's kinetic energy. Biomedical measurements show that the arms and hands together typically make up 13\(\%\) of a person's mass, while the legs and feet together account for 37\(\% .\) For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. In a brisk walk, the arms and legs each move through an angle of about \(\pm 30^{\circ}\) (a total of \(60^{\circ}\) ) from the vertical in approximately 1 second. We shall assume that they are held straight, rather than beingbent, which is not quite true. Let us consider a 75 kg person walking at 5.0 \(\mathrm{km} / \mathrm{h}\) , having arms 70 \(\mathrm{cm}\) long and legs 90 \(\mathrm{cm}\) long. (a) What is the avertage angular velocity of his arms and legs? (b) Using the average angular velocity from part (a), calculate the amount of rotational kinetic energy in this person's arms and legs as he walks. (c) What is the total kinetic energy due to both his forward motion and his rotation? (d) What percentage of his kinetic energy is due to the rotation of his legs and arms?