Chapter 9

(a) What angle in radians is subtended by an arc 1.50 \(\mathrm{m}\) long on the circumference of a circle of radius 2.50 \(\mathrm{m} ?\) What is this angle in degrees? (b) An are 14.0 \(\mathrm{cm}\) long 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?

An airplane propeller is rotating at 1900 TPm (rev/min). a) Compute the propeller's angular velocity in rad/s. (b) How many seconds does it take for the propeller to turn through \(35^{\circ}\) ?

The angular velocity of a fywheel obeys the equation \(\omega_{z}(t)=A+B t^{2},\) where \(t\) is in seconds and \(A\) and \(B\) are constants having numerical values 2.75\((\text { for } A)\) and 1.50\((\text { for } B)\) . (a) What are the units of \(A\) and \(B\) if \(\omega\) is in rad/s? \((b)\) What is the angular acceleration of the wheel at (i) \(t=0.00\) and (ii) \(t=5.00 \mathrm{s} ?\) (c) Through what angle does the flywheel turn during the first 2.00 \(\mathrm{s} ?(\text { Hint: See Section } 2.6 .)\)

A fan blade rotates with angular velocity given by \(\omega_{z}(t)=\) \(\gamma-\beta t^{2},\) where \(\gamma=5.00 \mathrm{rad} / \mathrm{s}\) and \(\beta=0.800 \mathrm{rad} / \mathrm{s}^{3} .\) (a) Calculate the angular acceleration as a function of time. (b) Calculate the instantaneous angular acceleration \(\alpha_{z}\) at \(t=3.00 \mathrm{s}\) and the average angular acceleration \(\alpha_{\mathrm{axz}}\) for the time interval \(t=0\) to \(t=3.00 \mathrm{s}\) . How do these two quantities compare? If they are different, why are they different?

A child is pushing a merry-go-round. The angle through which the merry-go- round has turned varies with time according to \(\theta(t)=\gamma t+\beta t^{3},\) where \(\gamma=0.400 \mathrm{rad} / \mathrm{s}\) and \(\beta=0.0120 \mathrm{rad} / \mathrm{s}^{3}\) . (a) Calculate the angular velocity of the merry-go-round as a function of time. (b) What is the initial value of the angular velocity? (c) Calculate the instantaneous value of the angular velocity \(\omega_{z}\) at \(t=5.00 \mathrm{s}\) and the average angular velocity \(\omega_{\mathrm{av}-\mathrm{z}}\) for the time interval \(t=0\) to \(t=5.00 \mathrm{s}\) . Show that \(\omega_{\mathrm{av}-\mathrm{z}}\) is not equal to the average of the instantaneous angular velocities at \(t=0\) and \(t=5.00 \mathrm{s},\) and explain why it is not.

At \(t=0\) the current to a de electric motor is reversed, resulting in an angular displacement of the motor shaft given by \(\theta(t)=(250 \mathrm{rad} / \mathrm{s}) t-\left(20.0 \mathrm{rad} / \mathrm{s}^{2}\right) t^{2}-\left(1.50 \mathrm{rad} / \mathrm{s}^{3}\right) t^{3} .(\mathrm{a}) \mathrm{At}\) what time is the angular velocity of the motor shaft zero? (b) Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity. (c) How many revolutions does the motor shaft turn through between the time when the current is reversed and the instant when the angular velocity is zero? (d) How fast was the motor shaft rotating at \(t=0,\) when the current was reversed? ( c) Calculate the average angular velocity for the time period from \(t=0\) to the time calculated in part (a).

The angle \(\theta\) through which a disk drive turns is given by \(\theta(t)=a+b t-c t^{3},\) where \(a, b,\) and \(c\) are constants \(t\) is in seconds, and \(\theta\) is in radians. When \(t=0, \theta=\pi / 4\) rad and the angular velocity is \(2.00 \mathrm{rad} / \mathrm{s},\) and when \(t=1.50 \mathrm{s},\) the angular acceleration is 1.25 \(\mathrm{rad} / \mathrm{s}^{2}\) , (a) Find \(a, b,\) and \(c,\) including their units. b) What is the angular acceleration when \(\theta=\pi / 4\) rad? (c) What are \(\theta\) and the angular velocity when the angular acceleration is 3.50 \(\mathrm{rad} / \mathrm{s}^{2} ?\)

A wheel is rotating about an axis that is in the \(z\) -direction. The angular velocity \(\omega_{z}\) is \(-6.00 \mathrm{rad} / \mathrm{s}\) at \(t=0,\) increases linearly with time, and is \(+8.00 \mathrm{m} / \mathrm{s}\) at \(t=7.00 \mathrm{s}\) . We have taken counterclockwise rotation to be positive. (a) Is the angular acceleration during this time interval positive or negative? (b) During what time interval is the speed of the wheel increasing? Decreasing? (c) What is the angular displacement of the wheel at \(t=7.00 \mathrm{s} ?\)

A bicycle wheel has an initial angular velocity of 1.50 \(\mathrm{rad} / \mathrm{s}\) . (a) If its angular acceleration is constant and equal to 0.300 \(\mathrm{rad} / \mathrm{s}^{2}\) , what is its angular velocity at \(t=2.50 \mathrm{s} ?\) (b) Through what angle has the wheel turned between \(t=0\) and \(t=2.50 \mathrm{s} ?\)

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev \(/ \min\) to 200 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 \(4.00-\) - 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)?

The rotating blade of a blender turns with constant angular acceleration 1.50 rad/s \(^{2} .\) (a) How much time does it take to reach an angular velocity of 36.0 rad/s, starting from rest? (b) Through how many revolutions does the blade turn in this time interval?

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

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

A high-speed flywheel in a motor is spinning at 500 \(\mathrm{rpm}\) 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 slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 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 computer disk drive is turned on starting from rest and has constant angular acceleration. If took 0.750 s for the drive to make its second complete revolution, (a) how long did it take to make the first complete revolution, and (b) what is its angular acceleration, in \(\operatorname{rad} / s^{2} ?\)

A safety device brings the blade of a power mower from an initial angular speed of \(\omega_{1}\) to rest in 1.00 revolution. At the same constant acceleration, how many revolutions would it take the blade to come to rest from an initial angular speed \(\omega_{3}\) that was three times as great, \(\omega_{3}=3 \omega_{1} ?\)

A straight piece of reflecting tape extends from the center of a wheel to its rim. You darken the room and use a camera and strobe unit that flashes once every 0.050 s to take pictures of the wheel as it rotates counterclockwise. You trigger the strobe so that the first flash \((t=0)\) occurs when the tape is horizontal to the right at an angular displacement of zero. For the following situations draw a sketch of the photo you will get for the time exposure over five flashes (at \(t=0,0.050 \mathrm{s}, 0.100 \mathrm{s}, 0.150 \mathrm{s},\) and 0.200 \(\mathrm{s} )\) and graph \(\theta\) versus \(t\) and \(\omega\) versus \(t\) for \(t=0\) to \(t=0.200 \mathrm{s}\) (a) The angular velocity is constant at 10.0 rev \(/ \mathrm{s}\) . (b) The wheel starts from rest with a constant angular acceleration of 25.0 rev \(/ \mathrm{s}^{2}\) . (c) The wheel is rotating at 10.0 rev \(/ \mathrm{s}\) at \(t=0\) and changes angular velocity at a constant rate of \(-50.0 \mathrm{rev} / \mathrm{s}^{2}\) .

At \(t=0\) a grinding wheel has an angular velocity of 24.0 rad/s. It has a constant angular acceleration of 30.0 \(\mathrm{rad} / \mathrm{s}^{2}\) until a circuit breaker trips at \(t=2.00 \mathrm{s}\) . From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. (a) Through what total angle did the wheel turn between \(t=0\) and the time it stopped? (b) At what time did it stop? (c) What was its acceleration as it slowed down?

In a charming 19 thcentury hotel, an old-style elevator is connected to a counter- weight by a cable that passes over a rotaing disk 2.50 \(\mathrm{m}\) in diameter (Fig. 9.28 ). The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it. (a) At bow many rpm must the disk turn to raise the elevator at 25.0 \(\mathrm{cm} / \mathrm{s}\) ? (b) To start the elevator moving, it must be accelerated at \(\frac{1}{8} \mathrm{g}\) . What must be the angular acceleration of the disk, in rad/s'? (c) Through what angle (in radi- ans and degrees) has the disk turned when it has raised the elevator 3.25 \(\mathrm{m}\) between floors?

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits \(10^{-7} \mathrm{m}\) deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 \(\mathrm{mm}\) and 58.0 \(\mathrm{mm}\) , respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 \(\mathrm{m} / \mathrm{s}\) . (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a \(\mathrm{CD}\) is 74.0 min. What would be the length of the track on such a maximum-duration \(\mathrm{CD}\) if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum-duration CD during its 74.0 -min playing time? Take the direction of rotation of the dise to be positive.

A wheel of diameter 40.0 \(\mathrm{cm}\) starts from rest and rotates with a constant angular acceleration of 3.00 \(\mathrm{rad} / \mathrm{s}^{2}\) . At the instant the wheel has computed its second revolution, compute the radial acceleration of a point on the rim in two ways: (a) using the relationship \(a_{\mathrm{rad}}=\omega^{2} r\) and \((\mathrm{b})\) from the relationship \(a_{\mathrm{red}}=v^{2} / r\)

Utracentrifuge. Find the required angular speed (in rev/min) of an ultracentrifuge for the radial acceleration of a point 2.50 \(\mathrm{cm}\) from the axis to equal \(400,000 \mathrm{g}\) (that is, \(400,000\) times the acceleration due to gravity).

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 \(60.0^{\circ} ;\) (c) after it has turned through \(120.0^{\circ} .\)

An electric turntable 0.750 \(\mathrm{m}\) in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev \(/ \mathrm{s}^{2} .\) (a) Compute the angular velocity of the turntable after 0.200 \(\mathrm{s}\) . (b) Through how many revolutions has the turntable spun in this time interval? (c) What is the tangential speed of a point on the rim of the turntable at \(t=0.200 \mathrm{s} ?(\mathrm{d})\) What is the magnitude of the resultant acceleration of a point on the rim at \(t=0.200 \mathrm{s} ?\)

Centrifuge. An advertisement claims that a centrifuge takes up only 0.127 \(\mathrm{m}\) of bench space but can produce a radial acceleration of 3000 \(\mathrm{g}\) at 5000 rev/min. Calculate the required radius of the centrifuge. Is the claim realistic?

(a) Derive an equation for the radial acceleration that includes \(v\) and \(\omega,\) but not \(r .\) (b) You are designing a merry-go-round for which a point on the rim will have a radial acceleration of 0.500 \(\mathrm{m} / \mathrm{s}^{2}\) when the tangential velocity of that point has magnitude 2.00 \(\mathrm{m} / \mathrm{s}\) . What angular velocity is required to achieve these values?

Electric Drill. According to the shop manual, when drilling a \(12.7-\mathrm{mm}\) -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 rev \(/ \mathrm{mm}\) , find (a) the maximum linear speed of any part of the hit and (b) the maximum radial acceleration of any part of the bit.

At \(t=3.00 \mathrm{s}\) a point on the rim of a \(0.200-\mathrm{m}\) -radius wheel has a tangential speed of 50.0 \(\mathrm{m} / \mathrm{s}\) as the wheel slows down with a tangential acceleration of constant magnitude 10.0 \(\mathrm{m} / \mathrm{s}^{2}\) . (a) Calculate the wheel's constant angular acceleration. (b) Calculate the angular velocities at \(t=3.00 \mathrm{s}\) and \(t=0\) . (c) Through what angle did the wheel turn between \(t=0\) and \(t=3.00 \mathrm{s} ?\) (d) At what time will the radial acceleration equal \(g ?\)

The spin cycles of a washing machine have two angular speeds, 423 rev/min and 640 rev/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 maxi- mum radial acceleration, in terms of \(g .\)

You are to design a rotating cylindrical axle to lift \(800-\mathrm{N}\) buckets of cement from the ground to a rooftop 78.0 \(\mathrm{m}\) above the ground. The buckets will be attached to a hook on the free end of a cable that wraps around the rim of the axle; as the axle turms, the buckets will rise. (a) What should the diameter of the axle be in order to raise the buckets at a steady 2.00 \(\mathrm{cm} / \mathrm{s}\) when it is turning at 7.5 \(\mathrm{rpm} ?\) (b) If instead the axle must give the buckets an upward acceleration of \(0.400 \mathrm{m} / \mathrm{s}^{2},\) what should the angular acceleration of the axle be?

While riding a multispeed bicycle, the rider can select the radius of the rear sprocket that is fixed to the rear axle. The front sprocket of a bicycle has radius 12.0 \(\mathrm{cm}\) . If the angular speed of the front sprocket is 0.600 rev/s, what is the radius of the rear sprocket for which the tangential speed of a point on the rim of the rear wheel will be 5.00 \(\mathrm{m} / \mathrm{s} ?\) The rear wheel has radius 0.330 \(\mathrm{m} .\)

Small blocks, each with mass \(m\) , are clamped at the ends and at the center of a rod of length \(L\) and negligible mass. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through (a) the center of the rod and (b) a point one-fourth of the length from one end.

A 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 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 perpendicularto the bar through one of the balls; (c) an axis parallel to the bar through both balls; (d) an axis parallel to the bar and 0.500 \(\mathrm{m}\) m from it.

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 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.

An airplane propeller is 2.08 \(\mathrm{m}\) in length (from tip to tip) with mass 117 \(\mathrm{kg}\) and is rotating at 2400 \(\mathrm{rpm}(\mathrm{rev} / \mathrm{min})\) 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?

You need to design an industrial turntable that is 60.0 \(\mathrm{cm}\) in diameter and has a kinetic energy of 0.250 \(\mathrm{J}\) when turning at 45.0 \(\mathrm{rpm}(\mathrm{rev} / \mathrm{min})\) . (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?

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?

A light, flexible rope is wrapped several times around a hollow cylinder, with a weight of 40.0 \(\mathrm{N}\) and a radius of 0.25 \(\mathrm{m}\) , that rotates without friction about a fixed horizontal axis. The cylinder is attached to the axle by spokes of a negligible moment of inertia. The cylinder is initially at rest. The free end of the rope is pulled with a constant force \(P\) for a distance of \(5.00 \mathrm{m},\) at which point the end of the rope is moving at 6.00 \(\mathrm{m} / \mathrm{s}\) . If the rope does not slip on the cylinder, what is the value of \(P ?\)

Energy is to be stored in a 70.0 \(\mathrm{kg}\) flywheel in the shape of a uniform solid disk with radius \(R=1.20 \mathrm{m}\) . To prevent structural failure of the flywheel, the maximum allowed radial acceleration of a point on its \(\mathrm{rm}\) is 3500 \(\mathrm{m} / \mathrm{s}^{2} .\) What is the maximum kinetic energy that can be stored in the flywheel?

A frictionless pulley has the shape of a uniform solid disk of mass 2.50 \(\mathrm{kg}\) and radius 20.0 \(\mathrm{cm}\) . A 1.50 \(\mathrm{kg}\) stone is attached to a very light wire that is wrapped around the rim of the pulley (Fig. 9.32\()\) , and the system is released from rest. (a) How far must the stone fall so that the pulley has 4.50 \(\mathrm{J}\) of kinetic energy? (b) What percent of the total kinetic energy does the pulley have?

A bucket of mass \(m\) is tied to a massless cable that is wrapped around the outer rim of a frictionless uniform pulley of radius \(R\) , similar to the system shown in Fig. \(9.32 .\) In terms of the stated variables, what must be the moment of inertia of the pulley so that it always has half as much kinetic energy as the bucket?

How I Scales. If we multiply all the design dimensions of an object by a scaling factor \(f\) , its volume and mass will be multiplied by \(f^{3}\) . (a) By what factor will its moment of inertia be multiplied? (b) If a \(\frac{1}{48}\) - scale model has a rotational kinetic energy of \(2.5 \mathrm{J},\) what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

A uniform \(2.00-\mathrm{m}\) ladder of mass 9.00 \(\mathrm{kg}\) is leaning against a vertical wall while making an angle of \(53.0^{\circ}\) with the floor. A worker pushes the ladder up against the wall until it is vertical. How much work did this person do against gravity?

A uniform \(3.00-\mathrm{kg}\) rope 24.0 \(\mathrm{m}\) lies on the ground at the mtop of a vertical cliff. A mountain climber at the top lets down half of it to help his partner climb up the cliff. What was the change in potential energy of the rope during this maneuver?

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass \(M\) and radius \(R\) about an axis perpendicular to the hoop's plane at an edge.

About what axis will a uniform, balsa-wood sphere have the same moment of inertia as does a thin-walled, hollow, lead sphere of the same mass and radius, with the axis along a diameter?

A thin, rectangular sheet of metal has mass \(M\) and sides of length \(a\) and \(b\) . Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one comer of the sheet.

A thin uniform rod of mass \(M\) and length \(L\) is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet and (b) the midpoint of the line connecting its two ends.

Trip to Mars. You are working on a project with NASA to launch a rocket to Mars, with the rocket blasting off from earth when earth and Mars are aligned along a straight line from the sun. If Mars is now \(60^{\circ}\) shead of earth in its orbit around the sun, when should you launch the rocket? (Note: All the planets orbit the sun in the same direction, 1 year on Mars is 1.9 earth- years, and assume circular orbits for both planets.)