A bicycle odometer (which measure) +pb( k9xj(oleesgqpc4e; n a1 d)8uns distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use9s4 1)je+k8e)e(p p gl; uncn(xqo bad it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constaiaw 86 :,q 8 mjro oaxcbwa(kt5*6r/bhnt 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 acceleratiw/r8atj*bc5q b: ixwm(,6k o8r6haaoon change?

Could a nonrigid body be described by a single value of the angular velocity 1iq0i:dg (pc l$\omega$ Explain.

Can a small force ever exert a greater torque than a laevs,mt. ,vl 6pwawr7 6rger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind youogt078c 9bqb3 j8 2kedx*ca9ic. jcon r head than when your arms are stretched out in front3jbdk.i0 xge 79ncc 9c8q8joc2ob *at of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and threez 5d(9 wrllfwsb;vi+v0r-i7 b 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 f -lwbs+597w(drrl ;0vizvbilarge front sprocket? Why?

Mammals that depend on being able to run fast have slen awkb *e qj-p4t8(c,jpder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribut pp8 -ewqj4ct,*a bjk(ion of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carren wz01v) u;zqy a long, narrow beam?

If the net force on a system is zero, is the u 4 sj6eishxz5e p-+1knet torque also zero? If the net torque on a system is zero, sux1ekphsz 54-eji 6+is the net force zero?

Two inclines have the same height but make different anglesnh 0 bleih;*a ;-tmvl9 with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be l em*-hv ;ba lh0t9i;ngreater? Explain.

Two solid spheres simultane*edf (z;p5j jnously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches thj5( nj ;zfpe*de 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 cylinder3 :wf prwsx0 - bj 5:huo..zmm35vacyn have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the botto-b y3mv: 05.pufmo:cs x n.5rzww 3jham? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are con+4 w pzypb 6b:mub3r0wserved. Yet most moving or rotating objects eventuallw bzymr4ubp+63b wp :0y slow down and stop. Explain.

If there were a great migration oel 5x )nknwd.b0 (efp;ptt,r6f people toward the Earth’s equator, how would this affect the length of thenxf pkel t)en50; prb(dw ,t6. day?

Can the diver of Fig. 8–29 do a somersault without havi6 n87yx kj/d c)e(dsyq5xc.zt ng any initial rotation when she leaves thy xscz/.yd7j etk5x6c)q8n d (e board?

The moment of inertia of a* h1*wvnw oncs/ze; u:n d*mk3 rotating solid disk about an axis through its center of mass i*n 3dw/* s wh1nuvkzn;oc* e:ms $\frac{1}{2}WR^2$ (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 0 4u 7tu5+nu jsutx-)+61zeamt5lwi 9ok o gtla 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or st1tkn5jtt +lw u5g auu0 -)os7o4t+6xlmzuei9 ay the same? Explain.

Two spheres look identical and have the same mass. However, j) csiq6-( uc0,bu fuucd):z hone is hollow and the other u6(i cbu)fc0h,dq)s -z: ujcuis solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rowfg; xagf h6)ej79eys u5;wrac(,nf2tating 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, descrie9fg7 r)f nha,2;6cg (sjwaeu5 x;fwybe 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 rotating turntabl s:xj8 inay1tio8:r),h9fxji e. What happens if you walk toward thoitjji ,i )n sf8hxaxy1r ::89e center?

A shortstop may leap into the air to cat*fc0q,oywm t i ly;+7 lvwrt*3j2l:vich a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you lo3ll20trmf j;qy7: *i,wti yv+ *lvcwook quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discu to f+,r):dgvvss why a helicopter must have more than oofgvv,t r :d+)ne rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followin-be8d zhbbriw feg 4 1;mg2f3;g angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earlead (s pve 1f9p, ymqne*nyob18.*+rth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameelm9 o sv1.n*n*pb+ q1(yed8fp,eayrters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mooc hsn960k- rc2ry)va(j ) stvgn, 380,000 km from Earth. The beam diverges at anv(c6)rr) ayk-j9cg0sts2h vn 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 roteutqld3w7 7v-+ u18b9qv5n 4aq4pjbys, x ixvate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blaips5q7n b4qae vl,8q-j w19t7u v3 udx4yx+vb des slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anoabf:iid2s ynlz18ib q 1)ld-/ther child. If the ball makes 15. 2b)fina11l/sybii- dd zq 8l:0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How8*s2og+c- d:l h tiukples n;/ many revolutions do the wheels ma;s /+*ults:ld- 2e8igonh kpcke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atc5qvm+j 57kp+ g,eqn v9 -/ vql,mefvu 2500 rpm. Calculate its angular v gjpnvlfe vm vq+59u+k v,7-,/m5cqe qelocity 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 onb k+5 qzov::pp.hyu7j e complete revolution in 4.0 s (Fig. 8–38). (a) What is the l.ybq5pju z7kh:ov+p: inear 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)5s)pbf6y t*bh 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 poi3vo6cc.1skctrn( o f0nt
(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) mus-rei 39+)d p bxu0hx 2wkujt(at a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an ac kb2dext-uwup j0(a x9+ hir3)celeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from o/lr 7lbpk6r tdql .j.1 ;61pmvn6zbu 130 rpm to 280 rpm in 4. r6tq rj;vl.6p1b.pnb zkm 6lol/du71 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 radius fdw-:jqm4 l7y$R_1$ 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 Apo/ i9so:/7;rxsfw4 xa plj)lqkllo 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-m9r fla 7 ox/qx;wss:ji/lk)p4in time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 154iuj 9.vx: k(cizt0gn ,000 rpm in 220 s. Through how many revolutions didi 9iu0t(xn jc4zv.:g k it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcujwu69w, -xc bylate
(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-e(9lkur 5yhyyi; cs1 jets in a whirling “human centrifuge,” which takes 1.0 min tok;1y leschu5y-9i r (y 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 accem c* b)no(y5tuml ;vi:lerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this t mu my5)*o;n(b lvit:cime?    m

  A cooling fan is turned off whenmj2f z7,tg67,bou-k nr,r kp p it is running at 850rev/min It turns 1500 revolutionkf k,6um,7rb, 2z-ntojp 7prgs before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uni*ds) (a mgb* t,;uq*d*nry0 we9nukqqformly from 95km/h to 45km/h The tires havw ed *(r *gqtu*q*;0mbqdn a)k,ynus9e 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 reu/cbw:vv u0 (yvolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0ub:vw0v yc/u( .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 climbing adfgtf:07gc/1 xr 9m5p m) vcja hill. The pedalsf9 5c v gxarmm/pt0cf17)jd:g rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wido 0vh4xjow4hn8 s u-/we. What is the magnitude of the torn 8oxsh/w4uw04h-vojque 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 whee1:ez fg7nd5sk l shown in Fig. 8–39. Assume that a fricti z e75fn1:dskgon torque 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 end/w oyw +lii/2ks of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this sy2/ ow/wl +ykiistem.

  The bolts on the cylinder head of an vkpb 6x9/i bvq.*fyj1 engine require tightening to a torque of 38 bk.*j1bp/vf6ivx y9 q$m \cdot N$ 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 ofw( a*x/va9 nvi* 2ab,. mik h4nyxspa+ a 10.8-kg sphere of radius 0.648 m when the axis of rotation ism**v i2sa(yia wanx+ hbp ,kv4/.ax9n through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter.zv l53*a(qho(+xsnyx 0wur9x The rim and tire have a combined mass of 1.25 kg. The mass of ther w( a3sqxxn +vo(yluhz*905x hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball o eg-c, ,o 4illg9anf7an the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. ,g,4 -oaln7lf9eg a ciCalculate
(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 pottermqe)u .pos l4b br6(v-’s wheel rotating at constant angular speed (Fig. 8–42). The friction forc.6m-pv r4qb )ules (obe between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig2 01unyba h5q9qp -rh h:wfm:a. 8–43 abom0ura :5-q2nw qpf1ha9by: hhut
(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 atoms8y3 *zw:iyi/b0zygxi whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, of i w8*pi*8wjengineers 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 forcfi ipwwj*88o*e 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 cylinore *dd,hu15bd0: u qhder with a radius of 8.50 cm and a mass of 0.580 kg. Calcu,e*dhur0 5uo dhd bq1: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, accelerating it bgeo ,. pd gx/0gb477jwe )htbfrom 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 smallgvrmmc.+d8 w2h1 g;+v1l,tz 3o tjrvt 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 tt2 1vc++ wtohmv8vjg3md.,;lg r1zrof radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is s 7pp)lqr ujyckf 69x m2y)o2:uhut off and is eventually brought uniformly to rest by a f2:2qmxfjyp uc)u 6lpo k9)yr7rictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at u5l(y)(gzgt e4 u2utg7 $m/s^2$ 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 erb1daw p/ )n 1:t6aobof the forearm, which b)rao pbt116w:a /endrotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a*+ 4vpdt4tv*/u oodco ;, duil long thin rod, as shown in Fig. 8–46*; vd+l*,4ootdui u4pcd t/ ov.
(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 odk: 0yet2c 7cmn:lr 5 /s8hfdgmsu a+-f two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates th ,lwzyipj9i(0* zrl0g e hammer from rest within four full turns (revolutions) and releases it i*l glr00j z(wy,zip9at 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 (8sak3w 5;rgoizxar7 inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque o40 6xb7gx5ge5rhq e f ca9mvh1f 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 withoutek (ms5p5*r lc slipping down a lane at c m5lkr(ps e*53.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect t7u u1b as*up j8x7s6fio the Sun as the sum of two terpujaxbu68 7f s7u s*i1ms,
(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 164zvt)h97 jtz f*0 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation ratz7zht*vf 9tj) e of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fro i:rv w3wvlr-ob7ge8) v6sl:d m rest and rolls without slipping down )o7w3ivw: l:rlrv8s egb 6vd- 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 bak8f3ane4 w h mcvt7t r+np wvi)58:xr:lanced vertically on its tip. It starts to fall and its lower end does noti: wrw a8f8ek7c3+5tp n4xvnrhmv: t) slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mp/ier:575hbhd r l+b) tgr )zjomentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular s7:zbde5btl) h gprj5 rh)r+/ipeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular m1gj2d x:,y+w8zb) mdlef al r:omentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatingfwml :xde 1:r 28j+,da)z yblg 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 x) ,.qizg*):b oi gtsev))tf sat his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speedt. sbi)q z)eo)g:f,gsixtv*) 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 sv7w4ppb1e yk26 :unk of inertia by a fact vub71k 4pe :yp2kn6wsor 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 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rot.ku 5wejg9ppf(vop 5 u( knqm,y)z9u.ation rate from an initial rate of 1.0 rev every 2.0 s to a finkyjeu ..f( kpo(m,p)95up wn v95gquzal 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 7gj 9/wodytp )xo 2/scq:54iub rcm g1around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then thrc7 q/wx gb2 /1m)c p9r5dj:u s4ootgiyows 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 of a figure sn/xq34:irojb kater 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 angularmkls 4(xz-c5 jo bq--u momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropi 4cfh.)zz nz;ped onto an identical disk r znc )i.h;z4zfotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 iwzeo2nf1+3fh;kin.9l 27l n .xdiwf$rev/s$ 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 standsaof3 a z bpk5:(z4d-xk at the center of a rotating merry-go-round platform of radiusax4a(okp5 :3bzzk- df 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 velocr,tw 39ectax.vu 2q n( uuiee/+ yf+*tity of 0.8 re,x weuuu v 2a+t3eqy.9(*fi/+tc nt$rad/s$ 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 about half ir.+j mq j+xy:te(ks(15aq/ fk o ms3tkts mass in the process, mjkk3qs+1(e.y+kx:ja m (s tfqo/r5 tand 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 involnf ev5 y6(1rdzve winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass cozng 4i 7gxu:,8o+wz $ 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 resthg2yt, mp6 j3 8l;gdfg, that can rotate freely without friction. The moment of inertia of the personl8gmf;2g 6jg,hpy dt3 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 stand;f42i6j zlz:k d3jded s at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings 6dfd dlz2kzi3:;jje4and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on the 2b )c;fbb lix3top edge of the spool. A person grabs the end of the rope and walks a distance Lilb)c; f2bx 3b, 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 that the same side always faq lg::ad.v/0o agl k2mces the Earth. Determin2l:koa0a.dvg / g: lmqe 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 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest-k v-6weriq:1r ,qojt at a 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 shape of a uniform cylinder of radius 0.20 m acquiresyqk 70p vyjg+5 a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the yvjq05py7k+g motor.    $m \cdot N$

  (a) A yo-yo is made of two solxc1u9 xtnq/tn l40r19awko. lid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 1l/ 4x.90nktt1rxuo9 nawq cl kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the revu+lf. 8wms ygavw)u1nvq, * .ar 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 times as great, wertlu igmp4/c4.km5 6yd rya5 8te rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the s8d l4gkr4 t55y mp6mau.it y/ctar is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a6x3 nh vqo.l4n low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywhenqvx6nh43 .loel 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-mug0sfinu36e80opk4e, mty $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface qm 87dz wr)9fwat speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e.cskdt6.tj 3. 4k )bms 9lfzcf2, we ignore friction) about a hingez4 ms 92f)lctktd.6ck.b 3fjs 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 shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we wcykehn hq-;wy9sq -0 9ant 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 hc -9w0qh-e yyh9k ;qnsas height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with2 x5pxoewmmu,6z83 k/ u2hvdh a radius The forces acting on the cyclist anuvw8e5ko 2mx,d3/hxhz 6 mpu2d 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 lendnxhy 1 du5 r9gmh:8md9g11tj gth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. Wh1n d:1rymtgjm x89d19 hdu5 ghat 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 777 iyjl; 3ht. k5yp.;jab xynk.docxarms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning . p7.di;kyx7n t;b7 5olj k3cxjyhy.a skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two xa55+v funv0 m38 qmw;fuva;t cylindrical plates, of mas n tu 5q;mvva3f;fa 5+wxvu0m8s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 5 2pgwz :5qf.zxk s8shtrack of Fig. 8–58. What is the minimum value of the vertical height h that q. f:5sw 28psxzghz5 kthe marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dpf+l2mu/m6iw2a qra.ep7fddzvf( 41o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85lp mrm6vgz 2+5 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the anguzmpg6m+lrv2 5 lar 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