A bicycle odometer (which measures distan9xl o;qgh,. :w5 hryjtce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicy5 tj,: ghywq.ol rhx;9cle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point opn6,+9 igsg+ufakk 2b2 v:cxpn the rim have radial and/or tangential acceleration? If the disk+ sn 9pp2 ,k26igkafcgu:+bvx’s angular velocity increases uniformly, 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 single value of the angzrwqlu2rbyz 4lt4p +f14;s )o ular velocity $\omega$ Explain.

Can a small force ever exert zks dcpr.hb22j3j(7x05 ch7wbdm*j k a greater 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 t)7b:onc hqt p49qx c)po do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you q:o7 pqt)hcbc4x)9 np to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pey 9btx+x.kqm/dal cranks. In which gear is it harder to pedal, a small rear smx+qy9bkt x /.procket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fy4g a a(- nmdqygo*1qdy. 9ro.ast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advd-yyy. 1o*aq mqan49rgg d.o(antageous.

Why do tightrope walkers (Fig. 8–35) carry a long, n h(zdcg9:6sz*bjwbj3q/0f jz.- cj uxarrow beam?

If the net force on a sys)77vv kos m/jg:n+.m5(c cp7mpbhf zstem is zero, is the net torque also zero? If the net torque on a system is zero, 5p/.+sbsvn7g)(o j7mk mh mzpcc 7vf:is the net force zero?

Two inclines have the same height b2+vny u(jiacngwdw2z1yuih(2 p1z,: ut make different angles with the horizontal. The same steel ball is rolled down each inclinez2u1 i+ywh 2 ndycg(jnizp 1v(u: ,wa2. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fddg,cojbwt )uf8m(iz /+1e i sr;c2 s1rom 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? Whizwsmd2 1ct) 8idur j;goe+1sbci(/f ,ch has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder hav v 2t y;fprjx-7xzp2a mtxo:(.e the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at p7.(a:2x2p v ;xrtmtj-xyfzo 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 most movk;u,8 rbtcw3 cing or rotatin3 rbcctk,uw;8 g objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equat (hlnci eh82zt c,z0e4or, how would this affect the lengh(z lc,eznti 8e0h42c th of the day?

Can the diver of Fig. 8–29 do a somersapi*qe. y1 dpzq6wq b1.ult without having any initial rotation when*ei.61w. ypq1d bpqzq she leaves the board?

The moment of inertia of a r iy 2jno,hc, hgee+b93otating solid disk about an axis through its center h9+3e,yie bgjnc 2,hoof mass 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+:j1j;c2j g xs.othdg a rotating stool holding a 2-kg mass in each outstretcdhg;j+txo 1.j:gj2 s ched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howeverru 3 zq *bt/zj5zh fqhr(2(+vj, one is hollow and the other is sobvr+q (5 /h*t2jqzh( jzf3uzrlid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. 0 -9qx(jvm-gf hke n2vIn what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed in-jn9hmk2g qe-fxv0( vcreasing or decreasing?

Suppose you are standing on the edk2h5 rw3c lgy0/g ar8rge of a large freely rotating turntable. What /gr0wg 5rakc 3h 2ylr8happens if you walk toward the center?

A shortstop may leap ipad+v7o 4;cvw89 b pxknto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that hispak9do8+x7p c bwv; 4v hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lj1qdm -5n z7pbaw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep tm5- n7q1j zdbphe helicopter stable.

  Express the following angles in radians: (a no4.yil /ro:i) 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 coincidence. Calculate, using the4x 0eldy z*p; +ex3xbx9uj*sk information inside the Front Cover, the angular diameters (in radd; 3u* 9zye0xk* xlej4ps+xxbians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moorr26hmdhlsc 8 sh7 :+pn, 380,000 km from Earth. The beam diverges at an angle 78h +rh 6rs cm:ph2lsd$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender ro :w,8ky l rdi,z4doc4dtate at a rate of 6500 rpm. When the motor is turned off during operation, the blakc ,ordd44yzdi8, w:ldes slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothem wg z4rj:xom6m e(-j-**kmch o8 27f 3nvjsitr child. If the ball makes 15.vwj*7mo:n(6f8*c2m -mjz-s m34thi g xkreoj0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter trav*onf6f5,dr 8ywy( rsnx, 0b6ckf o8so els 8.0 km. How many revolutions do theb x58 ,c ffyw,oyrk0 r66(nsf8o nsd*o wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates sae np+81+cp pat 2500 rpm. Calculate its angular velocitys 1p+ ec8+panp 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 copwfktk,wo q33 vfe(kp(6 1x. emplete revolution in 4.0 s (Fig. 8–38). (a) What is e v k(wp,o1t ke6p3xwf3.k(qfthe linear 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 velocityrfssko6* o .-d of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofk,0j . q;dptf *2egupf 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) must a centrifuge rotate if a particle 7.0 cm from the ; , vr5,mnvhg gu)*7frzrg8hr1 onny 1axis of rotation is to exper8vrun1yf gg,hr r,mvh 7nrz;n*o g)5 1ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter whees pnj bn*l9xjjyo 9oy(:bgj d3: 4vt.5l accelerates uniformly about its center from 130 rpm to 280 rpm bos:t.jv:b *jj xly(g5y9 dn9n3pj4oin 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 radiusr 9xgd1kafe/w ; 6vta* $R_1$ 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 spaceckj m8u g wr87yr-s1bn4cb,qm2raft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution82rnm y u81rmgc7ks-bq4w bj , 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 uniformly from rest to 15,000 rpm qgasnl na k+vgk8 ym4l,* 8n8:in 220 s. Through how many revolutions dy* 8nvn:lkn,+sa ag4lm 8kg8qid it turn in this time?    $rev$

  An automobile engine slows down from 4 zgc/edf l2lt(9j7l8 j8ysp;r +c dme0500 rpm to 1200 rpm in 2.5 s. Calculate
(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 highspeed jets in a w5jv2x 0wc qieu;n** j8:nvz othirling “human centrifuge,” which takes 1.v0q *8nuz: *iect;j2 wjo nvx50 min to turn through 20 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 360p4zg kbn-hcuhu80aq+f9r (pw1uy7 t + rpm in 6.5 s. How far wilugwh8 u+ph70 uc-k+f4b9ranzypq1t (l a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850j wakolb6gr0 vqg; 9+(rev/min It turns 1500 revolutions befor;gkg9q + borl(vawj 06e 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 as9atv lx9mkd mu h*4.:z the car reduces its speed uniformly from 95km/h to 45kmka4 xzdt.humv9:9l *m/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

  The tires of a car make 65 5(t a0 5;d0vrdgsgdsrrevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires ;0d(s g5rdrsv a0t5gd 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 a bike puts all her weight on each pedal wh.)l p8*chu lsken climbing a hill. The pedals rotatec8 l)h*plk.s u in a circle of radius 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 N on the end of a door 74 cm wide. What is the ma(h ddrlnv;:qtbsa:)bf c+0i 7 gnitude of the torque if 7 ft0lcbbin +q:sv); (ar d:hdthe 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 axle of the wheel shown in Fig. 8–39. xd q/1 mdwtbw9f:03la Assume that a frictdw/ 0x m3:dfw9ltqba 1ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massl ka/6x -0-fjfy/ vdwnt8/hhfwv6b*mness rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then rele/ mjkd/6-whfwf f0 vtxya*-vnn /b6h 8ased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine rext/3ehxbj. ,uy rh,n r1-p yf,quire tightening to a torque of 38n u1yr.pe,yr3b,j ,-xh/f htx $m \cdot N$ 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 1038/k hem n( 0fnisxhq:.8-kg sphere of radius 0.648 m when the axis of rof0n: qe3m ix/ks 8(nhhtation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bii m*pad71 bu5bcycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of pi57b du bma1*1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated rm2n tep47 r,o*vgnp* in a horizontal circle of radius tn r*ov*,grmne p724p 1.2 m. Calculate
(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 constxfuv.gqvmn 30-oao2(ant angular speed (Fig. 8–42). The friction forca2gvumx o-3 (.novq0fe 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 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 the array of point n ft6o(-d 0nyhw86sd* c*vffkobjects shown in Fig. 8–43 an0nw6h sf8f6- tov(dk*yf* d cbout
(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 oxygoo 28/yuhiw. ren atoms whose total mass is $5.3 \times10^{ -26}$ 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 rau.kyi61m,au ite, engineers fire four tangential rockets as shown in Fima.u yi ,u6k1ig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required 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 uniform cylinder with a radius of 8.50 cm and a mass of 0.qnqi+ rz7 ff-,580 k+q, rqzif 7fn-g. 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 fro +q4cuggd9wxm ;un43wm 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 tangentially3y8*guv ws r;hg ac9q7v,l 5jb 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 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, negychjgq wub r v7al9v*; s538,glecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 r o zc7(h53sx/4qjocuz pm is shut off and is eventually brought uniformly tsxzc3jc/ ( z7uh o4o5qo rest by a frictional 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.( bx9z+xd;iij6-kg ball at 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 bal -xvwfdu4*pq/rewg j1og/z3 / l is thrown solely by the action of the forearm, which rotates about the elbow joint under the action op3w4vzo// qggjfdr*x/ - uew1f 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 ca solknddkfga(uzb 4-29 3dz8) n be considered a long thin rod, as shown in Fig. 8–46ou)gakf84kl 92s-(zdb3 zndd.
(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 twq3z4 dgsrqpd:,qp2 ghs :d9a4 o masses, $m_1$ 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 rg+caygu; v/p x(jj2a/est within four full turns (revolutions) and relea;ug yajvcp/ j+(gxa2/ ses 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 mom l m;bac d6s(y2t:i(prent of inertia 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 develops 1r/ 6r1 yryaplmiw(a9a 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 7.3 kg andbc . ;,wr1hzst bgy0t) radius 9.0 cm rolls without slipping down a lane at rwz)gbs t. h,y;t0b1c3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energywvtv14g (o 5)8 vusr-aecd ,bw of the Earth with respect to the Sun as the sum of two 4)ud1gvt rs8a v5e oc,w-vw( bterms,
(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 mass of 1640 kg and a r8 gzvc(lf1 .eqadius of 7.50 m. How much net work is requirz1e.g ql8f(c ved to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and masu*kq qggz)t gb709 0hcs 1.80 kg starts from rest and rolls without slipping dowk u0gb cqq)9h*0ggt7 zn 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 ist )vhfnwy 7a5- balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end 57han) f -vtwyof the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatinm/ 0nf :utqcv9g on the end of a thin string in a circle o9 vucqm0ftn: /f radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uh g;0o iogvb:9niform cylindrical grinding wheel of radius 18 cm when rotating ath 9o:gvi; 0bgo 1500 rpm?    $kg \cdot m^2$
(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 ratixyvmba bv)9x5x4m:ab+ef1 / e of 1.3rev/s If he raises his arms to a1 /meybm9b xivx5a+bfx4v):a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce hcvr au*bu wyu.*d8 y5 fvd3.n h;e0js7er moment of inertia by a factor of about 3.5y37hedu.nyfb*u 05wvr a sjvu8 *;cd. 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?   $rev/s$

  A figure skater can increase her spin4i vz.-j 4zbg4jwlpu* rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of g-4uzz.pw bi4jjvl* 43 $rev/s$ 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 8ul jl3uw2eg+axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of 82jeu3+lgulwclay, 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 of a3( 4fhe/)xnqe*y ck e9rpla 8m 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 momentum of the Ear ;yt9pdcp o1hpfe pz6 j5.()yvth
(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 oheulva0dm zm/;1tia3 24 v 8t(avwj 2ff moment of inertia I is dropped onto an identical disk rotating at angulai me4 h djam2(vu;t328flw0ta/vzav 1r speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsgy9wkob2m d;/ at 2.4 $rev/s$ 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 stands at the center of1ai 62t)tstihh 9mp c) a rotating merry-go-round platform of radius 3m16)sh hitt2ipat 9)c.0 m and moment of inertia 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 rotating freely with0q7;hkuui f ccr -0*k 2jj)pbd an angular velocity of 0 ;ukj -f0c *)kru70dqbhc2pji .8 $rad/s$ 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 collapses into a wtdv 7.t6. ak2vfm 8h,rjjm4cfhite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Asm kmvf4fchdja 2 ..r8v 7tj6,tsuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate 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 i:fy/u 2tw84 nmn8hte vn excess of 120 $km/h$ 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 rq j(qkw8h j1g8/d)y-b ld/ oy$ 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 freely wi.6vfjbu9 ct 7tthout friction. The moment of itb6u fc7 9.vtjnertia of the person 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 h;( pkpq)hhzb57 2p) 7yskgsk stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia o( pspqkkyz7hhk72s;gh5 bp)) f 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 rolls on the ground w:d) cd:als+ hf d-l,ffith the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distancef )dal:s,:fd +d-clhf 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 sihcgs5 2fvl 6 sf5l7sni7jl ;4hde 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 o7lgfh cvfsln45h5s7i26 s;jlrbiting the Earth.)    $\times10^{ -6}$

  A cyclist acceleratex 1i(n nvvyv+ :hfl)q5zgl/g7 i0azg3s from rest at a rate 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 the shape of a uniform cy+ qmhkm0ce4a0 linder of radius 0.20 m acquires e 0+4qk0ammh ca rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made 4m 3.7*d r8icvh lykuqfo .8uwof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.07 i3u lufv.c8q*rwmd4h7 oky8.5 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 from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is pfpwbd o (i7*hxd2+.vthe angular 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 8.0 times as great, wn*xe* b)6dz l-ena3 8vi cdxr7ere rotating at a spee i dx)v8ner *7ba 6d-zxncl*3ed of 1.0 revolution every 12 days. If it were to undergo 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 use energy stor6 07.e idfeof-v09 4c,zv 1o c;ta jbvsdmn.dred 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, an1.e7dez0a,csf 4r.ti d-9vvcf nj6d0bovom;d should be 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 aw g3bl,qo44 n$H_2O$ 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) is rol sjg3ubnf+-4tling 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 ignorin 2vz cfj,75l+5 wrbze friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held hornc55 rz+f,b 2i zv7ljwizontally 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. 8–21g.]

A wheel of mass M has radius R. It is standing vertically ojm;b locf/,usf6, lm,n 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. 8–55). The step has hei;s,u ,mf /jfm,l6l cboght h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat rv4iwesn 1.3 v;ort9sqoad is making a turn with a radius The forces acting on the cyclist and cycle at9nri;s qw13o4.esvvre 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 ao,mlut,9e dl y5u5olp3+x -c rock into a sling of length 1.5 m and begins whirling 53lu9te+u-dc,m5o po, yxa ll 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 feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s1tma(vg.( xi i body as a solid cylinder and her arms as thin rods, making reasonable estimates forixtvg(mai(.1 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.   

  You are designing a clutcsq7y xcv -(i;qh assembly which consists of two cylindrical plates, of mas7cs (ix -;vyqqs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 along the looped rough track of Fig. 8–5kt u;o um33i-qv(viv+8.3qkum-v(;+ o iuvvi3t 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 assumg ccjyu47lkv16z 51( d4waavlv 04spoe $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly f8,;3 rqltdu2 7kgsine rom 90km/h to 602rlgn 8sitqd 7;3 eku,km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration 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