A bicycle odometer (which measures distance traveled) is attachtb k/vu-c )cu,ed near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a v-u,/ uk)cctbbicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point o.x p;5nebm c+:y- pznqn the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration chan5 .xn;mypz+nbp -c :qege?

Could a nonrigid body bxai5 p;bg4hw vd;ku5 7e described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? 6sxct. cjf*r 4 xyd8ax. .uzd4Explain.

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 hand(6y,a+id wdhgd ewlh/ 4rbo*-s behind your head than when your arms are stretched out in front of you? A diagram may help you t*+gdwl -4,orh (y/dd6bhea w io answer this.

A 21-speed bicycle has wb3 0o fqdu(edl1+dd9seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a largodfd+30u d(ld q1b9 wee front sprocket? Why?

Mammals that depend on being able to run fl*w v-89 (jq slc5duiyast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationa (vj i5*8qw-dscy lu9ll dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longp w2 jlxl /)t8d*:l *gmls+fjf, narrow beam?

If the net force on a system is zero, is be;ud+(wg+kithe net torque also zero? If the net toui(g+ek +wb d;rque on a system is zero, is the net force zero?

Two inclines have the same height lo: rf* mi )v5jxmu7.bh.e*fe but make different angles with the horizontal. The same steel bamxl )i:uevje .rf.*5hmf* bo7ll is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down atrett78o0* t,x-x g ibn incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greate-0to ,i txr7 et*xb8tgr speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the samqig5d i5g;xmq9o 0klge) ;8fze 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 bottomol) 8g59figed imqk xz;;qg50? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conservllx;t;k ov 6h(ed. Yet most moving or rotating objects eventually k ;(oh;6xl vtlslow down and stop. Explain.

If there were a great migration of people toward the Earth’s 7g3y w53ziko kequator, how would this affect the length of the gwiy5k okz373 day?

Can the diver of Fig. 8–29 do a somersault without having an( fgy w87/3txz rthgp8y initial rotation when she leavesyh/ f(7r3t zgwxtp8g8 the board?

The moment of inertia of a rotating solid disk about an axis through .bzdvhrh;vb)2snp1 + its center of mass isbnbh.s1v +dhv2 zr)p; $\frac{1}{2}WR^2$ (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 rotatiojyi5 v0mc7iy ,cd-jan1+0 vzng stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, orn,zcv0acjo -ji07y +i5v1my d stay the same? Explain.

Two spheres look identical and have the same mass./ym2n m2,j nrv However, one is hollow and the other is solid. Describe annr vm2 /2jymn, experiment to determine which is which.

The angular velocity of a in mqjp w :is/5k,k+5nwheel 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 pointsp+:s , kmkn55njqi /iw 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 xs9 /xl 0+8hslfbda+erotating turntable. What happens ifhde9+xal f/b xsl80 +s you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Asx;fs/kb er 2la iq53/hi2w3 fd he throws the ball, the upper part of his body rotatb5/fhkl if;i 32sadwe r3/2xqes. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angu*gxlu/) bj oa6q . u(;gyx7quclar momentum, discuss why a helicopter must have more than one rotor (or propeller /uql)6*q;y. jgu g(ocx7ba xu). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) z.73z+1gq mhrmmlh hemtp2w;bi8 ;8k30 $^{\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 amazin(vxvj5piv 55q+ t3fykg coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, +5ki t5fp5jxv 3 v(yvqas seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. ::ll;bsm, sf lThe beam diverges at an ang;f:bs m,:ls llle $\theta$ (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 rpm. f;9hf9 jsew9 onkk3/3m tf/ mvWhen the motor is turned off during operation, the blades slow to rest in 3.0 s. Wh; jn9omksv/hfft mwf9k9 e33/ at 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 biw) j1hsnywm/t8tu farr- *+g4a,1amakes 15.0 revolutions, what is its ds -g ta/1ira),n a+rj*8wm hwfut4 1byiameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions dj,clp ;:9zvumo tjcv;, m:9zupl he wheels make?    $rev$

  (a) A grinding wheel 0.35 s ema/fs( z 42mssq0s7m in diameter rotates at 2500 rpm. Calculate its angular veezsm72mssa4q/f s (s0 locity 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(g ry faf58yv- mih,.k makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is , 8a5(mfg-yvkiryhf. the 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 ; so46 6me.uerkj/bmb e-pgp0 the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a ri1a:54 zaxr xrpra0sqn3:4- tufbo8ok(u7 ipoint
(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 a6uc9 exia bn;2 particle 7.0 cm from the axis of rotation is to experience an acceleration 6ecbn;ix2 9au of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformlyvl-i arsqp99qw w1dur6u8+ 2 r about its center from 130 rpm to 280 w8a1 +r2w-dsrplqui9 u96r vqrpm in 4.0 s. 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 radiubt7 nl2)+f+twon l;cv s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astrofd wf3dl:mui:,zty b u,hfb,vw b kh1*7+v*y5 nauts 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 spacecraft can be d:fwvbw* fhhvubi,fy ,dy1 375m+:,l ukb *ztthought 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 in 220 s. Through a1g6 ep,mih4k(fp9n1 guit0vhow many revolution01me ,t ippf1k g h9av4u6gi(ns did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5n v93jlqdwya6(nq;mwc(5 +fa 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 inwy,r.mm 7 sv8(bypb1xe(u j) o a whirling “human centrifuge,” wjmwb,uy1yx7 e o8(v. psmbr()hich 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 accelerates uniformly from 24yv 55w+qgxfqr9qh+ j4 *y oo3u0 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tq+r f3 ooqq 5+y5v uy4*jwxhg9ime?    m

  A cooling fan is turned off when it is running at 850rev1 pk( gh(2gbn0qmr4op t4gakex8 i3v5 /min It turns 1500 revolutions before it com851gep4v(p gqmo2(0i k kxr 4ghtban3es 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 cacl c1;a1:nvu /oizqy;r reduces its speed uniformly fro ;v;iy:qc/1 au zconl1m 95km/h to 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 65 revolutions as the car reduces its speed uniforr rg ,fzh 5+bjrz.p63imly from +z3 5 ,fhgizrrbr. p6j95km/h to 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

  A 55-kg person riding a bike puts all her8h. /-w efjdd; de/ddk weight on each pedal when climbing a hill. The pedals rotate in a circlee/wefdj ;kd.8h- d/ dd 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 forc)tqo 92au ) p9xa0lb taa*kx-2idc. bee of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque -axxkl ba9p*.t9doe a)q2 b0)it2cua 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 torq81kqkgecj kak j9 9wr3: -z9dhue about the axle of the wheel shown in Fig. 8–39. Assume thaw893ehad rjj-cgkzk: 19k9qk t a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachedzji) sogo+9g;ij6 4 1kq tkfs) to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horioj)9iz 4s 6jfoggs) ki1 k+q;tzontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of /s8 ao2oz p /qum74bp.nx m(n qwx-3qkan engine require tightening to a torque of 38o-p3au b.x8 pk2 mzno4w(sqq/nmqx7/ $m \cdot N$ 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 sphe x37)k b4xbs n7,urtrpre of radius 0.648 m when the axinkb,)prt3ub7 7rxs4xs of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclelfe.unl2 a;-j wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignore2u-. aj;nlf eld (why?).    $kg \cdot m^2$

  A small 650-gram ball on pvfs)9gy j7;m ca 1*njthe end of a thin, light rod is rotated in a horizontal circle of 1g ajpfsy)jnvm c;* 97radius 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 o)+ah8dmsva5dfz s 85 tn a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands atd 8s+ha s8d5 )fva5zmnd 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 inertiz;aswk:uj- i eeki, ey,boh f+x);/+ka of the array of point objects shown in Fig. 8–43 aii)b; ;,fj+kz-+:ekehsuwke /,oxay bout
(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 consist,g;4s.tlu:d (rf *b * o,nohzuv *sitks of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical sateq h fg*;gyz-b 6lda1b.llite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 bzq- *d g; yhflba.1g6kg 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 cylin-ljtaeu:-u ismx) ; b/der with a radius of 8.50 cm and a mass of 0.580 kg. Calcumi- ;xl-tj:ube/au)s late
(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, acceler-y vhvpzg95 4aating 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 han f wqa/-(wgk+ye n7,9*cbk o*) jreypnd-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit yyrow(,q 7np*j /a gwb-e+kk9c)ef n* 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 and is eventualzfgb +r )n.tc:ly brought uniformly to rest by a frictional torque of 1.2.r c)z+f nb:tg $m \cdot N$ 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 ac2sz (v q )jrp sltskhcs42na w2l(tqwaq )90+(celerates 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 of the forearm, which ir+d ba( h/pm-e 5rjzo r: ;en(qel27fo6a8n trotates about thjan+:a (tpz/e5rr nmb-i2ql 7deo6( or 8fhe ;e elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Figbg50a25b()xgsq e1nbkl4f c y. 8–46g)1k be04 n(bxyqafgl255b c s.
(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 twol y.yg6mm 7v892ffp vf 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 4dj o+dxdqhxn jqpo3 h2v0(-) four full turns (revolutions) and releasxd( 0qjxo2+dpv4ndj-o3q h)h es it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment .,fat9 g i u3hjw(:yxzof 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 nkvvjujfgcb7x/43 /2 fg9ueu o8ox87 develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg u,/ ,u)cc0i/gs qv n luatx:f5and radius 9.0 cm rolls without slipping down a lane at 3.cnq5a/t:lgu, c0 )/ ,fiuvsux 3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum wfh ir1k5hvr91ix x/j9 /t:fm of twfwi/xr19m j5 1hi/:9kh fxtr vo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg dgr9 ( s: ibd0i48wuw9w13ez l(hwyf fand a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 109 z : h81iwwyg(f wri wbfld9u4s(d3e.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 r.w9jw0 q:o7eoy/.w hrwx1vo dest and rolls without slippin/w.vh q:xoo 7rwd w y.e91o0wjg 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 on its tip. It starts g zjl(c6y0 0j.va6hsyf 1 ovy)-xvt,wto fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the grfox(j6vy h ,j6ygw10 ztavly v.s)-c0ound? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating ob*g.n4: y8bbt5rpzd 1tb:ujf6 dc6izn the end of a thin string in a circle of radius 1.10 m at an actfd6gjbi b .65zut1pz:n4:8 b ry*dbngular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular- 6ohxffq6n1 f momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rota6fh-6 q1fo fnxting 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 pla;hw+g4n e:c ge2l rc- /ugqu(ytform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fgg/lq-4e2 nr u:gu;+cwec y h(ig. 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. ba4ut ae * ye ie:wq4f6mww8:-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 tmaw-b 8 wwe fyt6:uq*:a4ie4ehe 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 ratq5rr*nftnt4, (ksgz,rk3 e9xe of 1.0 rev every 2.0 s to a final rate onxzn 4tr gf*59sr q ,rt,ek(k3f 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 centeh/s5s+7+o hawfk- ovx )v,4; ndlz rgzr 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 arhfa 5)gzvns kl h,so -+o7z;vd4xw/+s 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 angulaztwdijzzg hr+nb)o5:+;vl-n 7 uj4 *gxz1-q tr momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angulark vh*zshc ls,55(3wfe2 3 doqn 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 3 9 blraagn5 jeug(vn8o r3z11of inertia I is dropped onto an identical disk rotating at azbrjan81o3 r5n9e lug 1v(3g angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.+vv284wwgqo)qs ,s7 9s1hcgl9 w uy nj4 $rev/s$ 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 merr5z (6rylrn7ke qh*d0m y-go-round platform of radius 3.0 m and moment *k5 de 0qzrnrh67my( lof 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 ol5q9wz;9 dq2hy*xw kif 0 iwk9*hqld9zyq2;5 xw .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 w4kppqt3 p1,jt hite dwarf, losing about half its mass jppq t4p 1t3,kin the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of/ :;opc5bt6oykshol7 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 ur(2ie i x )ll 9on9:x.;zngym$ 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 : c,cgwldg y/:ivskz3:fk0qk(lf; )d freely without friction. The moment of inegq /)y kd;c,0lv g:d:3:c kli(zwffksrtia of the 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 dq9n)+9cofnl j :pgtd- iameter merry-go-round turntab 9:qd g)o-j9n tnfpl+cle that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on ln+-mae(cj/:;c 3d0 s y4aqbspw7ymg the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. a:ygb;sn/-pa3yc7jq4+ e lmm wds c(0What 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 always,j f)eqy 37y0,nwjh f6jd8rsswgjv . / 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 j6sej0 y)w/n83vs qwjr.7,gdffyj, h Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of b8lg24a+,lfskm w( o u1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a unifo/n2xbi etj-r ca:d: :f ep53vbrm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered byef3ei-pac 2xrtb 5v:d ::j/bn the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 bu6t hf3b;8t c+:ctgh 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 tobct68+tg 3 tf b:cuhh; 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 ofp1wcw39 z c*ps ik.kl7dxj ms-j71-fb the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times7zy x+duvy dn mfo*w7/ 3w-bb, as great, were rotating at a speed of 1.0 revolution every 12 days. If it we+yxbo7,fn*d dz-w u7/v3m ybwre to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-:(hqxjp, eg tx(.vj3m0 ubvu8pollution automobile is for it to use energy stored in a he,p: uuvhjg0(xm8 .j ebt3vqx(avy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “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 illustra6wha ec: d n,oz7y)te5tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a ho vmt0kz;6c *n/rcp6keuy4 - :k(v.wv wi;dbu yrizontal 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 )grqwxl4bpz::tp 3*xd:z nui3fy0 s7(i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then relxp34 0fdy)stwbz :gqu3rz:np:x*l7ieased. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertical6er5yl1 tf2pily on the floor, and we want to exert a horizontal force F2tyr61 5el ipf 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 travelin7 o1qucll6zo awzc0-9 g with speed v=4.2m/s on a flat road is making a turn with a 1l67l-wzcz co9uao0q radius 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 kxzfunc ( 8l.kx* 0nf.whirling the rock in a nearly hok.lk n8.uxfxncz(0 f* rizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, mjsa*mv/xka 3k-n f5 )i,g/s00t txiydaking reasonable esti k, mn3sxj)y s/50*ka0afd /vxit-igtmates 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 clutchda,3nxu s ysc//lzk4* assembly which consists of two cylindrical plates, of massnux3syk// scz,4 l* da $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig. 8 yp2tf3fj*/ w9 znpwa2–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of p2ywp2f 3*f9ta w /znjthe loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asvo,hd;qoy, 1 /7bk ilksume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed ranar2su p08fie5 :bvoe, up0 hr;p1 3 pw1tv/uniformly from 90km/h to 60km/h The tires have a /p: pr pr i2r,vu s001p 1fwe baan;tvuo5e8h3diameter 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