The sweeping second hand on your wall clock is 20 cm long.

What is (a) the rotational speed of the second hand, (b) the

translational speed of the tip of the second hand, and (c) the

rotational acceleration of the second hand?

Vedad B.

Numerade Educator

You find an old record player in your attic. The turntable

has two readings: 33 rpm and 45 rpm. What do they mean?

Express these quantities in different units.

Salamat A.

Numerade Educator

Consider again the turntable described in the last problem. Determine the magnitudes of the rotational acceleration in each of the following situations. Indicate the assumptions you made for

each case. (a) When on and rotating at 33 rpm, it is turned off and slows and stops in 60 s. (b) When off and you push the play button, the turntable attains a speed of 33 rpm in 15 s. (c) You

switch the turntable from 33 rpm to 45 rpm, and it takes about 2.0 s for the speed to change. (d) In the situation in part (c), what is the magnitude of the average tangential acceleration of a point

on the turntable that is 15 cm from the axis of rotation?

Vedad B.

Numerade Educator

You step on the gas pedal in your car, and the car engine's

rotational speed changes from 1200 $\mathrm{rpm}$ to 3000 $\mathrm{rpm}$ in 3.0 $\mathrm{s}$

What is the engine's average rotational acceleration?

Salamat A.

Numerade Educator

You pull your car into your driveway and stop. The drive shaft

of your car engine, initially rotating at $2400 \mathrm{rpm},$ slows with a

constant rotational acceleration of magnitude 30 $\mathrm{rad} / \mathrm{s}^{2} .$ How

long does it take for the drive shaft to stop turning?

Vedad B.

Numerade Educator

An old wheat-grinding wheel in a museum actually works.

The sign on the wall says that the wheel has a rotational acceleration of 190 $\mathrm{rad} / \mathrm{s}^{2}$ as its spinning rotational speed increases

from zero to 1800 $\mathrm{rpm} .$ How long does it take the wheel to

attain this rotational speed?

Salamat A.

Numerade Educator

Centrifuge A centrifuge at the same museum is used to separate seeds of different sizes. The average rotational acceleration of the centrifuge according to a sign is 30 $\mathrm{rad} / \mathrm{s}^{2}$ . If starting at rest, what is the rotational velocity of the centrifuge after 10 $\mathrm{s?}$

Vedad B.

Numerade Educator

Potter's wheel A fly sits on a potter's wheel 0.30 $\mathrm{m}$ from its

axle. The wheels rotational speed decreases from 4.0 $\mathrm{rad} / \mathrm{s}$ to 2.0

$\mathrm{rad} / \mathrm{s}$ in 5.0 $\mathrm{s}$ . Determine (a) the wheel's average rotational acceleration, (b) the angle through which the fly turns during the $5.0 \mathrm{s},$

and (c) the distance traveled by the fly during that time interval.

Salamat A.

Numerade Educator

During your tennis serve, your racket and arm move in an approximately rigid arc with the top of the racket 1.5 m from your shoulder joint. The top accelerates from rest to a speed

of 20 m/s in a time interval of 0.10 s. Determine (a) the magnitude of the average tangential acceleration of the top of the racket and (b) the magnitude of the rotational acceleration of

your arm and racket.

Vedad B.

Numerade Educator

An ant clings to the outside edge of the tire of an exercise bicycle. When you start pedaling, the ant's speed increases from zero to 10 $\mathrm{m} / \mathrm{s}$ in 2.5 $\mathrm{s}$ . The wheel's rotational acceleration is 13 $\mathrm{rad} / \mathrm{s}^{2}$ . Determine everything you can about the motion of the wheel and the ant.

Salamat A.

Numerade Educator

The speedometer on a bicycle indicates that you travel 60 $\mathrm{m}$

while your speed increases from 0 to 10 $\mathrm{m} / \mathrm{s}$ . The radius of the

wheel is 0.30 $\mathrm{m}$ . Determine three physical quantities relevant

to this motion.

Vedad B.

Numerade Educator

You peddle your bicycle so that its wheel's rotational speed changes from 5.0 $\mathrm{rad} / \mathrm{s}$ to 8.0 $\mathrm{rad} / \mathrm{s}$ in 2.0 $\mathrm{s}$ . Determine (a) the

wheel's average rotational acceleration, (b) the angle through

which it turns during the $2.0 \mathrm{s},$ and $(\mathrm{c})$ the distance that a

point 0.60 $\mathrm{m}$ from the axle travels.

Salamat A.

Numerade Educator

Mileage gauge The odometer on an automobile actually

counts axle turns and converts the number of turns to miles

based on knowledge that the diameter of the tires is 0.62 m.

How many turns does the axle make when traveling 10 miles?

Vedad B.

Numerade Educator

Speedometer The speedometer on an automobile measures the rotational speed of the axle and converts that to a linear speed of the car, assuming the car has 0.62 -m-diameter tires.

What is the rotational speed of the axle when the car is traveling at 20 $\mathrm{m} / \mathrm{s}(45 \mathrm{mph})$ ?

Salamat A.

Numerade Educator

Ferris wheel A Ferris wheel starts at rest, acquires a rotational velocity of $\omega$ rad/s after completing one revolution and continues to accelerate. Write an expression for (a) the

magnitude of the wheel's rotational acceleration (assumed constant), (b) the time interval needed for the first revolution, (c) the time interval required for the second revolution, and (d) the distance a person travels in two revolutions if he is seated a distance $l$ from the axis of rotation.

Vedad B.

Numerade Educator

You push a disk-shaped platform on its edge 2.0 $\mathrm{m}$ from the

axle. The platform starts at rest and has a rotational acceleration of 0.30 $\mathrm{rad} / \mathrm{s}^{2} .$ Determine the distance you must run while pushing the platform to increase its speed at the edge to 7.0 $\mathrm{m} / \mathrm{s}$ .

Salamat A.

Numerade Educator

Estimate what Earth's rotational acceleration would be

in rad/s' if the length of a day increased from 24 $\mathrm{h}$ to 48 $\mathrm{h}$

during the next 100 years.

Vedad B.

Numerade Educator

A turntable turning at rotational speed 33 $\mathrm{rpm}$ stops in

50 $\mathrm{s}$ when turned off. The turntable's rotational inertia is

$1.0 \times 10^{-2} \mathrm{kg} \cdot \mathrm{m}^{2} .$ How large is the resistive torque that

slows the turntable?

Salamat A.

Numerade Educator

A 0.30 -kg ball is attached at the end of a 0.90 -m-long stick. The ball and stick rotate in a horizontal circle. Because of air resistance and to keep the ball moving at constant speed,

a continual push must be exerted on the stick, causing a $0.036-\mathrm{N} \cdot \mathrm{m}$ torque. Determine the magnitude of the resistive force that the air exerts on the ball opposing its motion. What assumptions did you make?

Vedad B.

Numerade Educator

Centrifuge A centrifuge with a $0.40-\mathrm{kg} \cdot \mathrm{m}^{2}$ rotational inertia has a rotational acceleration of 100 $\mathrm{rad} / \mathrm{s}^{2}$ when the

power is turned on. (a) Determine the minimum torque

that the motor supplies. (b) What time interval is needed

for the centrifuge's rotational velocity to increase from zero

to 5000 $\mathrm{rad} / \mathrm{s} ?$

Salamat A.

Numerade Educator

Airplane turbine What is the average torque needed to accelerate the turbine of a jet engine from rest to a rotational velocity of 160 $\mathrm{rad} / \mathrm{s}$ in 25 $\mathrm{s} ?$ The turbine's rotating parts have a $32-\mathrm{kg} \cdot \mathrm{m}^{2}$ rotational inertia.

Vedad B.

Numerade Educator

The solid two part pulley in Figure $\mathrm{P} 8.22$ initially rotates counterclock wise. Two ropes

pull on the pulley as shown. The inner part has a radius of 1.5a, and the outer part has a radius of

2.0a. (a) Construct a force diagram for the pulley with the origin of the coordinate system at the center of the pulley. (b) Deter of mine the torque produced by each force (including the sign) and the resultant torque exerted on the pulley. (c) Based on the results of part (b), decide on the signs of

the rotational velocity and the rotational acceleration.

Dading C.

Numerade Educator

The flywheel shown in Figure P8.22 is initially rotating clockwise. Determine the relative force that the rope on the right needs to exert on the wheel compared to the force that the left rope exerts on the wheel in order for the wheel’s rotational velocity to (a) remain constant, (b) increase in magnitude, and (c) decrease in magnitude. The outer radius is 2.0$a$ compared to 1.5$a$ for the inner radius.

Kishore M.

Numerade Educator

The flywheel shown in Figure $\mathrm{P} 8.22$ is initially rotating in the clockwise direction. The force that the rope on the right exerts on it is 1.5$T$ and the force that the rope on the left exerts on it is $T$ . Determine the ratio of the maximum radius of the inner circle compared to that of the outer circle in order for the wheel's rotational speed to decrease.

Salamat A.

Numerade Educator

A pulley such as that shown in Figure $P 8.25$ has rotational inertia 10 $\mathrm{kg} \cdot \mathrm{m}^{2}$ . Three ropes wind around different parts of the pulley and exert forces $T_{1 \mathrm{on} \mathrm{w}}=80 \mathrm{N}$ $T_{2 \mathrm{onw}}=100 \mathrm{N},$ and $T_{3 \mathrm{on} \mathrm{w}}=50 \mathrm{N}$ . Determine (a) the rotational acceleration of the pulley and (b) its rotational velocity after 4.0 $\mathrm{s}$ . It starts at rest.

Zachary W.

Numerade Educator

Equation Jeopardy 1 The equation below describes a rotational dynamics situation. Draw a sketch of a situation that is consistent with the equation and construct a word problem for which the equation might be a solution. There are many possibilities.

$$-(2.2 \mathrm{N})(0.12 \mathrm{m})=\left[(1.0 \mathrm{kg})(0.12 \mathrm{m})^{2}\right] \alpha$$

Salamat A.

Numerade Educator

Equation Jeopardy 2 The equation below describes a rotational dynamics situation. Draw a sketch of a situation that is consistent with the equation and construct a word problem for which the equation might be a solution. There are many possibilities.

$$\begin{array}{l}{-(2.0 \mathrm{N})(0.12 \mathrm{m})+(6.0 \mathrm{N})(0.06 \mathrm{m})} \\ {\quad=\left[(1.0 \mathrm{kg})(0.12 \mathrm{m})^{2}\right] \alpha}\end{array}$$

Zachary W.

Numerade Educator

Determine the rotational inertia of the four balls shown in Figure $P 8.28$ about an

axis perpendicular to the paper and passing through point A. The mass of each ball is $m$ .

Ignore the mass of the rods to which the balls are attached.

Salamat A.

Numerade Educator

Repeat the previous

problem for an axis perpendicular to the paper through point B.

Zachary W.

Numerade Educator

Repeat the previous problem for axis BC, which passes

through two of the balls.

Salamat A.

Numerade Educator

Merry-go-round A mechanic needs to replace the motor for a merry-go-round. What torque specifications must the new motor satisfy if the merry-go-round should accelerate

from rest to 1.5 $\mathrm{rad} / \mathrm{s}$ in 8.0 $\mathrm{s}$ ? You can consider the merry go round to be a uniform disk of radius 5.0 $\mathrm{m}$ and mass $25,000 \mathrm{kg}$ .

Vedad B.

Numerade Educator

A small $0.80-$ kg train propelled by a fan engine starts at rest and goes around a circular track with a $0.80-\mathrm{m}$ radius. The fan air exerts a $2.0-\mathrm{N}$ force on the train. Determine (a) the

rotational acceleration of the train and (b) the time interval needed for it to acquire a speed of 3.0 $\mathrm{m} / \mathrm{s}$ . Indicate any assumptions you made.

Check back soon!

The train from the previous problem is moving along the

rails at a constant rotational speed of 5.4 $\mathrm{rad} / \mathrm{s}$ (the fan has

stopped). Determine the time interval that is needed to stop

the train if the wheels lock and the rails exert a 1.8 -N friction

force on the train.

Vedad B.

Numerade Educator

Motor You wish to buy a motor that will be used to lift a 20 -kg bundle of shingles from the ground to the roof of a house. The shingles are to have a $1.5-\mathrm{m} / \mathrm{s}^{2}$ upward acceleration at the start of the lift. The very light pulley on the motor

has a radius of 0.12 $\mathrm{m}$ . Determine the minimum torque that

the motor must be able to provide.

Salamat A.

Numerade Educator

A thin cord is wrapped around a grindstone of radius 0.30 $\mathrm{m}$ and mass 25 $\mathrm{kg}$ supported by bearings that produce negligible friction torque. The cord exerts a steady $20-\mathrm{N}$ tension force on the grindstone, causing it to accelerate from

rest to 60 $\mathrm{rad} / \mathrm{s}$ in 12 $\mathrm{s}$ . Determine the rotational inertia of the

grindstone.

Vedad B.

Numerade Educator

A string wraps around a $6.0-\mathrm{kg}$ wheel of radius 0.20 m. The wheel is mounted on a frictionless horizontal axle at the top of an inclined plane tilted $37^{\circ}$ below the horizontal.

The free end of the string is attached to a $2.0-\mathrm{kg}$ block that

slides down the incline without friction. The block's acceleration while sliding down the incline is 2.0 $\mathrm{m} / \mathrm{s}^{2} .$ (a) Draw separate force diagrams for the wheel and for the block.

(b) Apply Newton's second law (either the translational form or the rotational form) for the wheel and for the block. (c) Determine the rotational inertia for the wheel about its axis of rotation.

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Elena, a black belt in tae kwon do, is experienced in breaking boards with her fist. A high-speed video indicates that her forearm is moving with a rotational speed of 40 rad/s when it reaches the board. The board breaks in 0.0040 s and her arm is moving at 20 rad/s just after breaking the board. Her fist is

0.32 $\mathrm{m}$ from her elbow joint and the rotational inertia of her forearm is 0.050 $\mathrm{kg} \cdot \mathrm{m}^{2} .$ Determine the average force that the board exerts on her fist while breaking the board (equal in magnitude to the force that her fist exerts on the board). Ignore the

gravitational force that Earth exerts on her arm and the force that her triceps muscle exerts on her arm during the break.

Vedad B.

Numerade Educator

Like a yo-yo Sam wraps a string around the outside of a 0.040 -m-radius 0.20 -kg solid

cylinder and uses it like a yoyo (Figure $P 8.38 ) .$ When released, the cylinder accelerates downward at $(2 / 3) g$ . (a) Draw a force diagram for the cylinder and apply the translational

form of Newton's second law to the cylinder in order to determine the force that the

string exerts on the cylinder.(b) Determine the rotational inertia of the solid cylinder.

(c) Apply the rotational form of Newton's second law and determine the cylinder's rotational acceleration. (d) Is your answer to part (c) consistent with the application of $a=r \alpha$ ,

which relates the cylinder's linear acceleration and its rotational acceleration? Explain.

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Fire escape A unique fire escape for a three-story house is shown in Figure P8.39. A $30 \mathrm{kg}$ child grabs a rope wrapped around a heavy flywheel outside a bedroom window. The flywheel is a 0.40 -m-radius uniform disk with a mass of 120 $\mathrm{kg}$ . (a) Make a force diagram for the child as he moves downward at increasing speed and another for the flywheel as it turns faster and faster. (b) Use Newton’s second law for translational motion and the child force diagram to obtain an expression relating the force that the rope exerts on him and his acceleration. (c) Use Newton’s second law for rotational motion and the flywheel force diagram to obtain

an expression relating the force the rope exerts on the flywheel and the rotational acceleration of the fly wheel. (d) The child's acceleration $a$ and the flywheel's rotational acceleration $\alpha$ are related by the equation $a=r \alpha,$ where $r$ is the flywheel's radius. Combine this with your equations in parts (b) and (c) to determine the child's acceleration and the force that the rope exerts on the wheel and on the child.

Zachary W.

Numerade Educator

An Atwood machine is shown in Example 8.5. Use $m_{1}=0.20 \mathrm{kg}, m_{2}=0.16 \mathrm{kg}, M=0.50 \mathrm{kg},$ and $R=0.10 \mathrm{m}$ (a) Construct separate force diagrams for block 1, for block 2, and for the solid cylindrical pulley. (b) Determine the

rotational inertia of the pulley. (c) Use the force diagrams for blocks 1 and 2 and Newton’s second law to write expressions relating the unknown accelerations of the blocks. (d) Use the pulley force diagram and the rotational form of Newton’s second law to write an expression for the rotational acceleration of the pulley. (e) Noting that $a=R \alpha$ for the pulley, use the three equations from parts $(c)$ and $(d)$ to determine the magnitude of the acceleration of the hanging blocks.

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A physics problem involves a massive pulley, a bucket filled with sand, a toy truck, and an incline (see Figure P8.41). You push lightly on the truck so it moves down the incline.

When you stop pushing, it moves down the incline at constant speed and the bucket moves up at constant speed. (a) Construct separate force diagrams for the pulley, the bucket, and the truck. (b) Use the truck force diagram and the bucket force diagram to help write expressions in terms of quantities shown in the figure for the forces $T_{1 \text { on Truck and }} T_{2 \text { on Bucket that the }}$rope exerts on the truck and that the rope exerts on the bucket. (c) Use the rotational form of Newton's second law to determine if the tension force $T_{1 \text { on pulley that the rope on the right }}$ side exerts on the pulley is the same, greater than, or less than the force $T_{2 \text { on pulley that the rope exerts on the left side. }}$

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(a) Determine the rotational momentum of a $10-\mathrm{kg}$ diskshaped flywheel of radius 9.0 $\mathrm{cm}$ rotating with a rotational speed of 320 $\mathrm{rad} / \mathrm{s} .(\mathrm{b})$ With what magnitude rotational speed must a $10-\mathrm{kg}$ solid sphere of 9.0 $\mathrm{cm}$ radius rotate to have the same rotational momentum as the flywheel?

Salamat A.

Numerade Educator

Ballet A ballet student with her arms and a leg extended spins

with an initial rotational speed of 1.0 $\mathrm{rev} / \mathrm{s}$ . As she draws

her arms and leg in toward her body, her rotational inertia

becomes 0.80 $\mathrm{kg} \cdot \mathrm{m}^{2}$ and her rotational velocity is 4.0 $\mathrm{rev} / \mathrm{s}$ . Determine her initial rotational inertia.

Vedad B.

Numerade Educator

A 0.20 -kg block moves at the end of a $0.50-\mathrm{m}$ string along

a circular path on a frictionless air table. The block's initial rotational speed is 2.0 $\mathrm{rad} / \mathrm{s}$ . As the block moves in the circle, the string is pulled down through a hole in the air table at the axis of rotation. Determine the rotational speed and tangential speed of the block when the string is 0.20 $\mathrm{m}$ from the axis.

Salamat A.

Numerade Educator

Equation Jeopardy 3 The equation below describes a process. Draw a sketch representing the initial and final states of the process and construct a word problem for which the equation could be a solution.

$$\left(\frac{2}{5} m R^{2}\right)\left(\frac{2 \pi}{30 \text { days }}\right)=\left[\frac{2}{5} m\left(\frac{R}{100}\right)^{2}\right]\left(\frac{2 \pi}{T_{\mathrm{f}}}\right)$$

Zachary W.

Numerade Educator

A student sits motionless on a stool that can turn friction free about its vertical axis (total rotational inertia $I ) .$ The student is handed a spinning bicycle wheel, with rotational inertia $I_{\text { wheel }},$ that is spinning about a vertical axis with a counterclockwise rotational velocity $\omega_{0}$ . The student then turns the bicycle wheel over (that is, through $180^{\circ}$ ). Estimate, in terms

of $\omega_{0}$ , the final rotational velocity acquired by the student.

Salamat A.

Numerade Educator

Neutron star An extremely dense neutron star with mass equal to that of the Sun has a radius of about 10 km—about the size of Manhattan Island. These stars are thought to rotate once about their axis every 0.03 to 4 s, depending on their size and mass. Suppose that the neutron star described in the

first sentence rotates once every 0.040 s. If its volume then expanded to occupy a uniform sphere of radius $1.4 \times 10^{8} \mathrm{m}$ (most of the Sun’s mass is in a sphere of this size) with no

change in mass or rotational momentum, what time interval would be required for one rotation? By comparison, the Sun rotates once about its axis each month.

Zachary W.

Numerade Educator

Determine the change in rotational kinetic energy when the

rotational velocity of the turntable of a stereo system increases

from 0 to 33 $\mathrm{rpm}$ . Its rotational inertia is $6.0 \times 10^{-3} \mathrm{kg} \cdot \mathrm{m}^{2} .$

Salamat A.

Numerade Educator

A grinding wheel with rotational inertia I gains rotational kinetic energy $K$ after starting from rest. Determine an expression for the wheel's final rotational speed.

Vedad B.

Numerade Educator

Flywheel energy for car The U.S. Department of Energy had plans for a 1500 -kg automobile to be powered completely by the rotational kinetic energy of a flywheel. (a) If the 300 -kg

flywheel (included in the 1500 -kg mass of the automobile) had a $6.0-\mathrm{kg} \cdot \mathrm{m}^{2}$ rotational inertia and could turn at a maximum rotational speed of 3600 $\mathrm{rad} / \mathrm{s}$ , determine the energy stored in the flywheel. (b) How many accelerations from a speed of zero to 15 $\mathrm{m} / \mathrm{s}$ could the car make before the fly-

wheel's energy was dissipated, assuming 100$\%$ energy transfer and no flywheel regeneration during braking?

Salamat A.

Numerade Educator

The rotational speed of a flywheel increases by 40$\% . \mathrm{By}$

what percent does its rotational kinetic energy increase? Explain your answer.

Vedad B.

Numerade Educator

Rotating student A student sitting on a chair on a circular platform of negligible mass rotates freely on an air table at initial rotational speed 2.0 $\mathrm{rad} / \mathrm{s}$ . The student's arms are

initially extended with 6.0 -kg dumbbells in each hand. As the student pulls her arms in toward her body, the dumbbells move from a distance of 0.80 $\mathrm{m}$ to 0.10 $\mathrm{m}$ from the axis of rotation. The initial rotational inertia of the student's body (not including the dumbbells) with arms extended is 6.0 $\mathrm{kg} \cdot \mathrm{m}^{2}$ , and her final rotational inertia is 5.0 $\mathrm{kg} \cdot \mathrm{m}^{2} .$ (a) Determine the student's final rotational speed. (b) Determine the change of kinetic energy of the system consisting of the student together with the two dumbbells. (c) Determine the change in the kinetic energy of the system consisting of the two dumbbells alone without the student. (d) Determine the change of kinetic energy of the system consisting of student alone without the dumbbells. (e) Compare the kinetic energy

changes in parts (b) through (d).

Salamat A.

Numerade Educator

A turntable whose rotational inertia is $1.0 \times 10^{-3} \mathrm{kg} \cdot \mathrm{m}^{2}$

rotates on a frictionless air cushion at a rotational speed of 2.0 $\mathrm{rev} / \mathrm{s}$ . A 1.0 $\mathrm{-g}$ beetle falls to the center of the turntable and then walks 0.15 $\mathrm{m}$ to its edge. (a) Determine the rotational speed of the turntable with the beetle at the edge. (b) Determine the kinetic energy change of the system consisting of the turntable and the beetle. (c) Account for this energy change.

Zachary W.

Numerade Educator

Repeat the previous problem, only assume that the beetle

initially falls on the edge of the turntable and stays there.

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Water turbine A Verdant Power water turbine (a “windmill” in water) turns in the East River near New York City. Its propeller is 2.5 m in radius and spins at 32 rpm when in water that is moving at 2.0 $\mathrm{m} / \mathrm{s} .$ The rotational inertia of the propeller is approximately 3.0 $\mathrm{kg} \cdot \mathrm{m}^{2} .$ Determine the kinetic energy of the turbine and the electric energy in joules that it could provide in 1 day if it is 100$\%$ efficient at converting its kinetic energy into electric energy.

Zachary W.

Numerade Educator

Flywheel energy Engineers at the University of Texas at Austin are developing an Advanced Locomotive Propulsion System that uses a gas turbine and perhaps the largest high speed flywheel in the world in terms of the energy it can store. The flywheel can store $4.8 \times 10^{8} \mathrm{J}$ of energy when operating at its maximum rotational speed of $15,000 \mathrm{rpm} .$ At that rate,

the perimeter of the rotor moves at approximately $1,000 \mathrm{m} / \mathrm{s}$ .

Determine the radius of the flywheel and its rotational inertia.

Salamat A.

Numerade Educator

Equation Jeopardy 4 The equations below represent the initial and final states of a process (plus some ancillary information). Construct a sketch of a process that is consistent with the equations and write a word problem for which the equations could be a solution.

$$

\begin{aligned}(80 \mathrm{kg})(9.8 \mathrm{N} / \mathrm{kg})(16 \mathrm{m}) &=\frac{1}{2}(80 \mathrm{kg}) v_{\mathrm{f}}^{2}+\frac{1}{2}\left(240 \mathrm{kg} \cdot \mathrm{m}^{2}\right) \omega_{\mathrm{f}}^{2} \\ v_{\mathrm{f}} &=(0.40 \mathrm{m}) \omega_{\mathrm{f}} \end{aligned}

$$

Zachary W.

Numerade Educator

A bug of a known mass $m$ stands at a distance $d \mathrm{cm}$ from

the axis of a spinning disk (mass $m_{\mathrm{d}}$ and radius $r_{\mathrm{d}} )$ that is ro-

tating at $f_{1}$ revolutions per second. After the bug walks out

to the edge of the disk and stands there, the disk rotates at $f_{\mathrm{f}}$ revolutions per second. (a) Use the information above to write an expression for the rotational inertia of the disk. (b)

Determine the change of kinetic energy in going from the initial to the final situation for the total bug-disk system.

Salamat A.

Numerade Educator

Merry-go-round A 40 -kg boy running at 4.0 $\mathrm{m} / \mathrm{s}$ jumps tangentially onto a small stationary circular merry-go-round of radius 2.0 m and rotational inertia 80 kg # m2

pivoting on a frictionless bearing on its central shaft. (a) Determine the rotational velocity of the merry-go-round after the boy jumps on it. (b) Find the change in kinetic energy of the system

consisting of the boy and the merry-go-round. (c) Find the change in the boy’s kinetic energy. (d) Find the change in the kinetic energy of the merry-go-round. (e) Compare the kinetic energy changes in parts (b) through (d).

Zachary W.

Numerade Educator

Repeat the previous problem with the merry-go-round initially rotating at 1.0 $\mathrm{rad} / \mathrm{s}$ in the same direction that the boy is running.

Salamat A.

Numerade Educator

Repeat the previous problem with the merry-go-round

initially rotating at 1.0 $\mathrm{rad} / \mathrm{s}$ opposite the direction that the

boy was running before he jumped on it.

Zachary W.

Numerade Educator

Another merry-go-round A carnival merry-go-round has a large disk-shaped platform of mass 120 kg that can rotate about a center axle. $A 60$ -kg student stands at rest at the edge of the platform 4.0 $\mathrm{m}$ from its center. The platform is also atm rest. The student starts running clockwise around the edge of the platform and attains a speed of 2.0 $\mathrm{m} / \mathrm{s}$ relative to the ground. (a) Determine the rotational velocity of the platform. (b) Determine the change of kinetic energy of the system consisting of the platform and the student.

Alexander L.

Numerade Educator

A rough-surfaced turntable mounted on frictionless bearings initially rotates at 1.8 rev/s about its vertical axis. The rotational inertia of the turntable is $0.020 \mathrm{kg} \cdot \mathrm{m}^{2} . \mathrm{A} 200-\mathrm{g}$ lump of putty is dropped onto the turntable from 0.0050 $\mathrm{m}$

above the turntable and at a distance of 0.15 $\mathrm{m}$ from its axis of rotation. The putty adheres to the surface of the turntable. (a) Find the initial kinetic energy of the turntable. (b) What is

the final rotational speed of the system (the lump of putty and turntable)? (c) What is the final linear speed of the lump of putty? Find the change in kinetic energy of (d) the turntable, (e) the putty, and (f) the putty-turntable combination. How do you account for your answers?

Zachary W.

Numerade Educator

Stopping Earth's rotation Suppose that Superman wants to stop Earth so it does not rotate. He exerts a force on Earth $\vec{F}_{\text { son }}$ at Earth's equator tangent to its surface for a time interval of 1 year. What magnitude force must he exert to stop Earth's rotation? Indicate any assumptions you make when completing your estimate.

Salamat A.

Numerade Educator

BIO Triceps and darts Your upper arm is horizontal and your forearm is vertical with a $0.010-\mathrm{kg}$ dart in your hand (Figure $\mathbf{P} 8.65$ ). When your triceps muscle contracts,

your forearm initially swings forward with a rotational acceleration of

35 rad $/ s^{2}$ . Determine the force that your triceps muscle exerts on your forearm during this initial part of the throw. The rotational inertia of your forearm is 0.12 $\mathrm{kg} \cdot \mathrm{m}^{2}$ and the dart is 0.38 $\mathrm{m}$ from your elbow joint. You triceps muscle attaches 0.03 $\mathrm{m}$ from your elbow joint.

Zachary W.

Numerade Educator

BIO Bowling At the start of your throw of a 2.7 -kg bowling ball, your arm is straight behind you and horizontal (Figure P. 66 ). Determine the rotational acceleration of your arm if the muscle is relaxed. Your arm is 0.64 $\mathrm{m}$ long, has a rotational inertia of $0.48 \mathrm{kg} \cdot \mathrm{m}^{2},$ and has a mass of 3.5 $\mathrm{kg}$ with its center of mass 0.28 $\mathrm{m}$ from your shoulder joint.

Salamat A.

Numerade Educator

Leg lift You are doing one-leg leg lifts (Figure P8.67) and decide to estimate the force that your iliopsoas muscle exerts on your upper leg bone (the femur) when being lifted (the

lifting involves a variety of muscles). The mass of your entire leg is $15 \mathrm{kg},$ its center of mass is 0.45 $\mathrm{m}$ from the hip joint, and its rotational inertia is 4.0 $\mathrm{kg} \cdot \mathrm{m}^{2}$ , and you estimate that the rotational acceleration of the leg being lifted is 35 $\mathrm{rad} / \mathrm{s}^{2}$ . For calculation purposes assume that the iliopsoas attaches to the femur 0.10 $\mathrm{m}$ from the hip joint. Also assume that femur is oriented $15^{\circ}$

above the horizontal and that the muscle is horizontal. Estimate

the force that the muscle exerts on the femur.

Dominic A.

Numerade Educator

Punting a football Estimate the tangential acceleration of the foot and the rotational acceleration of the leg of a football punter during the time interval that the leg starts to swing forward in an arc until the instant just before the foot hits the ball. Indicate any assumptions that you make and be

sure that your method is clear.

Salamat A.

Numerade Educator

Estimate the average rotational acceleration of a car tire as you leave an intersection after a light turns green. Discuss the choice of numbers used in your estimate.

Zachary W.

Numerade Educator

Door on fingers Estimate the average force that a car

door exerts on a person’s fingers if the door is closed when the

fingers are in the door opening. Justify all assumptions you

make.

Salamat A.

Numerade Educator

A yo-yo rests on a horizontal table. The yo-yo is free to roll but friction prevents it from sliding. When the string exerts one of the following tension forces on the yo-yo (shown in Figure P8.71), which way does the yo-yo roll? Try the problem for each force: (a) $\vec{T}_{\mathrm{A}}$ s on $\mathrm{Y}$ (b) $\vec{T}_{\mathrm{B}}$ s on $\mathrm{Y} ;$ and (c) $\vec{T}_{\mathrm{CS} \text { on } \mathrm{Y}}$ Y. [Hint: Think about torques about a pivot point where the yo-yo touches the table.]

Zachary W.

Numerade Educator

Running to change time interval of day At present,

the motion of people on Earth is fairly random; the number

moving east equals the number moving west, etc. Assume

that we could get all of Earth’s inhabitants lined up along the land at the equator. If they all started running as fast as possible toward the west, estimate the change in the length of a day. Indicate any assumptions you made.

Check back soon!

White dwarf A star the size of our Sun runs out of nuclear

fuel and, without losing mass, collapses to a white dwarf star the

size of our Earth. If the star initially rotates at the same rate as

our Sun, which is once every 25 days, determine the rotation

rate of the white dwarf. Indicate any assumptions you make.

Zachary W.

Numerade Educator

What is the force that provides the torque that causes the toast to rotate?

(a) The normal force exerted by the plate on the trailing edge of the toast

(b) The force due to air resistance exerted by the air on the toast

(c) The gravitational force exerted by Earth on the toast when partly off the plate

(d) The centripetal force of the toast’s rotation

(e) The answer depends on the choice of axis.

Salamat A.

Numerade Educator

The toast is more likely to fall on the jelly side if it makes how many revolutions?

$\begin{array}{ll}{\text { (a) } 0.5 \text { revolutions }} & {\text { (b) } 0.8 \text { revolutions }} \\ {\text { (c) } 0.9 \text { revolutions }} & {\text { (d) No revolutions }}\end{array}$

Zachary W.

Numerade Educator

What does the number of revolutions that the toast sliding off the plate will make before it touches the floor depend on?

(a) The amount of jelly

(b) The height of its starting position

(c) The length of the toast

Salamat A.

Numerade Educator

Why does toast have a better chance of landing jelly-side up if it is quickly shoved off the plate or table?

(a) It falls faster than if slowly slipping from the plate and does not have time to rotate.

(b) It moves in a parabolic path.

(c) The torque due to the gravitational force about the trailing edge of the toast as it leaves the plate has very little time to change the toast’s rotational momentum.

(d) The hand probably gives it an extra twist and the toast makes a full rotation instead of a half rotation.

Zachary W.

Numerade Educator

The length of the toast is about 0.10 $\mathrm{m}$ and the mass is about

0.050 $\mathrm{kg}$ . Which answer below is closest to the torque about

the trailing edge of the toast due to the force that Earth exerts

on the toast when its trailing edge is just barely on the plate

and the rest is off the plate?

$\begin{array}{l}{\text { (a) Zero }} & {\text { (b) } 0.0025 \mathrm{N} \cdot \mathrm{m}} \\ {\text { (c) } 0.005 \mathrm{N} \cdot \mathrm{m}} & {\text { (d) } 0.025 \mathrm{N} \cdot \mathrm{m}} \\ {\text { (e) } 0.05 \mathrm{N} \cdot \mathrm{m}}\end{array}

Salamat A.

Numerade Educator

If the La Rance tidal basin station in France could produce power 24 hours a day, which answer below is closest to the daily amount of energy in joules that it could produce?

$\begin{array}{ll}{\text { (a) } 240 \mathrm{J}} & {\text { (b) } 240 \times 10^{6} \mathrm{J}} & {\text { (c) } 6 \times 10^{9} \mathrm{J}} \\ {\text { (d) } 2.5 \times 10^{10} \mathrm{J}} & {\text { (e) } 2 \times 10^{13} \mathrm{J}}\end{array}$

Zachary W.

Numerade Educator

Suppose a tidal basin is 5 $\mathrm{m}$ above the ocean at low tide and that

the area of the basin is $4 \times 10^{7}$ sq $\mathrm{m}$ (about 4 miles by 4 miles).

Which answer below is closest to the gravitational potential energy change if the water is released from the tidal basin to the low-tide ocean level? The density of water is 1000 $\mathrm{kg} / \mathrm{m}^{3}$ . $[\text { Hint: The level does not change by } 5 \mathrm{m} \text { for all of the water.] }$

(a) $5 \times 10^{8} \mathrm{J}$

(b) $5 \times 10^{11} \mathrm{J}$

(c) $1 \times 10^{12} \mathrm{J}$

(d) $5 \times 10^{12} \mathrm{J}$

(e) $1 \times 10^{13} \mathrm{J}$

Salamat A.

Numerade Educator

The La Rance tidal basin can only produce electricity when what is occurring?

(a) Water is moving into the estuary from the ocean.

(b) Water is moving into the ocean from the estuary.

(c) Water is moving in either direction.

(d) The moon is full.

(e) The moon is full and directly overhead.

Zachary W.

Numerade Educator

Why do water turbines seem more promising than tidal basins for producing electric energy?

(a) Turbines are less expensive to build.

(b) Turbines have less impact on the environment.

(c) There are many more locations for turbines than for tidal basins.

(d) Turbines can operate 24 hours/day versus for only 10 hours/day for tidal basins.

(e) All of the above

Salamat A.

Numerade Educator

Why do water turbines have an advantage over air turbines (windmills)?

(a) Air moves faster than water.

(b) The energy density of moving water is much greater than that of moving air.

(c) Water turbines can float from one place to another, whereas air turbines are fixed.

(d) All of the above

(e) None of the above

Zachary W.

Numerade Educator

Which of the following is a correct statement about water turbines?

(a) Water turbines can operate only in moving tidal water.

(b) Water turbines can produce only a small amount of electricity.

(c) Water turbines have not had a proof of concept.

(d) Water turbines cause significant ocean warming.

(e) None of the above are correct statements.

Salamat A.

Numerade Educator