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Chapter 8

ROTATIONAL MOTION

Educators

AH

Problem 1

(I) Express the following angles in radians: $(a)$ 45.0$^{\circ}$, $(b)$ 60.0$^{\circ}$, $(c)$ 90.0$^{\circ}$, $(d)$ 360.0$^{\circ}$, and $(e)$ 445$^{\circ}$. Give as numerical values and as fractions of $\pi$.

AH
Averell H.
Carnegie Mellon University

Problem 2

(I) The Sun subtends an angle of about 0.5$^{\circ}$ to us on Earth, 150 million km away. Estimate the radius of the Sun.

AH
Averell H.
Carnegie Mellon University

Problem 3

(I) A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle $\theta$ (Fig. 8-40) of $1.4 \times 10^{-5}$ rad. What diameter spot will it make on the Moon?

AH
Averell H.
Carnegie Mellon University

Problem 4

(I) The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 4.0 s. What is the angular acceleration as the blades slow down?

AH
Averell H.
Carnegie Mellon University

Problem 5

(II) The platter of the $\textbf{hard drive}$ of a computer rotates at 7200 rpm (rpm $=$ revolutions per minute $=$ rev/min). $(a)$ What is the angular velocity (rad/s) of the platter? $(b)$ If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? $(c)$ If a single bit requires 0.50 $\mu$m of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

AH
Averell H.
Carnegie Mellon University

Problem 6

(II) A child rolls a ball on a level floor 3.5 m to another child. If the ball makes 12.0 revolutions, what is its diameter?

AH
Averell H.
Carnegie Mellon University

Problem 7

(II) $(a)$ A grinding wheel 0.35 m in diameter rotates at 2200 rpm. Calculate its angular velocity in rad/s. $(b)$ What are the linear speed and acceleration of a point on the edge of the grinding wheel?

AH
Averell H.
Carnegie Mellon University

Problem 8

(II) A bicycle with tires 68 cm in diameter travels 9.2 km. How many revolutions do the wheels make?

AH
Averell H.
Carnegie Mellon University

Problem 9

(II) Calculate the angular velocity $(a)$ of a clock's second hand, $(b)$ its minute hand, and $(c)$ its hour hand. State in rad/s. $(d)$ What is the angular acceleration in each case?

AH
Averell H.
Carnegie Mellon University

Problem 10

(II) A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8-41). $(a)$ What is the linear speed of a child seated 1.2 m from the center? $(b)$ What is her acceleration (give components)?

AH
Averell H.
Carnegie Mellon University

Problem 11

(II) What is the linear speed, due to the Earth's rotation, of a point $(a)$ on the equator, $(b)$ on the Arctic Circle (latitude 66.5$^{\circ}$ N), and (c) at a latitude of 42.0$^{\circ}$ N?

AH
Averell H.
Carnegie Mellon University

Problem 12

(II) Calculate the angular velocity of the Earth $(a$) in its orbit around the Sun, and $(b)$ about its axis.

AH
Averell H.
Carnegie Mellon University

Problem 13

(II) How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 g's?

AH
Averell H.
Carnegie Mellon University

Problem 14

(II) A 61-cm-diameter wheel accelerates uniformly about its center from 120 rpm to 280 rpm in 4.0 s. Determine $(a)$ its angular acceleration, and $(b)$ the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating.

AH
Averell H.
Carnegie Mellon University

Problem 15

(II) In traveling to the Moon, astronauts aboard the $Apollo$ spacecraft put the spacecraft into a slow rotation to distribute the Sun's energy evenly (so one side would not become too hot). At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. Think of the spacecraft as a cylinder with a diameter of 8.5 m rotating about its cylindrical axis. Determine $(a)$ the angular acceleration, and $(b)$ the radial and tangential components of the linear acceleration of a point on the skin of the ship 6.0 min after it started this acceleration.

AH
Averell H.
Carnegie Mellon University

Problem 16

(II) A turntable of radius $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1/\omega_2$?

AH
Averell H.
Carnegie Mellon University

Problem 17

(I) An automobile engine slows down from 3500 rpm to 1200 rpm in 2.5 s. Calculate $(a)$ its angular acceleration, assumed constant, and $(b)$ the total number of revolutions the engine makes in this time.

AH
Averell H.
Carnegie Mellon University

Problem 18

(I) A centrifuge accelerates uniformly from rest to 15,000 rpm in 240 s. Through how many revolutions did it turn in this time?

AH
Averell H.
Carnegie Mellon University

Problem 19

(I) Pilots can be tested for the stresses of flying high-speed jets in a whirling "human centrifuge," which takes 1.0 min to turn through 23 complete revolutions before reaching its final speed. $(a)$ What was its angular acceleration (assumed constant), and $(b)$ what was its final angular speed in rpm?

AH
Averell H.
Carnegie Mellon University

Problem 20

(II) A cooling fan is turned off when it is running at 850 rev/min. It turns 1250 revolutions before it comes to a stop. $(a)$ What was the fan's angular acceleration, assumed constant? $(b)$ How long did it take the fan to come to a complete stop?

AH
Averell H.
Carnegie Mellon University

Problem 21

(II) A wheel 31 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.8 s. How far will a point on the edge of the wheel have traveled in this time?

AH
Averell H.
Carnegie Mellon University

Problem 22

(II) The tires of a car make 75 revolutions as the car reduces its speed uniformly from 95 km/h to 55 km/h. The tires have a diameter of 0.80 m. $(a)$ What was the angular acceleration of the tires? If the car continues to decelerate at this rate, $(b)$ how much more time is required for it to stop, and $(c)$ how far does it go?

AH
Averell H.
Carnegie Mellon University

Problem 23

(II) A small rubber wheel is used to drive a large pottery wheel. The two wheels are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of $7.2 rad/s^2$, and it is in contact with the pottery wheel (radius 27.0 cm) without slipping. Calculate $(a)$ the angular acceleration of the pottery wheel, and $(b)$ the time it takes the pottery wheel to reach its required speed of 65 rpm.

AH
Averell H.
Carnegie Mellon University

Problem 24

(I) A 52-kg person riding a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. $(a)$ What is the maximum torque she exerts? $(b)$ How could she exert more torque?

AH
Averell H.
Carnegie Mellon University

Problem 25

(II) Calculate the net torque about the axle of the wheel shown in Fig. 8-42. Assume that a friction torque of 0.60 m$\cdot$N opposes the motion.

AH
Averell H.
Carnegie Mellon University

Problem 26

(II) A person exerts a horizontal force of 42 N on the end of a door 96 cm wide. What is the magnitude of the torque if the force is exerted $(a)$ perpendicular to the door and $(b)$ at a 60.0$^{\circ}$ angle to the face of the door?

AH
Averell H.
Carnegie Mellon University

Problem 27

(II) Two blocks, each of mass $m$, are attached to the ends of a massless rod which pivots as shown in Fig. 8-43. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system when it is first released.

AH
Averell H.
Carnegie Mellon University

Problem 28

(II) The bolts on the cylinder head of an engine require tightening to a torque of 95 m$\cdot$N. If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm across (Fig. 8-44), estimate the force applied near each of the six
points by a wrench.

AH
Averell H.
Carnegie Mellon University

Problem 29

(II) Determine the net torque on the 2.0-m-long uniform beam shown in Fig. 8-45. All forces are shown. Calculate about $(a)$ point C, the $_{CM}$, and $(b)$ point P at one end.

AH
Averell H.
Carnegie Mellon University

Problem 30

(I) Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center.

AH
Averell H.
Carnegie Mellon University

Problem 31

(I) Estimate the moment of inertia of a bicycle wheel 67 cm in diameter. The rim and tire have a combined mass of 1.1 kg. The mass of the hub (at the center) can be ignored (why?).

AH
Averell H.
Carnegie Mellon University

Problem 32

(II) A merry-go-round accelerates from rest to 0.68 rad/s in 34 s. Assuming the merry-go-round is a uniform disk of radius 7.0 m and mass 31,000 kg, calculate the net torque required to accelerate it.

AH
Averell H.
Carnegie Mellon University

Problem 33

(II) An oxygen molecule consists of two oxygen atoms whose total mass is $5.3 \times 10^{-26} kg$ and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $1.9 \times 10^{-46} kg\cdot m^2$. From these data, estimate the effective distance between the atoms.

AH
Averell H.
Carnegie Mellon University

Problem 34

(II) A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.380 kg. Calculate $(a)$ its moment of inertia about its center, and $(b)$ the applied torque needed to accelerate it from rest to 1750 rpm in 5.00 s. Take into account a frictional torque that has been measured to slow down the wheel from 1500 rpm to rest in 55.0 s.

AH
Averell H.
Carnegie Mellon University

Problem 35

(II) The forearm in Fig. 8-46 accelerates a 3.6-kg ball at $7.0 m/s^2$ by means of the triceps muscle, as shown. Calculate $(a)$ the torque needed, and $(b)$ the force that must be exerted by the triceps muscle. Ignore the mass of the arm.

AH
Averell H.
Carnegie Mellon University

Problem 36

(II) Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8-46. The ball is accelerated uniformly from rest to 8.5 m/s in 0.38 s, at which point it is released. Calculate $(a)$ the angular acceleration of the arm, and $(b)$ the force required of the triceps muscle. Assume that the forearm has a mass of 3.7 kg and rotates like a uniform rod about an axis at its end.

AH
Averell H.
Carnegie Mellon University

Problem 37

(II) A softball player swings a bat, accelerating it from rest to 2.6 rev/s in a time of 0.20 s. Approximate the bat as a 0.90-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.

AH
Averell H.
Carnegie Mellon University

Problem 38

(II) A small 350-gram ball on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculate $(a)$ the moment of inertia of the ball about the center of the circle, and $(b)$ the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore air resistance on the rod and its moment of inertia.

AH
Averell H.
Carnegie Mellon University

Problem 39

(II) Calculate the moment of inertia of the array of point objects shown in Fig. 8-47 about $(a)$ the $y$ axis, and $(b)$ the $\chi$ axis. Assume $m = 2.2 kg$, $M = 3.4 kg$, and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the x axis. $(c)$ About which axis would it be harder to accelerate this array?

AH
Averell H.
Carnegie Mellon University

Problem 40

(II) A potter is shaping a bowl on a potter's wheel rotating at constant angular velocity of 1.6 rev/s (Fig. 8-48). The friction force between her hands and the clay is 1.5 N total. $(a)$ How large is her torque on the wheel, if the diameter of the bowl is 9.0 cm? $(b)$ How long would it take for the potter's wheel to stop if the only torque acting on it is due to the potter's hands? The moment of inertia of the wheel and the bowl is $0.11 kg\cdot m^2$.

AH
Averell H.
Carnegie Mellon University

Problem 41

(II) A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 560 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?

AH
Averell H.
Carnegie Mellon University

Problem 42

(II) A 0.72-m-diameter solid sphere can be rotated about an axis through its center by a torque of 10.8 m$\cdot$N which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere?

AH
Averell H.
Carnegie Mellon University

Problem 43

(II) Let us treat a helicopter rotor blade as a long thin rod, as shown in Fig. 8-49. $(a)$ If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. $(b)$ How much torque must the motor apply to bring the blades from rest up to a speed of 6.0 rev/s in 8.0 s?

AH
Averell H.
Carnegie Mellon University

Problem 44

(II) A centrifuge rotor rotating at 9200 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.20 m$\cdot$N. If the mass of the rotor is 3.10 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest, and how long will it take?

AH
Averell H.
Carnegie Mellon University

Problem 45

(II) To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8-50. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 32 rpm in 5.0 min, starting from rest?

AH
Averell H.
Carnegie Mellon University

Problem 46

(III) Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia $I$. The blocks move (towards the right) with an acceleration of $1.00 m/s^2$ along their frictionless inclines (see Fig. 8-51). $(a)$ Draw free-body diagrams for each of the two blocks and the pulley. $(b)$ Determine $F_{TA}$ and $F_{TB}$, the tensions in the two parts of the string. $(c)$ Find the net torque acting on the pulley, and determine its moment of inertia, $I$.

AH
Averell H.
Carnegie Mellon University

Problem 47

(III) An Atwood machine consists of two masses, $m_A = 65 kg$ and $m_B = 75 kg$, connected by a massless inelastic cord that passes over a pulley free to rotate, Fig. 8-52. The pulley is a solid cylinder of radius $R = 0.45 m$ and mass 6.0 kg. $(a)$ Determine the acceleration of each mass. (b) What % error would be made if the moment of inertia of the pulley is ignored? [Hint: The tensions $F_{TA}$ and $F_{TB}$ are not equal. We discussed the Atwood machine in Example 4-13, assuming $I = 0$ for the pulley.]

AH
Averell H.
Carnegie Mellon University

Problem 48

(III) A hammer thrower accelerates the hammer $(mass = 7.30 kg)$ from rest within four full turns (revolutions) and releases it at a speed of 26.5 m/s. Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate $(a)$ the angular acceleration, $(b)$ the (linear) tangential acceleration, $(c)$ the centripetal acceleration just before release, $(d)$ the net force being exerted on the hammer by the athlete just before release, and $(e)$ the angle of this force with respect to the radius of the circular motion. Ignore gravity.

AH
Averell H.
Carnegie Mellon University

Problem 49

(I) An automobile engine develops a torque of 265 m$\cdot$N at 3350 rpm. What is the horsepower of the engine?

AH
Averell H.
Carnegie Mellon University

Problem 50

(I) A centrifuge rotor has a moment of inertia of $3.25 \times 10^{-2} kg\cdot m^2$. How much energy is required to bring it from rest to 8750 rpm?

AH
Averell H.
Carnegie Mellon University

Problem 51

(I) Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.20 m high. Assume it starts from rest and rolls without slipping.

AH
Averell H.
Carnegie Mellon University

Problem 52

(II) A bowling ball of mass 7.25 kg and radius 10.8 cm rolls without slipping down a lane 3.10 m/s. at Calculate its total kinetic energy.

AH
Averell H.
Carnegie Mellon University

Problem 53

(II) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, $(a)$ that due to its daily rotation about its axis, and $(b)$ that due to its yearly revolution about the Sun. [Assume the Earth is a uniform sphere with $mass = 6.0 \times 10^{24} kg$, $radius = 6.4 \times 10^6 m$, and is $1.5 \times 10^8 km$ from the Sun.]

AH
Averell H.
Carnegie Mellon University

Problem 54

(II) A rotating uniform cylindrical platform of mass 220 kg and radius 5.5 m slows down from 3.8 rev/s to rest in 16 s when the driving motor is disconnected. Estimate the power output of the motor (hp) required to maintain a steady speed of 3.8 rev/s.

AH
Averell H.
Carnegie Mellon University

Problem 55

(II) A merry-go-round has a mass of 1440 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 7.00 s? Assume it is a solid cylinder.

AH
Averell H.
Carnegie Mellon University

Problem 56

(II) A sphere of radius $r = 34.5 cm$ and mass $m = 1.80 kg$ starts from rest and rolls without slipping down a 30.0$^{\circ}$ incline that is 10.0 m long. $(a)$ Calculate its translational and rotational speeds when it reaches the bottom. $(b)$ What is the ratio of translational to rotational kinetic energy at the bottom? Avoid putting in numbers until the end so you can answer: $(c)$ do your answers in $(a)$ and $(b)$ depend on the radius of the sphere or its mass?

AH
Averell H.
Carnegie Mellon University

Problem 57

(II) A ball of radius r rolls on the inside of a track of radius R (see Fig. 8-53). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping?

AH
Averell H.
Carnegie Mellon University

Problem 58

(II) Two masses, $m_A = 32.0 kg$ and $m_B = 38.0 kg$, are connected by a rope that hangs over a pulley (as in Fig. 8-54). The pulley is a uniform cylinder of radius $R = 0.311 m$ and $mass 3.1 kg$. Initially $m_A$ is on the ground and $m_B$ rests 2.5 m above the ground. If the system is released, use conservation of energy to determine the speed of just before it strikes the $m_B$ ground. Assume the pulley bearing is frictionless.

AH
Averell H.
Carnegie Mellon University

Problem 59

(III) A 1.80-m-long pole is balanced vertically with its tip on the ground. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [$Hint$: Use conservation of energy.]

AH
Averell H.
Carnegie Mellon University

Problem 60

(I) What is the angular momentum of a 0.270-kg ball revolving on the end of a thin string in a circle of radius 1.35 m at an angular speed of 10.4 rad/s?

AH
Averell H.
Carnegie Mellon University

Problem 61

(I) $(a)$ What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 28 cm when rotating at 1300 rpm? $(b)$ How much torque is required to stop it in 6.0 s?

AH
Averell H.
Carnegie Mellon University

Problem 62

(II) A person stands, hands at his side, on a platform that is rotating at a rate of 0.90 rev/s. If he raises his arms to a horizontal position, Fig. 8-55, the speed of rotation decreases to 0.60 rev/s. $(a)$ Why? $(b)$ By what factor has his moment of inertia changed?

AH
Averell H.
Carnegie Mellon University

Problem 63

(II) A nonrotating cylindrical disk of moment of inertia $I$ is dropped onto an identical disk rotating at angular speed $\omega $. Assuming no external torques, what is the final common angular speed of the two disks?

AH
Averell H.
Carnegie Mellon University

Problem 64

(II) A diver (such as the one shown in Fig. 8-28) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed (rev/s) when in the straight position?

AH
Averell H.
Carnegie Mellon University

Problem 65

(II) A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of 2.5 rev/s. If her initial moment of inertia was $4.6 kg\cdot m^2$, what is her final moment of inertia? How does she physically accomplish this change?

AH
Averell H.
Carnegie Mellon University

Problem 66

(II) $(a)$ What is the angular momentum of a figure skater spinning at 3.0 rev/s with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 48 kg? $(b)$ How much torque is required to slow her to a stop in 4.0 s, assuming she does $not$ move her arms?

AH
Averell H.
Carnegie Mellon University

Problem 67

(II) A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia $820 kg\cdot m^2$. The platform rotates without friction with angular velocity 0.95 rad/s. The person walks radially to the edge of the platform. $(a)$ Calculate the angular
velocity when the person reaches the edge. $(b)$ Calculate the rotational kinetic energy of the system of platform plus person before and after the person's walk.

AH
Averell H.
Carnegie Mellon University

Problem 68

(II) A potter's wheel is rotating around a vertical axis through its center at a frequency of 1.5 rev/s. The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 2.6-kg chunk of clay, approximately shaped as a flat disk of radius 7.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it? Ignore friction.

AH
Averell H.
Carnegie Mellon University

Problem 69

(II) A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0.80 rad/s. Its total moment of inertia is $1360 kg\cdot m^2$. Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. $(a)$ What is the angular velocity of the merry-go-round now? $(b)$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

AH
Averell H.
Carnegie Mellon University

Problem 70

(II) A uniform horizontal rod of mass $M$ and length $\ell$ rotates with angular velocity $\omega$ about a vertical axis through its center. Attached to each end of the rod is a small mass $m$. Determine the angular momentum of the system about the axis.

AH
Averell H.
Carnegie Mellon University

Problem 71

(II) Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, $(a)$ what would the Sun's new rotation rate be? Take the Sun's current period to be about 30 days. $(b)$ What would be its final kinetic energy in terms of its initial kinetic energy of today?

AH
Averell H.
Carnegie Mellon University

Problem 72

(II) A uniform disk turns at 3.3 rev/s around a frictionless central axis. A nonrotating rod, of the same mass as the disk and length equal to the disk's diameter, is dropped onto the freely spinning disk, Fig. 8-56. They then turn together around the axis with their centers superposed. What is the angular
frequency in rev/s of the combination?

AH
Averell H.
Carnegie Mellon University

Problem 73

(III) An asteroid of mass $1.0 \times 10^5 kg$, traveling at a speed of 35 km/s relative to the Earth, hits the Earth at the equator tangentially, in the direction of Earth's rotation, and is embedded there. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.

AH
Averell H.
Carnegie Mellon University

Problem 74

(III) Suppose a 65-kg person stands at the edge of a 5.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of $1850 kg\cdot m^2$. The turntable is at rest initially, but when the person begins running at a speed of 4.0 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.

AH
Averell H.
Carnegie Mellon University

Problem 75

A merry-go-round with a moment of inertia equal to $1260 kg\cdot m^2$ and a radius of 2.5 m rotates with negligible friction at 1.70 rad/s. A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation, causing the platform to slow to 1.35 rad/s. What is her mass?

AH
Averell H.
Carnegie Mellon University

Problem 76

A 1.6-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of 24 rev/s from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.

AH
Averell H.
Carnegie Mellon University

Problem 77

On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius $R_1 = 2.5 cm$ and winds its way out to radius $R_2 = 5.8 cm$. To read the digital information, a CD player rotates the CD so that the player's readout laser scans along the spiral's sequence of bits at a constant linear speed of 1.25 m/s. Thus the player must accurately adjust the rotational frequency $f$ of the CD as the laser moves outward.
Determine the values for $f$ (in units of rpm) when the laser is located at $R_1$ and when it is at $R_2$.

AH
Averell H.
Carnegie Mellon University

Problem 78

$(a)$ A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.013 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest. $(b)$ What fraction of its kinetic energy is rotational?

AH
Averell H.
Carnegie Mellon University

Problem 79

A cyclist accelerates from rest at a rate of $1.00 m/s^2$. How fast will a point at the top of the rim of the tire $(diameter = 68.0 cm)$ be moving after 2.25 s? [$Hint$: At any moment, the lowest point on the tire is in contact with the ground and is at rest-see Fig. 8-57.

AH
Averell H.
Carnegie Mellon University

Problem 80

Suppose David puts a 0.60-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle, accelerating it from rest to a rate of 75 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?

AH
Averell H.
Carnegie Mellon University

Problem 81

$\textbf{Bicycle gears:}$ $(a)$ How is the angular velocity $\omega_R$ of the rear wheel of a bicycle related to the angular velocity $\omega_F$ of the front sprocket and pedals? Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively, Fig. 8-58. The teeth are spaced the same on both sprockets and the rear sprocket is firmly attached to the rear wheel. $(b)$ Evaluate the ratio $\omega_R/\omega_F$ when the front and rear sprockets have 52 and 13 teeth, respectively, and $(c)$ when they have 42 and 28 teeth.

AH
Averell H.
Carnegie Mellon University

Problem 82

Figure 8-59 illustrates an $H_2O$ molecule. The $O-H$ bond length is 0.096 nm and the $H-O-H$ bonds make an angle of 104$^{\circ}$. Calculate the moment of inertia of the $H_2O$ molecule (assume the atoms are points) about an axis passing through the center of the oxygen atom $(a)$ perpendicular to the plane of the molecule, and $(b)$ in the plane of the molecule, bisecting the $H-O-H$ bonds.

AH
Averell H.
Carnegie Mellon University

Problem 83

A hollow cylinder (hoop) is rolling on a horizontal surface at speed $\upsilon = 3.0 m/s$ when it reaches a 15$^{\circ}$ incline. $(a)$ How far up the incline will it go? $(b)$ How long will it be on the incline before it arrives back at the bottom?

AH
Averell H.
Carnegie Mellon University

Problem 84

Determine the angular momentum of the Earth $(a)$ about its rotation axis (assume the Earth is a uniform sphere), and $(b)$ in its orbit around the Sun (treat the Earth as a particle orbiting the Sun).

AH
Averell H.
Carnegie Mellon University

Problem 85

A wheel of mass $M$ has radius $R$. It is standing vertically on the floor, and we want to exert a horizontal force $F$ at its axle so that it will climb a step against which it rests (Fig. 8-60). The step has height $h$, where $h < R$. What minimum force $F$ is needed?

AH
Averell H.
Carnegie Mellon University

Problem 86

If the coefficient of static friction between a car's tires and the pavement is 0.65, calculate the minimum torque that must be applied to the 66-cm-diameter tire of a 1080-kg automobile in order to "lay rubber" (make the wheels spin, slipping as the car accelerates). Assume each wheel supports an equal share of the weight.

AH
Averell H.
Carnegie Mellon University

Problem 87

A 4.00-kg mass and a 3.00-kg mass are attached to opposite ends of a very light 42.0-cm-long horizontal rod (Fig. 8-61). The system is rotating at angular speed $\omega = 5.60$ rad/s about a vertical axle at the center of the rod. Determine $(a)$ the kinetic energy $_{KE}$ of the system, and $(b)$ the net force on each mass.

AH
Averell H.
Carnegie Mellon University

Problem 88

A small mass $m$ attached to the end of a string revolves in a circle on a frictionless tabletop. The other end of the string passes through a hole in the table (Fig. 8-62). Initially, the mass revolves with a speed $\upsilon_1 = 2.4 m/s$ in a circle of radius $r_1 = 0.80 m$. The string is then pulled slowly through the hole so that the radius is reduced to $r_2 = 0.48 m$. What is the speed, $\upsilon_2$ , of the mass now?

AH
Averell H.
Carnegie Mellon University

Problem 89

A uniform rod of mass $M$ and length $\ell$ can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8-63. The rod is held horizontally and then released. At the moment of release, determine $(a)$ the angular acceleration of the rod, and $(b)$ the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [$Hint$: See Fig. 8-20g.]

AH
Averell H.
Carnegie Mellon University

Problem 90

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing \(\frac{3}{4}\) of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrownoff mass carries off either $(a)$ no angular momentum, or $(b)$ its proportional share $(\frac{3}{4})$ of the initial angular momentum.

AH
Averell H.
Carnegie Mellon University

Problem 91

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance $\ell$, holding onto it, Fig. 8-64. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool's center of mass move?

AH
Averell H.
Carnegie Mellon University

Problem 92

The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moon's spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

AH
Averell H.
Carnegie Mellon University

Problem 93

A spherical asteroid with radius $r = 123 m$ and mass $M = 2.25 \times 10^{10} kg$ rotates about an axis at four revolutions per day. A "tug" spaceship attaches itself to the asteroid's south pole (as defined by the axis of rotation) and fires its engine, applying a force $F$ tangentially to the asteroid's surface as shown in Fig. 8-65. If $F = 285 N$, how long will it take the tug to rotate the asteroid's axis of rotation through an angle of 5.0$^{\circ}$ by this method?

AH
Averell H.
Carnegie Mellon University

Problem 94

Most of our Solar System's mass is contained in the Sun, and the planets possess almost all of the Solar System's angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar System's total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass $1.99 \times 10^{30} kg$, radius $6.96 \times 10^8 m$) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planet's spin about its own axis.

AH
Averell H.
Carnegie Mellon University

Problem 95

Water drives a waterwheel (or turbine) of radius $R = 3.0 m$ as shown in Fig. 8-66. The water enters at a speed $\upsilon_1 = 7.0 m/s$ and exits from the waterwheel at a speed $\upsilon_2 = 3.8 m/s$. $(a)$ If 85 kg of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? $(b)$ What is the torque the water applies to the waterwheel? $(c)$ If the water causes the waterwheel to make one revolution every 5.5 s, how much power is delivered to the wheel?

AH
Averell H.
Carnegie Mellon University

Problem 96

The radius of the roll of paper shown in Fig. 8-67 is 7.6 cm and its moment of inertia is $I = 3.3 \times 10^{-3} kg\cdot m^2$. A force of 3.5 N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque of $0.11 m\cdot N$ is exerted on the roll which gradually brings it to a stop. Assuming that the paper's thickness is negligible, calculate $(a)$ the length of paper that unrolls during the time that the force is applied (1.3 s) and $(b)$ the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving.

AH
Averell H.
Carnegie Mellon University