A bicycle odometer (which measures distance traveled) is attaxfby28 3ta )cgched near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle w 3 2y8atxgfb)cith 24-inch wheels?

Suppose a disk rotates at constant angula-s 1xo2qfk463g nhselr 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 sk-g nx2 s4qolef6h13 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 anh jg6m xutr:k2m:,h)m gular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?d+sf *ks ) t;q/phkxe* 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 your hanx ,vyx;ei5(ukds behind your head than when your arms are stretched out in front of you? A diayi5xk; , ux(vegram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear u0zuohfiri :qr/ b8o.b1d( t9 wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocke urozbr ti1dui /f.(809o :bqht? 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 ha0 wocm73e0 rbwve slender lower legs with flesh and musc00wwmor bc 7e3le concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Figqwhn ek9k; 8/g. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is theb:;. o 3f4boum5 +:hrkgvtmtt net torque also zero? If the net torque on a system is zero.o:+4tbf3 ghkrto5bm; mvut:, is the net force zero?

Two inclines have the same height but make different angles with the bt5a 0ydp4u145h f2eh f2:b5rsfldzbhorizontal. The same steel ball isf 40zy5de u5 t bh2pbr:dl5fahb4 21sf 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 inclli.t *drgt./3p1q apaik( )mfine. One sphere has twice the radius k1*( 3pdpa mt.r)fq .tii/ag land 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 bottom?

A sphere and a cylinder have thstv): t3ga1 j0jwi; l3nat8xoe 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 ;ov0s lw) :aitnja3tg18tx j3has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conservednp6bf52vcec .o6.rp; c +ekhu . Yet most moving or roeu+ecfoknpb.hc.r vp2 6c6;5 tating objects eventually slow down and stop. Explain.

If there were a great migration of people-y /1 ctnq2h mqei,gnwnd +4t, toward the Earth’s equator, how would this affect the length tny,m1qin/ h-tq n2gew,+c 4dof the day?

Can the diver of Fig. 8–29 do a somersault without having alo(w 7b/ k xortx8kg9)ny initial rotation w/8kxgoolkb w (r9x7t) hen she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center rf)-lgcwjp tx/. ,b*c hd /7vr bxgdccr w/l. vjpfrt- ,h7*/)of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sittin +cyzptt98f-rhj-fx0.x id4 ng on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the maszr dtj9f--pc 0t xx +8fniyh.4ses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howeby gf:(sa8 b.over, one is hollow and the other is sofo:b8bay s(g.lid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatin+x5jge liu 9mcpb b591g on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular accelbm gl5 9p9b c+j15eixueration 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 rotatinoa/l.-y,v w org turntable. What happens if you walk towal-o wy,ov.r a/rd the center?

A shortstop may leap into the air to catch a ball and throw,0 kqg9-d,vg: t5co 3xutbyvo it quickly. As he throws the ball, the upper part of his body rotates. If you look vtbqdo9k,5, xv0c: -y3tgugo quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discrr1o7(xw r s9 gp;fm xc+,ics6uss why a helicopter must have more than one rotor (or prsx pixr7 9ro(c 1s+fc, 6rmw;gopeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followin9se 2hm:jidm.j) z*jv g angles 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 coiu2ihacav2*xt jk*e(m,,nz 2 sncidence. Calculate, using the information inside the Front Cover, tmes 2i(*jaz,*,h2autn2k xvc he 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 Moon, 380x7ti)b0n z,p w -lui8v31e hcg2a+xsp,000 km from Earth. The beam diverges at an pe,x8t zi wgb +vpu13)xis-7h0ancl2 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 rota 2 /uwdpo0q db2d(57n;(k/ gvcw b2ewdowh 0ixte at a rate of 6500 rpm. When the motor is turned off during operation, towo k7;bgwdehqpuwx22/nd 2(0 wd(/bdvc 5 i0he 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 ti)yrtt9dyk6 0 he ball makes 15r6)kti y 90tyd.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in r);v 31lgq1(i sksv4ciz pru3diameter travels 8.0 km. How many revolutions do the wheels make ) vr( l;qki1usp34vs1iczgr3?    $rev$

  (a) A grinding wheel 0.35 y s8- jb8ql.s4c* h/std:slqzm in diameter rotates at 2500 rpm. Calculate its angular velocity in 8sjhbq :-l 4*sdz/yq t.css8l $rad/s$ $\omega$ =    $rad/sec$
(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 my:t0; 1 sxch+ uk4fp,vkz(wgi akes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from +1 :uxpkg;hsctk( ivf40y w ,zthe center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around kc:d 7 gkr0t0t mka+p4the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spsqkr+ a3pil+a0yo0 4seed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a parthq/ 0udns( nj/ch5 .n owkk4:pcnd*b4icle 7.0 cm from the axis of rotation is to experience an acceleration o/n upkcjcqbwnk.h d0nn/ ( *ds45:h4 of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 g jd kecl.ixjacdhi xam71c9/-.6a, :rpm to 280 rpm in 4.0 s. Determikx6jd/ 7 .cl-cda:em.icj 9ah i1,axgne
(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 1,m ddouc3y spz.oc,fe e*j.7 $R_1$ 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 spacecrnxvwxn)1xo/ ) wxm( j+aft 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-() /vwx) mn+ xn1joxxwmin time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpmk;dge+u f3o u-rsl 2nre 8/rq. in 220 s. Through how many revolutir/ eerqnd lk.8g2 s ;orf3uu-+ons did it turn in this time?    $rev$

  An automobile engine slows 3bd3 vi n;x9pi0s8)7ieb sql2r nzt(v down from 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 flying highspeedh105vsohtcru()3)zvc+ x :axddx 5 z s jets in a whirling “human centrifuge,” which takes 1.0 min to tux15rx0h3zs+do)u: vh()s 5acvt dxzcrn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rp3s-yzf6yt 9++a bmxw9s uz/mtm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveledym3 tfzz6mb+sua9sx + y-9/t w in this time?    m

  A cooling fan is turned off when it is running at 850rek2q;gkqb9 ;yn;hy 5;zarv;w z fxv6j9v/min It turns 1500 revoluti;9gwqqkb29ja hzzr k yyn;;fx6;;5vvons before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformlu 0 g(c8dpg0voy from 95km/h to 45km/h The tirgp(80vcdg o u0es 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 a *j+qhudb0ce9gfo d;7j/w/ jxs the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamet*9cxhqdjo ebu/g 0fjd / j;7+wer 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 ridin. 6ex q3,seyxog a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle6 s e3xe.oqx,y 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 forcekdq4 57jj;weaydab1 d3i:6by of 55 N on the end of a door 74 cm wide. What is the magnitude b d5ijq;a3:d aeyd6b41yw k7jof the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39b1r5ma 8chgr2torm86 q r6h7r. Assume that a friction tor mohq25rbr rc6rm8ra 17t86g hque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends o22tvgzp;1 ka af a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on th2a21 pakzg ;vtis system.

  The bolts on the cylinder heasssx+w 3 hncf 98lqkua883d m;d of an engine require tightening to a torque of 38 q3kd wl sc;8fx mn3ass+88h9u $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when g n h;qcvgga jizbcw)74 i/w :ahk9)7+the axis of rotation is/cvjz);h77+b)w aiqgh wigcgk 94 an: through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The r0* ,z kd;dnh nqo*8tsq6whi ,rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored nh*wstiodz 8qh;d kr ,0q *,6n(why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a c*)ic z-8xt6dvah 2b/g* u2hcp n 4dpqthin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculapvc4q2bxht*-2 za/gpdci8d6 n)c h u* te
(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 an o7 l0zz1,yg i1frd6hh28ldqrgular speed (Fig. 8–42). h l 2z,groy1rid 76f810lqdhz 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 inertia of the array of poiz /pq(nngw/d 3nt objects shown in Fig. 8–43 abzw(g nq/dp 3/nout
(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 of4s ( a1xq +qbq/qn97 fx*spohr 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 satellite spinning at the correct 9m ,wz6hk*jwqrate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satell6z k,w9 jmhwq*ite 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 i: f5,q6o+:m,fg5in air d .jun;d0a*rxwgr g8.50 cm and a mass of 0.580 knodi m+ rf 6wg5:a:u 5x,ig0jqgnr ; .d,*riafg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rrpr sqp)c8a;(z(r e.t 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 tangenmg zo n ,p.tv.9s,o7or qp,p)atially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform zg ro9o, q. p, avsn,ot.p7p)mdisk 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off 84)0w/hh.jj fhls btl and is eventually brought uniformly to rest by a frictional torque of 1 lhbh f .0stj)l/4wj8h.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) ivdbw5s;:fb 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 ball is thrown solely by the action o3r )iozqo lbp t9em6+h: p)aj7f the forearm, which rotates abouto3 el 7bzp)thm6:+rap )qi9jo 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 canbh ni)+2jn u/k be considered a long thin rod, as shown in Fig. 8–46.k 2bi/+h)nunj
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of twoo*pe(iqbs 2 8sbchxjb +6 )v(h 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 (rl 96o gzw(mml*evolutions) and releasemwlo*z 9g( 6lms 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 mo tk :z14mzk3xoment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a tor 0opng;)mqiwx2 7kgn/que of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of massk1; z89xg9zkcm0va kk 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3km80gvk kx c;9z 1ak9z.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energr2zmhg tonjfsakn*z 9gba/5y1r 2 ury,i4 44/y of the Earth with respect to the Sun as the sum of two terms,/2 z*sg 2ayuynn ati9h51zkb4j4mgf r/ or4,r
(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 7)a ninvdeen s* ys,4d 8*(rfv y(bo(wkg and a radius of 7.50 m. How much net work is required to accelerat(iw*v s7(vd)a y rsfny8 nn*ee4,(odbe 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 and 1w r7kk1e:t3qhi +wqjrolls without slippwjik1:+wk7 tr1ehq3 qing down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically oncw4,t cs*md (s2byil;q1y :ty its tip. It starts to fall and its lower end does not sltywidbcl,t cy 1y;4(q mss :*2ip. 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 momentum of a 0.210-kg ball rotating on the end of a thinauqwzc-( vkt7z4./ b jq5 yed8 string in a circle of radius 1.10 m at an angular speeeu/q kb5w-cy 7d(.z8q4 ajztvd of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical , ,x q-m gr9yv4ut4xowgrinding wheel o4 mrt4,o9vxqx ,gyuw- f 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-8 5qath 1*wca. ufzkrkq6xv: rotating at a rate of 1.3rev/s If 6 rqwh v1au-*5 kqax8z.kf :tche 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 shown in Fig. 8–29) can reduce her moment of inertia b(1.5ymewx 9ep q4vz tay a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotati4a1 wpxzyme.5 (q9vt eons 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 initial rz+ pmuh f1h2:mate of 1.0 rev every 2.0 s to a final pzu hh :m2mf1+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 arounds,l-u vsw3j* d a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg l*d-w ssjv,3u chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figu- 9qlx dzo+n.*0vdxqere skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Er.yuyb7jnjr:0 sj:m: arth
(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 duqc:u+ gfu 8/nw2bh5pq zci0: ropped onto an identical disk rotating 5ip:c2/n ufhqzu 8 q buc0+:wgat angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2(awvb ) s:p+l667ox rvfn+cft.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 cenc h11ly8chy5gl 1bt,cter of a rotating merry-go-round platform of radius 3.0 m and mom1 h1tgh,y5clc bycl81ent 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-li y4, 0u,lag6f;lwbugo-round is rotating freely with an angular velocity of 0l0 aub ,wfulg ;yl4,i6.8 $rad/s$ 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 whit;yaz3b(vl )/rsiul/ :;jl is ie dwarf, losing about half its mass in the process, and wizi l/i/; j;(3iulr ys:bs vla)nding 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 ing*rm5gw1m,( cecilb g l4 c7q2qese.7 excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 2dpthdadbpv e859m.kr:,/w p5cpl 5u $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely 2 zdrbty .+q3w6kl/jjcr6us7 without friction. The moment of inertia of theq sc+3/ 2ybwur zdr. k66jtjl7 person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-goa 8og7ro7 dkfsbm 0 uxwu19*6 2vg*kqi-round turntable kgvqdi27kr f*90x us g1w ombuo* 678athat 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 the ground with the end of t6/ids h8ro :vk0ukx401 whwtlhe 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 without slipping. What length of rope unwinds from 0/kxokthis l6uv 0w: 18d4hrwthe spool? How far does the spool’s center of mass move?

  The Moon orbits the Endr/+qfewl1*nw( d5 iarth 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 orbitain5( dn*/wr+felqd w1 l 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 /) p7v6e ecwti nd3uugj;s8b+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 y7l-fhk* 3x3qzuwlz( 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. Calculateku -3w qlzyhx f7*z(3l the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two sov. pm4 t7;vr:lh c5gea/g)oi l l1pry5lid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentricegmr/a1vv; 4 lpci75: t.hg rlpyo5)l) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear whw+nj/-gbo u9e / 0z ult0ot7qheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, weri: pt.y fhle72q m 7.-jp0mrfre rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational colla rh -:pm.tmp.7rfe 7y0qfj2ilpse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energyzigdo 78ss6gfvz/2 g4 d xgf5: stored in a heavy rotating flywheel. 6fd i 8ss4zgdogx: v7z25ggf/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 “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an tbq+wocjmj9ud5: 4o d,q o5)j$H_2O$ 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 su0p:tym+lbu lnq l:q, r )zp0+vrface 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 y,x:( ,x sh)lyrmx3lowki z,2ignore friction) 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, determs,h( zmo :rkli23lxxy, wy,)xine (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 rak8dfln; prw /8dius R. It is standing vertically on the floor, and we want tpl fkr/n8; 8dwo 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=4ip gcvs27c9 0p.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist andipvpc7g0 s29c cycle 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 into a sling omtdvec) jwt7gtu9 4 -*f length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a djte )w7- m*t g49ctvurate 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 skate 0wl3hk y j;byvp tc28aa+u+*xr’s body as a solid cylinder and her arms as thin rods, makinj+ u+albkaw 02y*pcxy; v8t3hg reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylwtzryp7ifm:2 za(2x y tg:d7b -ht:;oindrical plates, of m at 7z:-7wxhofyt2bg rt(z2;p :m yi:dass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ro*1mc1:bf .lq y4t pepwlls along the looped rough track of Fig. 8–58. What is the minimumplb : we.4c*f tq11pym value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume- .kzl a:n1sto )ooef9 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifoaat o5y3wi muvrya919(cv )c+ rmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular a(+a vw9ti 1mc ay)yrauov59c 3cceleration 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