A bicycle odometer (which meast :mpzxfr wx,2gvl24g7sy,9 z ures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on amyx g,2pt79,z2wxz f4rgv:s l bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a powp-ys(r/irxc-1vl 7w6wb4np r (bk )zint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformlyci17rkw(/wys pbp 46l- n- wzrxv)b(r , does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a kuap kx 7ulw/ 8iy*a35tc ;0wssingle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greateeod nnpfn-i) or z/346r torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up wi9p52usmo 9ag vth your hands behind your head than when your arms are stretched out ina599mo 2g upvs front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedalrd7qphmg7 (c 0 cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a7cdqh(g 70p rm small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have swiz -si w/yx5v (z1c+wlender lower legs with flesh and muscle concentrated high, close to thy+wcwwi/z zv5ix- 1(se body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers mjx+ 0jrp4vqwn.. 3cc (Fig. 8–35) carry a long, narrow beam?

If the net force on a systemnv3 5z8kf p8bbw/9j 7 ujemaf0 is zero, is the net torque also zero? If the net torque on a system is zero, is the 3z85njf0ak 7b8w/f upjmbv 9enet force zero?

Two inclines have the same height but make different angles with the horizontalkzkms1awkp sqd, ;8j49q:8a c. The same steel ball is rolled down each ij9kakqs s,1a 4d;8qwzmpc8 k:ncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rol.q4u gr +61qtvj6ke acling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed thea6 1qevc6ku tj4rq.g+re? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have th hv yqpactwp914 vpj/.o;)s1,xv7o iye same radius and the same mass. They start from rest at the top of an inhoc4y91p1v jvyps. q/,t apvx ow7);icline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet movn)-ot9 jz+nvq 43usust moving or rotating objects eventually slow downnsq+ vz-3tu 9v) jonu4 and stop. Explain.

If there were a great migration d7):8wb7 c8 ozjtfjbd of people toward the Earth’s equator, how would this affect the lejod7: f jb8d 7zwcb)8tngth of the day?

Can the diver of Fig. 8–29 do a somersault without having a5i2t,uumnv8 k ocob)/ ny initial rotation when she leav,t)8 uov bn2u mk/i5coes the board?

The moment of inertia5w 5:+nzdulwj of a rotating solid disk about an axis through its center of m u55:znlj+w wdass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rota r 9jq2e;zl: u2prq;cbting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the rljq22:ur9q;c bz; pe same? Explain.

Two spheres look identical and have t mfgabpwj 958,he same mass. However, one is hollow and the other is sb8ma9 ,p 5gjwfolid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizo+wr+ *5xu, b8g qm7ayyox ,ksmntal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the anga brugk*y++mo msx5 xw, yq87,ular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on thepkta4:h;gs i: edge of a large freely rotating turntable. What happens if you walk toward the centsti phg;::ka 4er?

A shortstop may leap into the air to catch a ball and throw it qyx xj4mb2/kzdgo giqp fh9r97 x. g.*5uickly. As he throws the ball, the upper par92b*h goqixy7pr f4 jdxz 59/ ..xkmggt of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angulakr/;:z4d:1 of)q6vwa dthe dy r momentum, discuss why a helicopter must have mo)dr1 qv6k4; /y doah :zw:efdtre than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians:s4a 9xj8odgmch /7w 8j (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing lnhs58/ks1 wkc+9m mbcoincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and/nb 1w k+km58 9hslcsm the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moocx: jga;ok2z6 n, 380,000 km from Earth. The beam diverges at kgo2a 6;j cx:zan angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 r6 pxokl, 6mj*fvy11i fpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow dof61x lmkp i16yj*vof, wn?    $rad/s^2$

  A child rolls a ball on7;*zdagnulp(5i8 tt a a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diametertuztngap (5 ;d7al*i8?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutkjhqz):4 ;y yhions do the wheels: y ;khqjhzy4) make?    $rev$

  (a) A grinding wheel 0.35 m0r+hcw5; xu: (xlh/ bs;jjkty in diameter rotates at 2500 rpm. Calculate its angular velu(: tsx +jx/b0chjrk;yl5wh; ocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s k8 1us5e1t vm nc.vl7cmr;ws2(Fig. 8–38). (a) What is the lineas s5mu.;w17ke 2tcmnvvc 8lr1r speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Su5t- uuf 9xrha2 yhj7/fn    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speezp m-mt1k4iql-m6 l 3)urxea 0d of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) muspmsf6fpa /y+h /s3e;tt a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience anmps/;fy3 pet 6h+ s/fa acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifo a:y3qxyu8/t mrmly about its center from 130 rpm to 280 rpm i /3xaymq y:t8un 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(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. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu.x x( ptkikkh3h+t2pz/2 zv;ds $R_1$ 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, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecae(t ( n;qxyz9r 3m vla.r-,whraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accarvwz y(l3. 9n xt;qmrah(e, -elerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uke0y( n6d6zx uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thi0y6dnze (x6kus time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculateomv8 oefxs0/s2;q2i0sh y:vqh 3y p)7 ujb yp0
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying h)y 3 iq vl.eu:+:lfrsnighspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through :):qs+v .ulilry 3 nef20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in *aek pl ee6l3 s1*;tws6.5 s. How far will a point on t 1;e*psaels3 ktwe*l6 he edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when iqj enq xd.ur89m0 l/zsc2+(gft is running at 850rev/min It turns 1500 revolutixdrqf e0+uq/(zls2g98jn .mc ons before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly ff obotp8mb) v/v3eji69lb9, u 7qwjn-rom 95km/h to 45km/h T7n ,v9f o)9 / 38-blbwjvemoibjqpu6the tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fr ci2(( c7tjjlmom 95(ctij 72c (jlmkm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding pq u ,6-e+lyzra bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of raq-+zy6 rlu ,pedius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 d1n b)*fmh;u n2a4 mbiqymk k18gfc +2N on the end of a door 74 cm wide. What is the magnitude of u 8 b kb*+fk;2iyg2) m1mnmq4cfadhn1the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axlnmg9h- 4y vt1fe of the wheel shown in Fig. 8–39. Assume that a friction torque4tm -hyfgnv1 9 of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached tsca s:88vk3 95d qj qo-es/e4qc7pbvqo the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the n : obj9vsk- sceqaqq38/q5vs p c78e4det torque on this system.

  The bolts on the cylinder head of an engine 9dy;fqhcu; + oyep 8*wdt*we 7require tightening to a torque of 38 ; o;cu 8wpwfe*eyh*q +dt79y d$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when b2+(g p j-xw+wj mxz9y1sts:m7 c(zfathe axis of rotation is through its cenztzx:+- 1m9fg(y cm j2as7wjx( bspw+ter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66. x:)wve( 4dciqihu/g7 j8gbq,7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub 8q c,x/ii(h b7)e dqugvgw: 4jcan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a tquq.b)on- sq8n 3 k2sfrb--zhorizontal circle of radius 1.2 m. Calcularn-nuoq s )2-s q8tb-bk. z3qfte
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(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 the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angulzl-zn)rsr ;b8 ar speed (Fig. 8–42). The friction force between her handb;z)rr-sn l 8zs and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(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 hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of th8bts8z;8 mkn3wl. aghe array of point objects shown in Fig. 8–43 abo3 zkn 8t 8.;asmbgl8hwut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg 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 horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass ibjrlh/t) i,vq y- hv 2i;3,kmau(zug.s $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct r d1 3n2yr8uo02uefncz ate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the requiref0zn28rc u2u1 ne dy3od steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifkj4oo qx wkh33y--f7 sorm cylinder with a radius of 8.50 cm and a mass of 0.580 kg.-xo3k yqhjo47fws k-3 Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it(7,rqtgu3 j za from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a sm.p/onq l1mjewnx; lrix t,gr.*6o 82vall 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 ajr2wto/iq 6 eg po ;.n8*,l1vxnxrlm. mass of 760 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. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought fb6fi0f1ew: /i(li+qi 1dr zyuniformly to rest by a frictionalli b:0 1 +ifry/f zfqi16ed(wi torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 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,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball ak wkz q4(v3,kmc1g3wd t 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by thl q9h2ssde 62;2+azo.ua tu-c - lwatie action of the forearm, which rotates about the elbow joint under the zh ua-6a2l id2st2-t .ulq9s;wo+cae action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rotw hhgb fcwu(q)h ;1 9,;e trhuzs:mc416zjm9d, as shown in Fig. 8–4h1hz 9uhbqt ;twm9; (4c w,sjf gz)me6ru1:hc6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two mvd4kbvix6sd +cfj :m9w) vu+ +asses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within *-pa ros9ae/w8 q9pkofour full turns (revolutions) and releas eraoqwkopp*9-a/ 9s8es it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inerx)hpcbakd 3 43;(+ eq.z4lta hdjzqz7tia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develo2g*fndskj l 268l izf-ps a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7 r-sab)8j):dci0 wkw s.3 kg and radius 9.0 cm rolls without slipping down a la-)b8wdk )sa wji0:srcne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum offtt8z3ffzw 93 two tertz zf8 33w9tffms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a ms*p0w+y+lqn xhy:(;2bi u y tkass of 1640 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 8.00 s? Assu0pq (;kt:lwsuhyy+b*+ y2 xinme it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg *.a/utd *-* zjil zvfwstarts from rest and rolls without sliz *z -*jl v/utwdf*ia.pping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE 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?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts toqahy qnrd5.3ab)c: j u.ri/.d fall and its lower end does not slip. What will be the speed of ja3nqrahd.:u5i .) .rbc dq/y the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum):hgb:bc tgle 64o6sn of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10:hbgl sencb6:4 otg6).4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of2e4(o cowu (vo a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpmo24w(euo o(v c?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ch8 wfd u4i3+wrate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases t3fiuh+ d cww84o 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the oo27v+2cbj b/xe dg4v(e: fcpone shown in Fig. 8–29) 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 vc f2pbeo/(xvg jdo4cb2+e :7 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increastg+6k hax4m t/jkm7tn 9ve5+ ie her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate o/h6 it 9 xk7t5tnmage4kj mv++f 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a freqx zpcvjo5:(3yo2tuy.azl9e;k fy 0n,uency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and dia;looxc, yjp: en zt35zufyay.2 09kv( meter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum sf5;ffwl/: ah of a figure skater spinning at 3.5 $rev/s$ 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 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momenk e* (imn4mi1ztum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inerux a2)) o,xa-xvt tx9vtia I is dropped onto an identical disk rotat2vv )t xxxau-o),atx9ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 0.f8hx/.wkss q hj3xs$rev/s$ around a frictionless spindle. 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. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg standsqbdri oe4;e/*6 h3av-oj ay-j at the center of a rotating merry-go-round platform of radius 3.0 m and moment of i3-q-a eo/4y;he jdjbro*via6nertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is r:pyq4q :5x)a el: 5+s 0sskfqmw- osnpotating freely with an angular velocity of 0.8oxw :ap4:5fs)+em k:-sylpq0nq sq5 s $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually w9h q9gd fl b77-f(uzgybu)1fcollapses 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, what would the Sun’s new rotation ratqu (9f-1)fhbf 7g7u9z ydblgw e be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of nytxz-./t mn qw kr)wogypw+; l* /9zp*m,3ik120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass g0-vti dnfa ro so,(5+$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate f pwc,(thk 7vu+9 tthr0jrzj787 5gto *me2q pkreely without friction. The moment of inertia of the perutk (hh,owq j8ezg+vprp*2mkt r7 0cjt5t779son plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person staojzhto 0a9/ v)j9s;kynds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of in /aoj)t9yjs;v h0k9 zoertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope 4p 3ihuhbgo k6cy+3c) ufx7*lrolls on the ground with the end of the rope lying on the top edge of the sphp khly4 u6xcg37f3*icuo) +bool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. 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?

  The Moon orbits the Earth such that the same side always faces the Earth. Detexen8yec;(lro m6xf) 4rmine 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 ocy8(f ml nx)6o4 e;xrerbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a raqu 7tuga.tqei+r3ok6 tr+6 f6a 6oe5m te of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in qxlz3s4 u m7gimb-4: athe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratiq-:mlz4 7u bg iams4x3on. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so( ) 2sfnf v3nk/enbjr7lid 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 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released fr fnrfvb3 k )/(nsejn72om rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the anguc 1tlj6 6jw pw,7;ljq3vw5x.y vig -ygq+4rttlar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0g );zg( dko 72vhvyg9b times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undobd vgyz97hg ();2vk gergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to gf6xk :6q1kczd 8k7j4jr s(5nfgu ve2 y:mz*p use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should b:k 4rqk6*n:x ( 6vygz1pdefzsk2 mcjgju857f e able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an jm 0 e79g8xsixk) syz4ye/x7c $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) ilv(b mjj g8e(x +og *ebq8g)6ms rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore ffcsnj(d ,gw6bgw ah 9(3gu( rmx:e(l2 riction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the r9(s2g g nb jhuec(wm6xw(fa(3 l gd,: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. 8–21g.]

A wheel of mass M hasq69 9iyr q1 pp8qzlt0u 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 ag0rlqqiu1869py t9p qz ainst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a jr;m h. kv36buflat road is making a turn with a radius The forces acting on the cyclist and chvj 3 ru;mk.6bycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg ojp6 azsc:roi 4djwm4p5u) )7rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this jpc 4a6os) r dimuj)z54p o:7wfeat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as h u8l-y*;tshqthin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.-h; s*quhlyt 8   

  You are designing a clutch assembly which conscdk*dbd pp)b0izoc1l, a3 b :3ists of two cylindrical plates, of cb0 cpabd:i3p z)k,3*1dobldmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls giv7d ownfbw) 1(m(9a along the looped rough track of Fig. 8–58.ob9(1f)vwd nmiga(w7 What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not * h-mnp5epgv4 assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the cb; gc min)rnmwg*8)5 qar reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular a*n)qw ingmcb 8mr)5 g;cceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s