A bicycle odometer (which measures distance traveled) is attached nyh e(t2.gccx k86:2ojrf8i n zear the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with j: ce86(ifn22 rx otch 8.yzgk24-inch wheels?

Suppose a disk rotates at constant aynyh ha2q3p/k5yh:d)ngular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which /nyq hy5dh3 a:yh2)kpcases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by igw i; -n.i;0vq aq pdr(w9q+ka single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explar0qaix;c0m*z 68q ifxin.

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 with your hands behind your head 18; tb sma;nhithan when your arms are stretched out in front of you? A diagram mb8hnasi1t; m;ay help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thp( ul3 dxv/nqfn55n)mree at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rvm3fuq5/( dxp 5nnl)near 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 fast have slp3q6+7a gu,axawmvb46 c mze,ender 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 advantb z, xv3a6qg+m c6a 7,wp4meuaageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narro8dvih-k*jh /f3 ti-p, pv1gf gw beam?

If the net force on a system is zero, is the net torqu74teppm k;h .+kqx fy4e also zero? If the net torque on a system is zero, is the ;ypx mk+7 kpq4h .4tfenet force zero?

Two inclines have the same height but make different anglo+rabr. )- wb:zziuwq) ,ea) gi:(rwtes with the horizontal. The same steel ball is rg )b tzrorw:w,ei+zbq-a): )ia.uw r( olled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouk 1,eaf .-f1uqotfm 5xsly start rolling (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 there? Which has the greater total kinetic energy at the a.ftfo 1 5mku-,eqx1fbottom?

A sphere and a cylinder have the same radius and the same mass. They start frio-vj7i: yeg-om rest at the top of an7 :i g-ej-iovy incline. 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 most mti0bkc .w,vn a5oc/ppu2n) c 0oving or rotating ktc5 /w ionn0c. ,apb)u2 c0pvobjects eventually slow down and stop. Explain.

If there were a great migration of people tou.xp 6i1yn:: k+z yitoward the Earth’s equator, how would this affect the length of the daxzu yi:. :16kyntop+i y?

Can the diver of Fig. li *aj0 tvbfo3*/ea:m 8–29 do a somersault without having any initial rotation wh*jfav3li et0oba */ :men she leaves the board?

The moment of inertia of a rotating solid disk about an6yb;z w cz7lg0 axis through its center of mass is 7b;z6w ygz0lc$\frac{1}{2}WR^2$ (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 rotating sw5p0.e 3w- no o3xkfko3mh6ootool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay themwop -3nx.3o woo5f3o ke60kh same? Explain.

Two spheres look identical and htpn7y) w-q6vjave the same mass. However, one is hollow and the other is solid. Describe an exper j 7ytq-)pvwn6iment to determine which is which.

The angular velocity of a wheel rouv3p3n se js4. mpfs9+tating on a horizontal axle points west. In what direction is the linears3pse m4u3fjs9 p. nv+ 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 increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What o1twr84m6fn .xl2 olv happ l6nvm fo1or84 xw.l2tens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it qpytx akp95n-0)vh q s(uickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). nvy5-khx )t 9ps0a(qpExplain.

On the basis of the laut,z* dvfiz/8pfjx . kh:e:(aw 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 cav d :k,ifhze8*tj/u:pz.fa(xn operate to keep the helicopter stable.

  Express the following angles in rt6x1-aquo;ivuuw./p 6gp0r aadians: (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. urv6m+mjy1wx dggc 7/;yrl. amazing coincidence. Calculate, using the information inside the Front Cover, thryc1mr+gy7. jl vm ./;xu6 gwde angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moonanx:(q d uj:-x, 380,000 km from Earth. The beam diverges at an :du aj: nxx-(qangle $\theta$ (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 rpp 9 g*dc/uhod1m. When the motor is turned off durio/dg cdh9u* p1ng operation, the blades 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 rmi,: jovhm1 *,f:oux on a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamet*: m rhivjxm oo,u1,:fer?    m

  A bicycle with tires 68 cm i7 uhfs juy4 t i;0xz3p6bho0t-n diameter travels 8.0 km. How many revolutions do the wheels mau37 xp -iuo 0b;h406tfjytzs hke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500k50 4uxr wkd*n rpm. Calculate its angulax0r n54*k dkuwr velocity 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 revoluti9 6sopigr)o rp ev:*) ok*sf3i8 ikj/pon in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child/8ooke9p6 k3vgrs:i*sporf*jp ii)) 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 velocitj:j hmwo/u /;oy 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 ; n* /4xu4rdk9awo rr u9qi,pcof 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 cw)txif r, lmb1d,q x4;entrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accx;4m)x b1 qifd,twl,releration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 53.bhgs3 xy5h27is do otxn j8280 rpm in 4.0 s. Ded5xo yj8bis.gh sh35 tox7n3 2termine
(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 radiusf 1.6dqkc)6x-ngyn4jmbo n)s $R_1$ 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 Apoll+*f kym7gp/urnhl7p x f3;vi,o spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they+ h gfvynxim*lpp;f ,kr73u /7 accelerated 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 uniformly from res fihm ns0e:awjj5:8a.t to 15,000 rpm in 220 s. Through how many revolutions did itehj.:f58m j siana:0w turn in this time?    $rev$

  An automobile engine slows down from 4500 mv;ptya. .p3srpm 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 whir5glz;z, fmal) rre: /aling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachin :/;lr)agem rl 5zf,azg 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 uni75tb,5wk;;ejomhtpsf+ n ,on formly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have trav5 tn b5 jowo+,7, ;fpstemkn;heled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It vqwlp0g ir.1x. cr 6.iturns 1500 revolutions before it covlr 0piq ir.gc.6wx.1 mes 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 frnf04.:q*ln .3yck9 irhu /)hnfxjv ig om 95km/h to 45km/h The tirern 9/cf0f :u.)i lvjihy*gqn4h 3kn.xs 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 g u8*rak6 vm/w65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.86v/m8ura*gwk 0 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 pi(yes+9j- qyxedal when climbing a hill. The pedals rjy9e x(-y qs+iotate 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 forftsa6h,g zgwx2:-1 . 1rt hemjce of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if ehf 1 : tgjx2w-r,zsh6.ag1tmthe 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 Figv- .zr0zv2t7;s o8: btvvzvf z. 8–39. Assume that a friction torque of 0;7vzs :tv .zozr-v vtb28f0vz.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 massless rod which pn5 e,7 5ro**/mztdi mqbg5xazivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then releasmq5x*,57gae zidro/n m *zt5 bed. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of c7,xl(p)g uny 0nnm v2an engine require tightening to a torque of 38 l u0g)cmpy(2nvn,7nx $m \cdot N$ 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-ksid/i,w1bz j3g sphere of radius 0.648 m when the axis of rotation di13szw /j, biis through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. T5mx,d qf,ew9rhe rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignor,m,5eqxd w9fred (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizontamlb5hk-z7 nvswp0 a) 3uc+.z rl cia.5v pul0+wczzksh) 3 mbn7r-rcle of radius 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 wcl:bf; m/ 77dy qsxtrp;m6m.vheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and m / l6vy.;7m:7msqd xprcbft;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 ofgjj qkb2. gp1; the array of point objects shown in Fig. 8–43 1q. jjggb2 ;pkabout
(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 tot febd *-tw .i57q3caxgal 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 8y4og ytyui 92at the correct rate, engineers fire four ta u8oig4yyt92y ngential 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 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 w4( ghgx;qqr-with a radius of 8.50 cm and a mass of 0.580 kg. Calculag rg-wqhq4x(; te
(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 swingsea7kxayzb;j21 5 c/rt a bat, accelerating it 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 small hand-vb--kjydrfo2: qgt(; 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, neglecting frictional t dby2v:-;(ro k- tfgjqorque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut xn,:vg6ty.0ee7 bgknawdz9/ off and is eventually brought uniformly to rest by a frictional torque oe 69dnz0k w xa:,vb.gn 7tyeg/f 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 b ;:06etl jy)k-p u h/k,hnts4wqdv-aa 3.6-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 ball is thrown solely by the action of the fobw: tg,4 lo s*ydt4/zirearm, which ro/o*d4,4y:sztwb gt litates about the elbow joint under the 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 rod, as show 8t( qw/r.vc,nvk anf/n in Fig. 8–4 8k fq w//,rvvc(ta.nn6.
(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 consistal/1 vud6tkgckz x69ux.qc*l;pk7j qxeb6-1s of two 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 rest within four full turns (0qv5zx qi+i h8 :yku(x y*8oelno/sw1 revolutions) and releases iy 8/ 5:xio1z+uy(nvolsqhw x*8 e0kq 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 inerti5,ra34)-fr 2ib -5fht trvyl3 jshmeca 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 vi5c(hg /c*te, k*9uacjr9 j6 /uhsjha 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 and radius 9.0 cm rolls without slxjv:guzm v6e,e slc490n a. v0ipping down a lane at 3,cv09e:4j6mav nuxlg z ve.s0.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the suk7im,kz2 y9hzm of two terms,k 7h,y9zi mk2z
(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 qx,/vn 4 e/ymvzf13 g.jad: pf*y6ataand a radius of 7.50 m. How much net work is required to * zfnaa1: eaty djg xm3v/v6 ,.qfy/4paccelerate 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 mass 1.80 kg starts from rest a daftbuy0* h9u7-ij7cnd rolls without slipping down a j7h0 *cau-yiu 9tb7df30.0 $^{\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 ik1weejr. ya pl4i0 1q-ts tip. It starts to fall and its lower end does not slip. What e.- 14we1qarlkyj p0i will be the speed of the upper end of 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 rotating on th*boi9s r 9k6ixe end of a thin string in a circle of radius 1.10 m at an angular speed of 6*ixkroi 99bs 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gn 8h,ep ub 8gn+b1clh)rinding wheel of radius 18 cm when rotatin,ugnp1) ce8 + 8bnlhhbg at 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 m-o/u)fys.l(n9ofmlb )ze j4 kgq4-d platform that is rotating at a rate of 1.3rev/s If he raises his arms o- 4ngb94)o (m.uq)/emdjzfsy k -lfl to 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 her moment op )zr;h7 i0waa:zifk5ax3ky b 218nbt f inertia by a factor of about 3.5 when changing from the straight position to the tufh7x);1bznary w2 ti :kpz308aaki5 bck 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)hwgvg;)v zo* )gm mr/ her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final r/)g*mgohmv w;vg)z r)ate of 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 throug7qdeom esh 6a/3 ;f rl1g:;dsqh 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 clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is t;rm:ae;1d37sq6el qsgo hd /fhe frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinnif do t/a1,u:)nw/ vcdeng 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 angulars a04c;ys6ii n momentum 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 inertia I is dropped onto anr9l7cov wj s /x5w*.uz identical disk rotatinw*rolsu7j/5xz v c.w 9g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at inh l idm8+1twq r b1l/m4qh).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 of a rotating merry-go-round plat/lx bldaaxjv7 /u:3)fform of radiu3 )7:ux/afba jllv dx/s 3.0 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 with an angular velocity of 0a0c+hi*. z)p:xvvtzhiw(9h d.xi h(z*zv0hd :.av+tip9wh)c8 $rad/s$ 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 intob 8hf6s4(v1zn gvw/or a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radh1zf4vs( o68rg b /wvnius. Assuming 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 in excess okii0rz;in, y: f 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 9 e ir 1(.euaz,g8xsi;m cpga-$ 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, initiao-fx*he m ,ifej 96f8slly at rest, that can rotate freely without friction. The moment of inertia of the person plus the ijh,-s6x*e m 8fffoe9 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 stands at the edge of a 6.5-m diameter merry-go-round tun7g+5ph+wb.slzco9pnt(v-n 74ocv fk*d e( lrntal*ot5 -(cp7hnd(kp o7v+c. n f+vbngl 9sew4 zble that is mounted on frictionless bearings and has a moment of inertia 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 rolls on-8sh *k p,jqxqy;kwl: the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person wiq,8yp*kj: qx;-hlswk thout slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Ear-yc d1on2x6d3kc n k1lth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbitin1no2-3xk6k 1 nlyddccg the Earth.)    $\times10^{ -6}$

  A cyclist accelerates;.t 46 mb ojxt8ad +uvoi i*6lpn*k.vs 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 ofmo +;ah/to h+pe( .ksx a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delk(+pm.+t ahs;oeh /xoivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical da m+ moz4;arrm.57yh1c-d -qpnyia u2isks, 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 7ayzpm2da4m-a5om+ -h cyqi . r1un ;rm. 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 the angular sppok a/9m:w(gnu) 1zfw eed 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 jb).w +z3p d6xrfj; axl +j2nlgreat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-qun+xdflj32 6 .;lrzwjxj b)pa+ arters 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 icvw/ ;bz10vrgt to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass 0grw1vz/ ;bcv of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and 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 /5l*miys h 2c20jlc jjan $H_2O$ 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 rolling on a horizontal surface k wr+iaid.nd-)re.1eat 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 friction) f/miwf nj,24j about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then fiwjfj 4n/m,2 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 standin+rn10 3 lubpm:qxz,fgg vertically on the floor, and we want to exert a horizontal force F at it3gx:,mz1n+flq bu p 0rs axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is pl)k7ll ) -4vpo1tq)2xk 2ilt8 iikbumaking a turn with a radius The forces acting on the cyclist and cycle are the normal force t)8l4 2bi i7xqp- t1p kli)lk2vo)ukl$\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 rock into a sling of length 1.5 m and begi9qi6r.ukj ib21pm , odns 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 feat, adb1,iru2 . 6qmiokjp9nd where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder/is23pe8m g htt5ce8ing;-v ona;o; k and her arms as thin 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 oc ;t-av8oi tn gk;/ h3ie8 g;m5sep2ntightly against her body.   

  You are designing a clutch assembly which ;tp;9mn lysz7consists of two cylindrical plates, ofsny; mp tlz;79 mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiv(52 v8i86v lrcv5cxxm fk2 tjus r rolls along the looped rough track of Fig. 8–58. 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 withov v68kf i(v52m82vlc j5trxxcut leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do noe k)p4pzzn 0 tv:n7g:ht assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car;+ouup).mk cr reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was thkcpo +umu. ;r)e 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