A bicycle odometer (which measures distance traveled) is attached nea5)mw gd1 pojv+r the wheel hub and is designed for 27-inch wheels. What h+jm 1 vdg)wpo5appens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at consta5hu83)hpd,ckwgs ct d. r5 tg1nt 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 acceleration? For which cases would the magnitude of either component of linear acceled,.1phkgs gdct)t r5 8hw c35uration change?

Could a nonrigid body be described by a singz0* zw24q19eiph pkem, m5qnj le value of the angular velocity $\omega$ Explain.

Can a small force ever ex3yyw7 k+qko: k+jk v7mert a 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 wc1 8 swuxd.3nwith your hands behind your head than when your arms are stretched out in front of you? A diagramw3cx u8.d1nw s may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whevb6cu82z y2js8p 67s gv+ tourkjln1 .el and three at the pedal cranks. In which gear is it harder to nlv68+j.ots jbs2cpur 6yzu7k82v1g 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 beinzz9 .aqd)dba 8g able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, expazd8) dza9 b.qlain why this distribution of mass is advantageous.

Why do tightrope waljel0ap4 q(3-j *litdl kers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque ai/tsq ,yksj8r xj*i.+lso zero? If the net torque on a system is zero, is the net force j*k/j yi,i.t+xqr 8sszero?

Two inclines have the same heightdhm-v 8hy8t3w but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the 83y8dht wm-vh bottom be greater? Explain.

Two solid spheres simultaneously startw0r dbrr 2ug jj*mv96f*n, *qe rolling (from rest) down an 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 greater speed there? Whijerjr q2 n*, *9urgb6fwd0m*vch has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They st;wt tibq eqw6g)/liy1 .4a1j*3avl jcart from rest aa1 bt4gw)ii.lq1lea vw6t c*/y3j q;jt the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or b+g);z 0l*0fmvr:5a rxgvrfi rotating objects eventually s*:mvgagb 0 0i)rfx 5;r fr+vlzlow down and stop. Explain.

If there were a great migratiow5g hy , is6;bat,uta88m;xfun of people toward the Earth’s equator, how would this affect the lengm;tb ;,s i wu8t6a f8,ghxau5yth of the day?

Can the diver of Fig. 8–29 1 izen3hmhb2 mc-z+7fdo a somersault without having any initial rotation when she leaves the c2-1ez3mhhn7ibmz f+ board?

The moment of inertia owvi,) uf ;( qfgfr*qd.f a rotating solid disk about an axis through its center of mass iiu)rv;gf( q *.,qwdffs $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in eadd 4p6z8fp :o jpqxet3ni/dbp4x5i9r.man. 0ch outstretched hand. If you suddenly8pir x mtd4x p6znq9 dfdpnb0p ..jae/34: i5o drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and td8bgij(o*xdp.bq z( yo8*/ii he other is solid. Describe an experiment to determine which / *(i bygibopq o*8d.d8 ji(xzis which.

The angular velocity of a wheel rotating on a horizontal axle poinpq7 4m11av6ljmr + ycxts 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 the angular speed increasing or decreasi xam47 1rjlypmv+ 61cqng?

Suppose you are standing on the edge of a large freely rotating turnt-e 9j3wqcwb zzbm xw )xk0e6/*able. What happens if you walk toward the center)wkz0w- we bxx b9e6m/* qj3zc?

A shortstop may leap into the a s(n.hny p.9ht 95m2 uhwzdun) b(pmg8ir to catch a ball and throw it quickly. As he th.h.9nz sm bw8n(hu yd2pph)5nt gm(9u rows 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. 8–36). Explain.

On the basis of the law of conv o+*u ,h;(zjyhzg1yd servation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeyzuh+jov zd 1( ,;y*ghller can operate to keep the helicopter stable.

  Express the following angles in 79*b z swh6v1vi0kxf oradians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using th b nug+jx-ats:lnmy-+(ig, kkk .)n4be information inside the Front Coverk tij-nyx.u b a)++snkkn:l(gb4m -,g, the 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, 380,000 km from Earth. The beam divergete tr*,dm/fl/ s at an ang/d m* ,/tletfrle $\theta$ (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. When the motor is turned ot74 7lr8wztn dff during operation, the blades slow to rest8tznrl77 w4d t 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 cl o(whv/p0.8o 9qbgj the ball makes 15.0 revolutions, what is its diamete( c8.j0w9h gbl/qpvoor?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ho0kxj a* +de ;lgq,pds(w many revolutions do t; kjsa0e+x*p lq gdd(,he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itbzc0),iqwj* 3 cugnw:u(t6 tm s angulazc0,wgwjq3im n6t*uctu )b :(r velocity 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 complem:*3rmx:g skb70 dn2o ijgcr 8te revolution in 4.0 s (Fig. 8–38). (a) What is 7jkmb :cxn2m83ord*0:ig s grthe 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 own7 ;+h1maiq(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(li;d/ib6 e0:vdooge*-ngu v 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 q q+2afn /s2q9n)9 yzzbei0zwcentrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accelerat+azzwfn q9qq e09inysb/ 2z2) ion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to -m)ozaf.q8i71 i rvxedq1da hhc 1o7 1280 rpm in 4.0 s. Determeaddhmira.qq oz17cv 7x8-f11 )i1 hoine
(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 radiusybqed y.u/ /4kc*rbq :o c:k7o $R_1$ 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 spa/befx16 ntp2 x,+kz ovcecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start ofox+f/6zk nt bv, 1xpe2 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 (l xikle/a j 1*8-t+bdsx8j atuniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it a (-li8d l1tbx*+a8 j ekj/stxturn in this time?    $rev$

  An automobile engine slows down from 450; .dpfiq ;)p;,misut.mrwj nt 0h k,b70 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*+lnqi 7uhjz81; 7t-ktx x ebib1pu-w of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final spej1 hn-k; x7 * tq1ei7uixz8 wlbubp-t+ed.
(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 ,*d/ni uq.4bcu/yj uz from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the whee cujiuu4,n/bqyd/*. z l have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min fnac w((g+,cd o9:a*p)a,nonca 4bi q It turns 1500 revolutions beforndaaw:+4* bfn ,9ac)gp,(ia(cq oc noe 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 reduced 1z24w+unf oyt cx*+ ugp2bo:c 8gu:os its speed uniformly from 95x4:p*o82u z d2bcuyw1+o ucgg:on+ tfkm/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 ungory(+bu kh8t; / muura 0 pcr13*kx,hiformly fro 3gac0troxy 8(m urur1k*/bkh,hu;+pm 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

  A 55-kg person riding a bike puts all her weight on each pedal when clima8gu+yy fezc ).nb.y1bing a hill. y eyca.uzg+) b.18fyn The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 ms6-xpol utp/v ;vye6- toyo02e(m.l N on the end of a door 74 cm wide. What is the magnitude of the torqu(m2y. /oe0 6tptvoupl ov--m6 y xel;se 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 i uooath- +9axh5vy*0xl)+ pzpn Fig. 8–39. Assume thva- 9o05o+*) pluthxz a+ xhypat 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,+ko+i8azo1r+ oq kt* f are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and th+zro tkq+f +1o*8ki aoen released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening t1obi;,jx; ta lh5pje;o a torque of 385,xt;jlj oh pb;1aei; $m \cdot N$ 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.j a,sj pvbutt 55wl/1bn9)c+e648 m when the axis of rotation is through its center. jbtj u ns591 w lbep,+)/c5vat   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The ccyj fly 17o*r: ;jg/u;wo boz2tg)vsb q*:p 0rim and tire have a combibljcrp 7ctug: 1*swy z;q;yb*fgo)vo 2j 0o:/ned 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 rod i3zzj. ui ku 9kap0wl:q2:,qqis rotated in a horizontal circle of radiuu9pajzi2w0:i:3, qkz qul.qks 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at o;gkau2-:r a hconstant angular speed (Fig. 8–4gruo:;2kah a -2). 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 ( :t5dquputd.-rk )dl of the array of point objects shown in Fig. 8–43 abddkpqutu (5l:dt )r. -out
(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 o)u gsr1 r0xdo+f 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 sat h) ;pqw1j6osxj2 a1eil31emzellite spinning at the correct 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 ifj1wejp1;z3 s)lmaoxe1 h i6q2 the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a i,i59wy(poxj 2t -vhsmass of 0.580 kg. Calcula5s(o,xvjip h it9-2wyte
(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 batq1ukv 9jev3 r0, accelerating 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 tangwxt *uyhu14vngg;k) zg58 m /:s6qmyqentially 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 disk of radiuzg 1* vgnmu)q 6y h;5 m/qw:tyx48sgkus 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 rotatin ybe/* 4(th f8t8pwwohg at 10,300 rpm is shut off and is eventually brought uniformly to rest by8/ph *4hye8b w ftt(wo a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accedkc6 d+ j4xr s+bij yild7foa5;aq /91lerates 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 thr z.gu+6sav uoc00xzj/+x1/q)at k ul aown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest toz xo/0acau1+ qz.vs/+kx g0u 6utjl)a 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 5- 4e:4 fhal:tnudw baconsidered a long thin rod, as shown in Fig. 8–464daafn:w-ut bhe4l5: .
(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 masses, m d:upt1rf (y4+*vwlz$m_1$ 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 accelc-qidm*6 fsg(erates the hammer from rest within four full turns (revolutions) and rel-i scqmg*(d 6feases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia q8er yy*lisni5 ,o zl0 y09c0rof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine developsv(uci kqt106em 8oz8c 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 withoul+1qr,o7 pfs ut slipping down a lane at 3.3 1plu fo7s,qr+ $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the s u.+wisc5x ah3um of two termsh5 c+s.u3iwx a,
(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 164l vpimd +hzar7de6s e(i5 z.310 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate o5 r (i6dva 3p7ies+1zdzlh e.mf 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 r n mb p6p8uxp) 6-v+3r(pcgs06vm dxybolls without slipping down a 30.0 0v (mpdbxyc6s+b xp6 pmv6u3p)gr-8n $^{\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. I1xdbb fte i94c9qkn1 8(f5r frt starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just brx 9 d59t (kf1fr1cf i8qbeb4nefore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotati(x+f deshio) i0z+9 tung on the end of a thin string in 9x) i0u+etf zh s+o(ida 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 cylindrical grinding wheekqw97h,dm0(rozu3*y vny rh7( dopc/l of radius 18 cm when r /3uw( rd9yhqy ozvch ,onr(k7m*0p7dotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating akvgds wa8:n53a /5pu-yd4f n sbj9+cct a rate of 1.3rev/s If he raises his v5/ 4 ydswncgbcu- a8f9 kspan:dj+ 53arms 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 inertro 17rd xu6nygr jl(n1 uixvkt7* 8e1,ia by a factor of about 3.5 when changrvik 7 nl,1*( yre1t18j dxrxnuogu76ing 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 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an in8twg4 l hgvvi3:2(s5v ae6 ofritial rate of 1.0 rev every 2.0 s to a (vi:asfhw 8v2ge5 3lo r 46vgtfinal rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through lywzvz9/w: s.z w 4vftb-s21nits center at a frequency v/w.tn29lvzy 1- wssf4wzzb :of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk 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 momentumyw 7a1ts02w*/ykpu lb pn1 3qz of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earthcp) q- wgy1 yrmkhru44 (-iqs3
(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 cylind otpj- is4d 9zlm6t9g93 yp5farical disk of moment of inertia I is dropped onto an identical disk rotating at angular speedz9ot3apy gt49 6 j-s m5pl9dfi $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 ;qb1f4eolx5kjd,) )s:ghmd q$rev/s$ 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 vki c.p8q1 /jxrotating merry-go-round platform of radius 3.0 m avxc.iq /81k jpnd 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 kx,g0x8palu t2ol j-1 with an angular velocity of 0.1ukl-alp0 xo ,jxg t288 $rad/s$ 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 ab*fsh5j +qr0tysqir.0cj,gv m muj3 xws59 +x9out half its mass in the process, and winding up with a radius 1.0% of its existing rx9 t0 m,cfshmg*q3jxsuj9 y0r5is+v j.r wq+5adius. 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 1z4b;g/dzq( ozj8lng77q zv :a20 $km/h$ 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 gww4)h2i + b:8e dsj u-x1azum3n*zsx$ 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 a6w dinp5*z j(xz;d 1zi platform, initially at rest, that can rotate freely without friction. The moment of inertia* ;56 zx( 1dnipzzjwid 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 i t/i/7jq77 l(g 7jlixc.ncqv edge of a 6.5-m diameter merry-go-round turixl ql77nvt /.iq/7cjg i(c7j ntable 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 the ground with the end of+. duuq h4z3zz 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. What lengzu+z43 zh q.udth 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 sirq6/ j ((c9v0ux vcsu7krwg e/csp54 gde 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 particle orbiting the Eart5(g4u v/pvcr9x(6kj qw cgs0c7eu/srh.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a g29y 4ecps 2ke ,rixe*rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the sh 8itjj96cy(/eh 0 virz-zjm*d ape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constanti9hv tji(e*m 8jzdcj-z/ r06 y 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 )qx i*szju0r (kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Uq ( rsxj*)uzi0se 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 wheel 4e;6m arhksp enx9. u5ryi .c:($\omega_R$) 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 massx 9b9sgefy u+t9d-. 7apas5ll 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 19+f7lybl te s59x usgpaad9-.1 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 awp-+6n3 kov7 ghkjqa(utomobile is for it to use energy stored in a heavy rotating flywheel. v(+7 kko-hag6jn qw3p 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 u2z1: d30b4s osnppzwkg9 /ol $H_2O$ 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 rollk,;jk24enxj5vs-s n + tjd 5oiv 4xaif(h4:eding 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 lecn.tijol2( mq2 hg ,3jngth L can pivot freely (i.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, as noh q 2( 3mi.t2lj,gjcshown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standi(g:0f w r ,t(wyp6xzding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height z,y rw 06tpw(i(x gf:dh, where h

  A bicyclist traveling wiz cs0 a5u-vf xt b)/ sk,pu/:qn vq0c2pe/xsj5th speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle arus2cva,jbt: uk q vfqp5/sxz0)//c5s-x0 enpe 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 zhu),8u wynad q;tj2f 3c-1svrock into a sling of length 1.5 m and begins whirling the rock in a nearl wvq f;d3 18tunuh-2cay,j)zsy horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as th 7tblxn /1pqv98.- ofx-vllxbin rods, making 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 a-p l-1nxv./ lb f79xxl8qb vtogainst her body.   

  You are designing a clutch assembly which consists of two cylindrical pl y.xaf5 6-nr8+1azf. eiifk 5dke k*zfates, of maf k.+fk.eeazk *f8nd -5r5f6xiz iay1ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopedxc* v b hu/qwubdd;, j*)(9zbm rough track of Fig. 8–58. What is the minimum value of the vertical height h that the ma (h);w *u 9cb/vbbumxj*dd ,zqrble 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 n;k9 5okuydpwrb v6*, kot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolution ts( 19ck,ahms w39qyrs as the car reduces its speed uniformly from 90km/h to 60km/h The tires hacraw(1sy k 9sm h,9tq3ve 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