A bicycle odometer (which measures distance traveled) is attached near the wq kq zxn4+6-7h5(c l qqvh/czjheel hub and is dkq 74h6 cv(jxq+lqncz /5 h-zqesigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotat40b dhh0fnwqp9z2 tp3/t7nz tes at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular 3thzf2q4d0/p7 t9nnzbwt 0p hvelocity 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 ang jfty l0vn9mehdv9** /*hf 6qoular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largerpobb7w7n, 1gmgn2gd 3 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 on3 tstf,e95 pk96 rupdo a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may hesftpe r996,t 5opku3nlp you to answer this.

A 21-speed bicycle has seven sjxsa /(io 7 mgpa 0qpz/9r3hz:prockets at the rear wheel and three at the pedal 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, a small fronz9 07zhmgo:rpj/sx/a(q a3pit sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with f+d qzq+q :(zr.hooqn/lesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dq zrz/hq+ .d+ :oq(qonistribution of mass is advantageous.

Why do tightrope walkers (Fbms 3 ++8 r:ui5mq2)w:y5+ueir+pido f fckhoig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? Ifl5 cnxh wog1-egf 1-b6t mq18c the net torque on a syst1xqm1 -h l6g8 ge1tc-nwf o5bcem is zero, is the net force zero?

Two inclines have the same height but muyhip;128 pptake different angles with the horizontal.28h y;piputp1 The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. 11qn7e xqio.;6zche(f h omv *2y5dz/s, pxjvOne sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the (c pef/;s6i,z5. 1v1v qh*x27y jz xmoohqd engreater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same ras16bf,td)a9oj 4 p lnldius 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 the bottom? Which has the greater total kinetic energy at the bott bfnaljp9641) sd,lotom? Which has the greater rotational KE?

We claim that momentum9qpjy z 71bx12owf5r p and angular momentum are conserved. Yet most moving or rotating objects eventually slow down and st1wxqozf 917y 5 pr2pbjop. Explain.

If there were a great migration of people towai)kolw vk)kb x4:5rr*y 1pjy 1rd the Earth’s equator, how would this affect the length of the dwrj:v1)xik y5)14pbo k* lkryay?

Can the diver of Fig. 8–29 do a som;v 1krh hlskt8kef /wjtr75z a9r57t1 ersault without having any initial rotation when she leaves thlk 8rhs5 ht1; 1tj9w5e a/kk rfvt7z7re board?

The moment of inertia of alrfik m8r .:+u rotating solid disk about an axis through its center of mass iu+m.ikfr8lr : s $\frac{1}{2}WR^2$ (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 stp7* *83bqys ufoxxj 7 4bxuyz6ool holding a 2-kg mass in each outstretched hand. If you suddenly drop the mu* jzoy43qxbps u77b6 xf*8yxasses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,q ;ruo;l )mhp9/rqq1x one is hollow and the other is solid. Describe an experiment to determine which is 1oqrpu;h9 lx/ mr;q)qwhich.

The angular velocity of a wh,1gqxq)c7;2n .if kb qu2tsyxeel rotating on a horizontal axle points west. In what direction isug q2.t; nysf2)cxq q 1ixbk7, 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 increasing or decreasing?

Suppose you are standing on the edge of anq4 e ltb2v ysjc+p-4ay1*ev6 large freely rotating turntable. What happens if you -vvs1644 ae y 2pecblynt*jq+ walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly.)ni;z sj5ao 0r f*ylu/ As he throws ouy asr ;/fl)nz* 5j0ithe 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). Explain.

On the basis of the law of consvp7)n e (ok:cfki 191*4orywj)xet0qpjnj .zervation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss oneoj7 wkc4:i jnnp.1q)kf09 tjxyvoz*)1 e(epr or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesgi lek;)n5e31v;jhjh in radians: (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 coincidence. Calc esb z8xnu1w2)ppo8o /ulate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moosxp8/z8 owo ub n12pe)n, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earthh4/2fc. lt ,qrxeb( aj. The beam diverges at an ljbh ,2r/e c.f qatx(4angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rota5zxlbks*us,8is5 e+vefy1eutcq 8( 5 te at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is svs liz,eeu5u+51 8f(q5 t*x8scyke bthe angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the banecg g/7i1l6nw:l4w8 geuj k0ll makes 15.0 revolutions, what is its diame7wlul c ee6jning 0/w18gg:k4ter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions *342suqzz2cc*nbgbx do the wheels sgz *c4b 22xb3n* uzcqmake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angul+uadrrk4157p,eg a)x d+arze ar velocity a ur 1 ra,pdad+reg4k5 +)xez7in $rad/s$ $\omega$ =    $rad/sec$
(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 4.5)7jlrd/c bp( )n izif0 s (Fig. 8–38). (a) What is the d (7l5i fbjp)iznr/c)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 velocity of the Earth (a) in its orbi c2ld205pc 2gk2uyvmm t around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofnln jldc: 4o,1:8c1z cja.a aatj-fx5 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 axis ofb4gos qp-e 2m- rotation is t po4 esq-mb2-go experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel ac .*3a kiojqtbsy 4r lf1u zl,o3k),v(pcelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determitkylpj14,3k,qo r i3bf)u sl* (.oa zvne
(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 radiu5* r(s b cgw/m9;jsrrds $R_1$ 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 spacecraft put thr4un)rygoudd ,e 00 b:emselves 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 revolution every minute during a 12-min time interval. The spacecradurrbeyn d:,o ug04 )0ft 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 dzqj 9xt.y)xk c/r8xpq/ 9rs6d. nel-15,000 rpm in 220 s. Through how many c/qjdx .xq-kpdn 9t/x 6zr8reys 9).l revolutions did it turn in this time?    $rev$

  An automobile engine slows down fro d6fq91 jzcn 5ol:qhh9*d9y+9 gnt kegm 4500 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 flydfy+ *kxcvqltm7rt( .9xwq9 : tp7 f 0/om(euling highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 compcewk vy t.*7 xr9l/fx+q(9lmq7:tdf otu0p(mlete 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 accelebypv goaly )085zn3 ;crates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tin z3b0;5v ycy8ogpa)lme?    m

  A cooling fan is turned off when it is rc 9x 0*fei hlf;gt6sad7/dsm6ru 1rx9unning at 850rev/min It turns 1500 revolution0 r7;g6i tsurc ha1s md/ fd99fxl6*xes 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 cxri( 5dwusc:db fr n, 4lr8a0*ar reduces its speed uniformly from 95km/h to 45km/h The t:rfnb450 *d(uc8 r xi,walrsdires 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) 2b/qofp21p . p5*rkwfrcy vs from 95km/h to 45km/h The tires have a diame fkf25rr./wov)2yspq 1pp *cbter 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 a ik +x:/7wmwzg8ivd9d ll her weight on each pedal when climbing a hill. The pedals rotate in a circle of7:+d9dikx z/ 8wgvwim 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 wid298ysfda uf33w1-cxagftr* p)pu 7wze. What is the magnitude of the torque if the force is exerte 7ufyuzxd9 *-23f pa8ratp gcww3sf1)d
(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 wheetug8 + eoloqo5 z2flf ue:/c35l shown in Fig. 8–39. Assume that a c8:e2t5u /5+oqffogzouel 3l friction 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 eshm8 :a) ,rzxvb0tg) d -e2rgjnds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position atds ,bv rxmj)z8g:-0a he2r g)nd then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylij,7sirw8pjr; +t ige:nder head of an engine require tightening to a torque of 38 8 itejp ,gwr:; +rsj7i$m \cdot N$ 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 radilt1mm-o,3.ok ue r0axus 0.648 m when the axis of rotation is through itm ru,tml.3kx0o 1o-ae s center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in di8 y rg/)vl1bxovldg 10ameter. The rim and tire have a combined mass of 1.25 kg. The mass of thev1g vlrd g80y1l)obx/ hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the endg)-)ak(bachp ;zmj. m of a thin, light rod is rotated in a horizontal circle opjmbck(gh ));-.a az mf 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 wheel rotating at constant angular spe 7sxj(( 7vorld89h/hcb j/ft wed (lxjdbh(htfc 9w/87v/ j( os7rFig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 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,/c5 2y 5efz 0wvmlfkre;eda. inertia of the array of point objects shown in Fig. 8–43 aw5l mf 0d 2/zrk5 yv.e,;eafecbout
(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 cons640 qvuyjhkt8ists of two oxygen 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 satellitn ex7q sqwop(cx1a) ;ke6uab6 9 w)n/he spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a maak;wuxneeo6nq71wc q69x) psa/)hb (ss 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 abdy 2hucck /). mass of 0.580 kg. Ca2u )hbkdcy.c /lculate
(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, acceleh.gfdv s*nsw*lc 66 0xrating 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-driven merry-go-rounmau,qkiv)kg qk +.vtx710 w.ad 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 2.7.vq1kv mu. t g ,wx+)0kqaika5 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 1q , fz;1.2vwhafqwu1z0,300 rpm is shut off and is eventually brought uniformly to rest by a friah,vqwu11w;z. fqfz 2ctional 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 accel*)y0pek01mcs p5rx 2 esbksj ,0lju b-erates a 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 m bz+2zg4z7tfball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action ofzt7gm+2zz4fb 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 shown in 7lrb- 1v+ g -ur,a8og8xf2r ocjtsp4l Fig. 8–4-r87 rlrfl,bgogx p+auo-c 4s2 tj81v6.
(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 consk cbwy- 46b248gndjj* r gp:xsala 8k5ists 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 f xft9-sp2w-a)ch gdp(ull turns (revolutions) an)9-sd c (fpgpawx2h-td releases 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 inertia of l468fi o). :8uemudi eg v0ykj$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine +,u pzssde 7 .p(*bqpoqie7+uj klmac8zw7.- develops 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 7.3 kg and radius 9.0 cm rolls without slipping down a 9+zngs3 p a0w acwa5e85ot(n-ebf7gr lanaa35rbn9+ 7g0o t -egfw8spe( nc5zwae at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earg j-asuc xoc,-6r2i+ o+fd5lath with respect to the Sun as the sum of tjo+ 2roi,xauc-gca 5- dl+sf6wo terms,
(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 rak/f0vs4 xi tvi u(vecofl7sr 5+qong4e6 7 4s1dius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 v x+ko(t scsgvnrof5elq4u7i 447/e6vifs0 1revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.nh *g-q -kbc9b0 cm and mass 1.80 kg starts from rest and rolls without slipping dow--bcq*gb kn 9hn 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 vy. 9t4- uw+jr,xl 8keubnd yz2ertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of-e d4uzwyt,xjn b y9k 28+l.ru energy.]    $m/s$

  What is the angular momentum of a 0.210-kg b;e8x6 s.tzo de.rzxc8k7m zz8all rotating on the end of a thin string in a circle of radiuz7ots; e 8zd6m.xrxkzze 8.8cs 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-bc6 mnqe n7,c/kg uniform cylindrical grinding wheel of radius 18 cm w ebcm67c qn,/nhen rotating 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 platform that is rotating at a rate x5aqqwu45 v:xk1qpr ta u ;+7t.ws8otof 1.3u8x5.:oqkxqpruw5a q +1awtv4t7t ;srev/s If he raises his arms 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 showu+ ni 8)u8jzsfb5s,fqoi--is ok .o;sn 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 rotations in 1.5 s when in the tuck position, what is8)n ouq js ob sssz f,i8ouf;i.i5k-+- her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial r3eplbb0h r . l/b *f)t0yvgv+ban4vx1 ate of 1.0 rev every 2.0 s to a final rate o4a/v b03 +nfxlhbb )b.tvl *r01g yvepf 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 throughca d6n .rzye507d0ylzs*f7zm its center at a frequency of 1.5rev/s The wheel can be yfm56*ed d0 ay0z srcz7l. 7znconsidered 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 the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular miud24h (pe(mx omentum 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 momentum ogqgl51gadv :y8, /tfp f 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 dro :rh/* phu4cvdhi/l,z pped onto an identical disk rotating at angula: 4v/h/uhc r,pldhz*ir speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns dp. 6wyuoe7 0iat 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 agn6 v4i.tcp 4tcb4mi/a3 6 blut the center of a rotating merry-go-round platform of radius 3.0 m and moment of inert 4u6ttab b46icm p4cvi.l3g/nia 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-/tb8do- a f, 3ey0yehcgo-round is rotating freely with an angular velocity of 0.8etyf-8/b oed 0ya,ch 3 $rad/s$ 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 collapsesinac51x8lylz +) ,9 lp ylh(es into a white dwarf, losing about half its mass in the process)5n1l(+h i9, lyl8 pyazcexls, 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 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 excessxi5( e+ sn6zxw5l;tan 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 *slxy2 o6fb5x$ 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 rotp0(kz e y-no 8g u3*wq-jxho5qe 9et)gate freely without friction. The moment of inertia of the person plus the platfop9oe8n*goj5hu(z3 xq y-t)g weke0-q rm 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 -xw fb ubqun7:dbri;n.;u1 )tof a 6.5-m diameter merry-go-round turntable tha-x bbu:.u7 n;f t);qd1u nrwbit 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 ;,1cwm n3 vrmig ,rj;drolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance Lj3vm ;,wngrdi,mr ;1c , 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 thti w3vnf;c;c1y-sql tg q7*0i e same side always faces the Earth. Determine the ratio of the Moon’s s1qvs;7ti c;wytq3f i*0lngc-pin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1+,u::2 : 4se4vhl pfu ibelucx 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 bxr ud.fn14:scp. /, swdr(pkcylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-dcb(susx.pf:.n pr4, /kwd1 rs interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrio-lcxc*, 1 qk0hj dy8tcal 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 co,yc xtl*hdc -8j1qo k0nservation 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 speed om l)fdj/l, i66p/p)c y2zuif jf 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 of2a(xu78ue gknnc;qe, our Sun, but with mass 8.0 times as great, 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-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 mas7c ee;xnqug( 2a 8nu,ks carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autom6gv8:gis a6a y6hk (miobile is for it to 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 be able to travel 350 km without needing a flywheel “spinua86vyhga 6g: kmsi 6(ip.”
(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 illustra 5onmzrggy:n/rqs,so --0 dp; tes an $H_2O$ 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 horizyvxky*a k+;92bz l2 ;fw*zrx rontal 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 (it4f*bfo(yr4qt 7bh,n( ib 7av.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and 4 (,4ortqyi7f*atbnf (hvb 7bthen 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 on the fl3to :1v vdx 5xu;w j2gqfy8c4joor, and we wv d21xv;5 o: 3gxj4qf ywtcj8uant 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 height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat rooy x17:e)p0l:bhvm)q8 tr7dogput5h ad is making a turn with a radius The forces acting on the cyclist and cycle are the normab:o1ryq5udt )phog7)e 0m 7lv:phtx8l 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 rock into a sling of length 1.5 mv8m+ohmzto* gwxb/5( 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 feat, and where does the xvmhobzo8 t+m (*g5w/torque come from?    $m \cdot N$

  Model a figure skater’s body *o0b8j qjyn4m)pm 3zoas a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning ska4bojo8 p) 0*nmq zym3jter with outstretched arms, and with arms held tightly against her body.   

  You are designing a cls3lgcfn+-pa , 0muo9x utch assembly which consists of two cylindrical plates, of mafnu c9,g -+mo0pl3xas ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r(nsbf u, +rtp.adius 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.rfs(bp+t ,un reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buzh 5dn: l+twd2t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifxzu7 tlkjy 2 ww9-a(3ral44ipormly from 90km/h tal2wrw4l x-(uap3jzi k4y79to 60km/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