A bicycle odometer (which measures distance traveled) is attached near the w;qip( l,q u +r97,a zhc .rbhfzi:hqx.heel hub a9ah c,.;qulh rx q7zi:+ fhqipbz(.,rnd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a p)l742tnf v-da r7(diuips u5m oint on the rim have radial and/or tangential acceleration? If the disk’s ang74i p7i5uda- vts)f(ulrnd m 2ular 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 angular veloci-c 8nar2 b6e zg+qfzc1ty $\omega$ Explain.

Can a small force ever exert a greater torque th eigq:nrzw;onbt0id 897 75bian a larger force? Explain.

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

Why is it more difficult to do a sit-up with yok3dg7 )jg jyj)ur hands behind your head than when your arms are stretched j 7gjj)d)3kgyout in front of you? A diagram may help you to answer this.

A 21-speed bicycle hi.v4ttrz a /p6 ji6a.pas seven sprockets at the rear wheel and three at the pedal cranks. In which 6 a t6p4zvpj i/r.ait.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 front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast h9,q whu cywbhp 0v4m83ave slender lower legs with flesh and muscle concentrated high, close to the body (Fig4wchm9,qv p yh 08wub3. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a ljh mv9 i:ax25zong, narrow beam?

If the net force on a system is zero.ums3/ik.p:,jovk 9 sh lm1qu, is the net torque also zero? If the net torque on a system is zero, is o/u..vmkk,i 3pm 1j:h9slqus the net force zero?

Two inclines have the same heitwd6o2xii4 ,li20vc dght but make different angles with the horizontal. The same steel ball is rolled down each incline. On dc,lodxivw6 t4i 0i22which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollhxt f;d.z8s e;kn2*mxing (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reache.txm;*2k xn;8h efdzss the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinggqqir6 4yti3/; p+order have 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 the bottom? Which has the greater total kinetic energy at the bottom? Whyi4gq/6qr; r+ go ipt3ich has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mostv ew g4pdv76j.2uii8e en1yz83we 8pg moving or rotating oyew8gjn ed 371v z2e84wpii6 g8u vpe.bjects eventually slow down and stop. Explain.

If there were a great migration osw z )dk1wveb,p1fkj/w :9;c uf people toward the Earth’s equator, how would dezwkwp;k:jf1c1s/ b,wv) u 9this affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rot+.qc drdn m+ .p;,zea4xo1 bxtation when she leaves the,t4d ; +d.meor a1zxp+qbx nc. board?

The moment of inertia of a rotp0 g 80s0lhss fv tn;.edf ;gg:0r2gemating solid disk about an axis through its center of mag;0hr 0eg0fmn20:e.gs pl gssf v; td8ss 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 sitti ,*)o- bfsjnvsxa.)9wa0hkz k ng on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop 0j n.9o,s)zwhvbxf ask- *ka)the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and thij b as :7:jsyq87ua6te other is solid. Describe : s7:byjsa 8t6qiaju7 an experiment to determine which is which.

The angular velocity of a wheel rotating on a hoji7,xq3: eqpr rizontal axle points west. In 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 acc,: i jq7qxrpe3eleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the u :;jq)- 7vfb6:azitmeii op2 edge of a large freely rotating turntable. What happens if you walk towar:i2fpo-qjzi7v b ;t)a6:e umid the center?

A shortstop may leap into the air rob25(pmgt.n.el 5tj 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 notiomr.ltn2e(.p5t 5g j bce that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mg.hztccb; +ut;h+drg + 0j (iu +;yirromentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep thtyihgdr++rr; 0iu+ zt;b( chjc;+gu . e helicopter stable.

  Express the following an0 -z hw/gomnyf m4;u-g9bke97 qjlmm- gles 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. Calculate, lvj :(,jd3h nm2yt,sj,xlk0f using the information inside the Front Cover, the angulal0 ,s,3 nm:t v(jldy2 kjf,hxjr diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The bi; 7jsg-7 fvt0+qsg yhtjuu/x9d2n 4 u(crqb;eam diverges at afg (du4bq7tgjs7-2/xh 9 i;uvnscjr q+ u;t0yn angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm56 bxvdlk*ge). When the motor is turned off during operatel)5vkdg6 xb *ion, 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 on a level floor 3.5 m to another child. If the ball makc5jxk5v- li7i es 15.il- 5ik7vjc 5x0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutiontdlt7)77qlem ea0 r epc12. vns do the wheels maqpv1 cl0lt 77tern ed7e.m)2ake?    $rev$

  (a) A grinding wheel 0.35 m in diamjd1z)0 ;h4iprg m x8 awjds.3peter rotates at 2500 rpm. Calculate its angular velocity idj) 3w g0j;rzip1mxad4 8.ph sn $rad/s$ $\omega$ =    $rad/sec$
(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 compledtn)k4zbn)h 277h yi rte revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cnh4 )7i )dryt7zb 2hnkenter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of theanhp 9p2g.h8ee ,sc4.mf j0+l jvc 17hhudz5n 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 of a j3p n v,w3wma9nrbs70point
(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 centridvj5g;9r:w i yfuge rotate if a particle 7.0 cm from the axis of rotation is to experieng:ird;9wvj y5 ce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2 nrhpt;p5m /n(80 rpm in 4.0 s. Det5(n rmhp n/tp;ermine
(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 radius(,bg ob)jbm hskk )yw3-.q +ee $R_1$ 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, astro-+ m-q nzqpv5z9bq 1fznauts aboard the Apollo spacecraft 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 revolution every minute during a 12-min time interval. The zmq-v-nqfp zb+1z59 q 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 inq8fhj+:i.4; w tqmp up 220 s. Through how many revolutions did mqt+p4 wi hfp;q8:uj. it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2z fq2 ks; lg7x895a4lxivo*et.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 whi1keyw7:c -gh xx vg/z473r6/cxabb pu rling “human centrifuca v c1h6yxb z4x7p3e-krg: x 7buwg//ge,” which takes 1.0 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 accelerat5gqb:r))1g ofags xzj.mg .g/es uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on) ag m:bgzg/oqr5.g. xf)s 1jg the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turrd5 fph:*o c2z-okz;-3v vytrns 1500 revolutions before it comes to avcr: *yk53h zfo-p-2vozd rt; 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 a0*0zba 5j vnbl4iy3cls the car reduces its speed uniformly from 95km/h toz5nba*j l0lvc4i3 y0b 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make mub ql(w+r09k65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80m rb(9lwu0 qk+ 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 pe / r8o3gbapz 9 -6odzntl3;cjndal when climbing a hill. The pedcar;o-t8z g9pnl3 /b6doj 3n zals rotate 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 meq9b;1yc. rsa9)a/tryirofe el;.x 3agnitude of the torque if t irqee rlr3xo;c1eby.)/9tsf .a;a9y he 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 to4qb:dn5 0umghie4 idy l*o,u4rque about the axle of the wheel shown in Fig. 8–39. Assume that a fricinu4h0o,y44 u: dlbed5 iqg*mtion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,td xapqivba:dkt g)3 86)g; d3 are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and ttaagp ):ib8)6q x ddv3g ;kd3then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening tol6)(rxel s lnu5pr7t/ a torque of 38 lr /(t)p7x5rlu lnse 6$m \cdot N$ 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 x1 .fzziwlgzt; n3 bebjp.8-kg sphere of radius 0.648 m when the axis of rotation is through pjznf. .xtw3; z*l8eg1 bzi b,its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6n*cxzg, ww(bwfl)2hkp3hw 3 + 6.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can *wwh3,)lp (nhcwxwg2+fk b z3be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball hfy r wgb6.n8((mzod7on the end of a thin, light rod is rotated in a horizontal cf g(b6.8(drwnzm hy7 oircle 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 ezys 3q*j+xr-4zrd *o+skja w, 2te5d on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her haq3r4zwjrtzy d+s* j2ke-,a+ sx5*deonds 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 of.wr-; r- bgsjb z45lnbjects shown in Fig. 8–43 abourf g .n-js4rlbbw-; 5zt
(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 t 1r9qlq8h7oplj r-:p 0pu8cywhose 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 rate,b1u;q*i ;mbhk6zu;nm pum3,d.j4tw r engineers fire four tangential rockets as shown in Fig. 8–44. If the nbu4i 1; pk wmqt*6br.m;uud 3hjz,;msatellite 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 )are r ek/4k gjza7s:gp*r*v,tg4/zd and a mass of 0.580 kg. Cal zjrzdrkg7/ *tvr/)aeg4 gekp *sa,4 :culate
(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 from r;0hzvylo . qb3.vwen7ka3 j5r est 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 me3g9x h59.owd 6hpztu krry-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 torqud thu95o3x k6gh. zpw9e. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off a3ehbt-,sb 9j k69d cbecieb;hvpi) * qyin62;nd is eventually brought unic 9j6y;t9bn,ee hk;* ) -pihv b3q6db2cbesi iformly to 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 3.6d (; rcsscn9ygd p 46cw2ie2,x-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 solel.h k3w:kt2jm pu wc.n8y by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 j:t.c uwkpk283 mn.wh $m/s$ 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 +hseura/p qc.q x (u;y:im9 w1thin rod, as shown in Fig. 8–46. 9+hq ; x.s1yau:u(wrp/eiqcm
(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 cons*dy*x xz(w oql3y t1(wists 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 (r s0ejm1f :73 wfreyuq7kw5c4hevolutions) and r kfwr yj0f1 e:5swcm7 43e7uqheleases 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 ohaj/)4 k5k +og8m.yg w uesk4a moment 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 developss-ftf, rxhm* 7z j7.rzsrh/g) 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 i(.mqd vn35 ola lane at (no3iq 5 .mldv3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect tojn(y ,r e15c.fm 60ytkgg srl) the Sun as the sum of two term (,rjl5) gt 6rgkmn1y.s0 fceys,
(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 radius of 7.50 m. How mucq labe8gl54k 2h net work is required to accelerate it from rest to a rotat4eba2 gkll 5q8ion 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 and ro k0 :1wlpz,jumna( c stj;9yqnf/(de-lls without slipping down cfma js;n ,:p(ket j1-9/0ylwqdznu(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 vertickidy 0 xe)4 krfp;3+mlally on its tip. It starts to fall and its lower k )4;m0 +i3kpdfxlrey 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 energy.]    $m/s$

  What is the angular mometm- n 1j:ta(v vqt(wh*ntum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at a(a: j*ttn h1t vmq(v-wn angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniformme t9pup r* f30y9x0tf+kd/og cylindrical grinding wheel of3f* mxr90 fytpp0dug/9 e+kto radius 18 cm when 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 rotg; w2vvnp)m zr*bo)+9u6 xnjxating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of r bonxvznw*2u6pgv;x)j+9rm)otation 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) 8 n.v)cc* 2 q7cek8x2ksktlhb can reduce her moment of inertia by a factor of about 3.5 when changing from the straight positio8s qt*n22)l b7ck8.vc ck khxen 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 spin rotation rate from an init*j3on(p8y0r-f rn4 cpkm dcqs *wn-y6ial rate of 1.0 rev every 2.0 s to a finalnrdr6np *f8wy*o (smqjc0k-cn4 -3 y p rate 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 through its c03;tbe9h u58:4 l z/ bsfola8ct2cvnhu8tl m wenter 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 tzn8lb cw3f5sc e h8lt9ltmbh:/4uv2a; 0 o8uof 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 a figure skak 0xq4nlqz27 fj*g p,uvb8x+zter 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/q-gh 6d9lb: tlbayx- 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 inertie8k p( yfbcjubos m1t/-(io 9(a I is dropped onto an identical disk rotating at an1/cyf beb ok((-p so9umtj(i8 gular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a-k lwdxu6+1 jqj z-z8tt 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-pa8h++g.f +dm 0y1dybr sy ok)go-round platform of radius 3.0 m and moment of fs .b)mghdya1+0opdy r y+8+ kinertia 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 a :8zs.mz8 elpe 4booy)n angular velocity of 0. pm84e .o:zso z8eybl)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 white q lr:pf;q1oh- dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its ex;q: r lpoq1f-histing 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 excesldyehv,1fclev h*2. 2s 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 k atsq,h ;td;:,/uz tx$ 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 atby 5k ; oz(+/wym9wtiz rest, that can rotate freely without friction. The moment of inertia of the persozy iwm/kb+y(o 9zw5; tn 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 stands a,wt)*9yucol cy u3/e. ;fn z4nx v/ixft the edge of a 6.5-m diameter merry-go-round turntable that is mounted)vo/yyw/if4,xuxc t9nlz 3f*n. c;u e 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 the ground with the end of the rope lyin vlhi87+vm:tf+aa 3c mw)/tcwg 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 perst7tc ahi/wmc l)vvf ma ++38w:on without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwayk wdj+fwj4bwyrv q9.h0;8bkead:( 6qs faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angu(+.jqkb;q6k948a0wjbd h:deyvww r f lar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate o-6+6c/jq(s2pf-nvmaa b vb0 oyw ihf0f 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 ykwl vedrbq -. ,2/quk*v2ux0zax9 5nof 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 9wde2n /y25u.-vakb0 z uq kv,xqxl*rtorque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mmr0z1 hy .sxejfnvxc917 /g:nass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when v1: j0c rx nxzn/m9.gsy71fheit 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 of the rear wheel (i 305ni s-fjej$\omega_R$) 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 ma9rq3:j.35z(h7 jxshtfrcjv c ss 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 of3x ( :v qz9c fjsjhhr5.tjcr73 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 uyfv.l1 730l e4hzuai yse energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrica4 zyu1aiv.l el3fh7y0l 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 illustrapc*rxx wr i0 )-+udh 3*1swbnftes 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 rolliij3t .66ogr ehng on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictstyz4q,se;u 89wr+w yc e)r9s. -w jqu7aj nk7ion) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular accssywqzu7.4c-uy)w8 +qs ,etnwj9ej k7r ; ar 9eleration 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 ve79w l-ebtaaksvv ; :4d5vub .vrtically on 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–u;.d b: a497ktselav5 vbvwv-55). The step has height h, where h

  A bicyclist traveling with speed v=4.wg 7lx n7.c8lv2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cyclel7 7 .nlgvxw8c are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock ieb s az74w8*tk0h8g+dtzg9umnto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, a ta9z m70*gzhb +4s88deg ukwtccelerating 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 skatedj-wrof/7f 7z y) bs)*w wh:ytr’s body as 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 spinnidf:w )h-jt)srfzb*w77 w yy o/ng skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plate4+ bvnhxn,hzi vin7u)2jpu- ,s, of mass vhu-in7,v n2nb) u4,xzhp +ji $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 F1hgjinqe1 b3fps3o 1 7ig. 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 without jeb 7g fo131s1hi3npqleaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuv.*qpy3+l xrw55a*wdlj vz h7me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly fu/va0oc nw)w:rom 90km/h to 60km/h The tires have a dinwuc0:/ owv) aameter 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