A bicycle odometer (which measures distance travekg b v0x 1x07*,mimblfled) is attached near the wheel hub and is designed for *kl b,fxm0m 1iv0 gbx727-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constanr/so i6yf727.tj6l *ah t5hx * zsnyzd se::ugt angular 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 accea:russ tj5z *6.z/n2 y*7hhdlt ysig6:e 7foxleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value ofu in-ar7 ,ae9a ; y*t-urok-vj the angular velocity $\omega$ Explain.

Can a small force ever exert a n)trt*)xak w/ greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up withmos7xtu6v) oq: 6qz 3j your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to anqx7 o)6o:6uv3jz s mqtswer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at do- 7hefu- yjy g,bo:o+p vz1*q d4p4kthe pedal cranks. In whic1o-7:*-e dvyqg 4fzphypdbujok ,4 +oh 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 have slender lower legs with .fubjja 9fo13l:7wyc flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rot yc:f wlj9b7f31ojua .ational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long8 8t3z d(,pv u0i 9 axtvzpp-m0haxm;l, narrow beam?

If the net force on a system is zero, is the sfbhb5i5hywyqx) tw ced.,/ g:4 i,-l net torque also zero? If the net torque on islqxfb:bhh)cwyt/5 yge i 5,4.,-dwa system is zero, is the net force zero?

Two inclines have the same height but make different angles withocublu,h + ;/(8krl :o g t,ngrdg2zf6 the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of ther g2orzub dng/h,; f8l u,k(gt c:+6ol ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from re*y+9+gcrd7y-1 iwgg + kcstsv st) down an incline. One sphere has twice the radius and twice the mass of td+v cir+*sg1 ygyt-+cw7kg9 s he 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 the same radius and the same mass. They st 0 lxpq62k mqh82ao;n/h tkc1vart from rest at the top x1p0cvakq 8/2 qn6 2hk o;lhtmof 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? Which has the greater rotational KE?

We claim that momentum o fu+)a2xvte)and angular momentum are conserved. Yet most moving or rotating objects eventually sl)u)av+ e o2xftow down and stop. Explain.

If there were a greatiy sn qj4f (9.u4naliwz2q0jxyr :,f; migration of people toward the Earth’s equator, how would this affect the leuij afyf0l9z ; :(44qyxj,ni.rw2snqngth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial wqfw 4 i)+; 0wrnxuw ir2w,sg9hd-j0s rotation whenir;4 sqwj0ud2+g h0 9 r )i-w,swnfwxw she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center lnq2j8nd:d3,-nk hl dn d2 nnl-3,kd8:l jqdhof mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rot(l *8hnc eloq zk6)3u8nro3nj ating stool holding a 2-kg mass in each outstretched hand. If you suddenly dropo38nn8)lceh j3u k l6nr*qoz( the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical *flz* b2v* zvjand have the same mass. However, one is hollow and the other is solid. Describe an experiment tob* * zz*j2lvfv determine which is which.

The angular velocity of a wheel7zs) g3ebw4kwz4nd r pb4jfo,u7) xu6 rotating 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 acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is thb4uwsd)nj ,r 6bxzg 77zwp4fu 3o4)kee angular speed increasing or decreasing?

Suppose you are standing on the edge of a la g8rb* o ab:evb1*-sdvrge freely rotating turntable. What happensev dvo1rb- :a8b g*sb* if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it qui/i8fxt ;h(-vx*df z7n 0rs rstckly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–3x7*s zr ts8 t-/iv0nhf;fr (dx6). Explain.

On the basis of the law of conservation of angular momentum,0y e, oy2*5o 4ygnhywz discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second weyy4nz5 g, yh *oy02opropeller can operate to keep the helicopter stable.

  Express the followinq7slvc7y8 v a38q* txmg 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 E3t.d k-db*b v2f4li1jeng; xzarth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in*v4zf-bi jkxdge; td2lb.31n 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 t,8tu;ax* r41r l srbvmj,hy+km from Earth. The beam diverges at an h8rbl,y jsra mvtu*4 1; xt+r,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 quq 5ex i,ud1dns 91h/hy k162v /vaisa rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angulanvih9qd,1 1 s51khis/u2 qv6ax/yeu dr acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on 7pu x mt,q/(hreg/ p7ta level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamex t7u(//mrh,7eq tp pgter?    m

  A bicycle with tires 68 3 3 q7wj1hxqoshdw(1kcm in diameter travels 8.0 km. How many revolutions do the wheels ma3wjd(7hqxqo1 1k s3whke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rt);, k6foi forpm. Calculate its angular vel;) fif too6k,rocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution94 vt0 fl:dm+5 wqpfbt in 4.0 s (Fig. 8–38). (a) What is th mdbv9ptw5ql:tf 04f+e 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 r(gpj0o:b:izb 32lo. - vwyqiorbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear s ) jh0yn3pl)ujpeed 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 3ekkr ys.xjop(.6uh .a1 ljw1centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleratiox1r.j pk u6 l syj1(aw..k3heon of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accep1x y*hpr.ev/uj. 0kf l*v*u klerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s rek. 0 .yvvhj*xk /*lpfuu1*p. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius uktbku /s6/4g x.bz8y $R_1$ 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 aboam e0 d;l n3;f:t3qjvisrd the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s ;jvtdies 033f: mq;l nenergy 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 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 v7*wrebct4aso+*/q cu (ur 5efrom rest to 15,000 rpm in 220 s. Through how many revolutions did it75 er vb*s/4ao (rue+u wqcc*t turn in this time?    $rev$

  An automobile engine slows down from 4500 yin ):,j m9:tjfqcrn2 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 hip4m2gb;k9p/ 8v mygqdy:z gnn1b**dqghspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching y dn2g**1kgng p4/b8z;myqq p 9vbm:d 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 lh/:if58sv e8kng,uw uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point onn,:uklfsg 58 iw/8hev 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 turns 1500 revolfnun: /zs *+ jsbx2ym1zn2u b1sys:n/f+jm* xutions 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 mae01+uoesm, hslz s8: zke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The+sze l 1s,zsu8o:m 0he tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformlb5. bwglnj+1oo:p*rjy from 95km/h to 45km/h The tires have a diameter o.lw1p gb r+jn:5o *bjof 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 put3w xe-sgwk6y(;q cn2p s all her weight on each pedal when climbing a hill. The pedals rotate in a se 2wgp;kyx6qnc3( w-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. Whatzh .g elmy/7 * b9orh9y5agf9q is the magnitude of the torque if the force h.5zy 99r a7 ey*loqhf9mbg/g is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume -e kwyg:tyq.gm z uvh/zd95:8 that a friction torquev9 :5 :e8- /dqtmhzy .kywgugz of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a i(6ml csi/fxf7 a39mc massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released7f sc3(f9iai m m/6cxl. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engineb -t.jaq2e h4q require tightening to a torque of 382qeaqhj4bt- . $m \cdot N$ 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 ine5-2f5oxt(lb ghk;6cs nsuzk 0rtia of a 10.8-kg sphere of radius 0.648 m when the axi5s cu 2k(knb;h -lzs05ogfx t6s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel tkq *2ci ep0s b9th.,s66.7 cm in diameter. The rim and ti kbqcph9stt 20,*si.ere have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light: (cxots;a1 ag4etvn.r0gf0x rod is rotated in a horizontal circle of radius(t xt rao4c0x gags:ev.;10 fn 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 b /btpl .afj01fowl on a potter’s wheel rotating at constant angular speed (Fig. 1bp .t0ljfa/f8–42). 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 inertia of the arracmc dg3i)3(n ly of point objects shown in Fig. 8–43 aboulg 3n)3(cimc dt
(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 whoshj:)a8f td:e he 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 cohz0 xr 5kfo4 pl-x/bf9rrect rate, 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 satellite is-45or /bk fh9xz0p fxl to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifovun,ez 5am 8l(rm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate 5l( 8eanvm,uz
(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, acceleratin;sg* a)nw k4psg 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-ro /5, gen9j(*rwh/ufx nc t:sjnund 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 mhtj /c ,/5n(r:ewugxns9 n*fjass 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 sblxtsv7 zb(pku162 2r0lx 9a hhut off and is eventually brought uniformly to rest by a frictional torque of 1.20a rsxl9tl h6vupz 1b27(x 2kb $m \cdot N$ 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 acceleratf:mm rva0w; hl 6.m3ymes 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 ba21en63u :tboac ui,4,h-lqpky9gha d ll is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The bao-adla2ig1 ,u:,yp9bq36teuk c 4 h hnll 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 k+z(k9m;nqyqi3c l 8 qthin rod, as shown in Fig. 8–43i(kc98m q +zq;qnyl k6.
(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 two m uk36trxl ;w;uasses, $m_1$ 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 withia g;8xrv 25:l8mb t hxkal,b8cn four full turns (revolutions) br,52 8tbacl8 g : a8vxhk;mlxand 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 mkif pe ev8 (h r+t)w03:xak3b7llit1ioment 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 develo- bec.itfr ;)w4m) kg4ludvh2ps 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 p/o +ustwa)hl4x,qbv 6vcq66hgqn -3 slipping down a la)qcqghsaq+6t ,6v4p w-ulh x3 /b6vonne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ol77w2teb tk* mf the Earth with respect to the Sun as the sum of two tetm7 lkb72ewt*rms,
(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 radiuswhdsb,ip7/c ( c 3b,ms of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.0bbcmws7p,,(sd hi 3c/ 0 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 rolc-m plr2qu; 0wiqx-9w- m *uaels without slipping uim*xpmu;l -rc 2eq aw0w9-q-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 balfqlks :j6(ae 9anced vertically 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 hiqse 9k j6:(lafts the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating od nxi xhk0e,7/n the end of a thin stringix0ndek/ 7hx, in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical0oobh 00pf 6at grinding wheel of rt h0060pf obaoadius 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 platfohbtc;t0se . (brm that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, thehbs; ( b0ttce. 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 s4nwd7p+w) zw9d6l ky g6lbet5 xfgb )9hown 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 is her angular xz ntk9fw4pw))wgylb b+7e566lgdd 9speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initu2dro 6b(m, aoial rate of 1.0 rr2(amou , b6odev every 2.0 s to a final 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 vertica,26mqbs.rco p l 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 sbm6 pc2q.,orthrows 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 momentum fr ju3i)o6.abof 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 mo*sl lm14jwm 843vl5 olwp lac2mentum 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 di-ospe) p l*g(51nnj8t wsj(ns sk of moment of inertia I is dropped onto an identical disk rotating at angular speeo*(s)jwp8n1l5 ensgsn( tp j-d $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns adv5 +y. i y4zj,f/4qw1ibuewqt 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 o diz9xzb.1tv)gtfn* po2*/vgd sha-5f a rotating merry-go-round platform h*5dvogg .b )izf td19 n/sv2p*-xztaof radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity ofe 63.r: h7iapi7op oe1vjl :jr 0.6e :ohr:7i3 1irjv7ppa.o jel8 $rad/s$ 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 dwarf, losing jt3wrjew0bt yzk6+0,about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no ang,wjwbzey 00 tjr 6tk3+ular 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 excefxt yee09-k:l ss 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 g8b:.olcmp/d$ 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 u6g ;ddq3a 1 ik9m2fcirest, that can rotate freely without friction. The moment of inertia of the person plus th3d c2agfd9ui i;k6 q1me 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 diame .c5g2rgt x/xmter merry-go-round turntable that is mounted on frictionless bearimr /gx5xc.g2tngs 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 tn+j;oe 3 i0csh;s9)rfpmf xa 7he end of the rope lying on the top edge of t ss e7i ;rmpxf90+3cfh;a)jonhe 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 the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thatj : kjmrb(hg8x0p bss3,8b gl) the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a parjbxg (lg0smb k 8:j3,s)h8bpr ticle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 fg ) ijy;)x9w gvs13xdm/$s^2$ 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 nbw3 sg)(ddl3r0 q8pguniform cylinder of radius 0.20 m acquires a rotational rate bws grd03qlng83) (dpof from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and d/j) se9,g v8u/l(xu0xp 9(wdftkv gwtke)n,yiameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 ,ksgx(lt9f(vguevywjxu08 te/ , )/kn) wdp 9kg 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 spk 99hyewn28u peed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of ourx 3d4wfqw*i1 i 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-quaiq3i 4wx1*wdf rters 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 zp4qp.cs2s6f,ht2s denergy 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 w,spc dtzp2 f2sh .64qsithout 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*s qwjz6o 2pl1i,oj qz/1 td0h $H_2O$ 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 (ho1hhscx 2j zsp0vs7u4( op) is rolling 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 (;:,v clls m9bji.e., we ignore 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, 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, a;: j,vlmb9c sls shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standin0o3le;,che,xchnq( r g9al9z 8ca1q r g vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against 9 ge,rr8la0,39xqlan ;hzch 1(qe c cowhich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn ;* w 59vckyfoaxxv327 84wnamn1 fuu qwith a radius 5m9nfc8 ;x 2a4u*u37qyo wnfvakv wx1 The forces acting on the cyclist and 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 of length 1.5 m and begins whirli:.u8 yhuhjyiy bn y-asu((e, 9ng the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm a e9,uby(.(yj-ui :h8 hnyusy after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin ro 8n xaojyn * vjmtjpr6(9(,el,ds, making reasonaby(tjpm a 9ve,rnnl (*jx,6o8jle 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 apf- 9r fjnyr0(7bmm llz4);wassembly which consists of two cylindrical plates, of mal(fzb-la mr rpn f9wjm4)y;07ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 gr4ylc; es3-b b7d ;ut6(l yemvlfj2/w+v5ql r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of ld vym s2lle urf45+ ;bj3eby/ 7twq6;c (gvl-the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as 2quha2ant1*q nulcqv 2r*1*bsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed u2stk c3 efpckav-s:8+ x))iqp niformly from 90k8- c+ta:ex3)csipv k s pq2k)fm/h to 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