A bicycle odometer (which3qtprz+8k( 8)t,uhxf kjj7d c measures distance traveled) is attached near the wheel h)jk ,xt88uk q3hzjfd(7t c+p rub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity.a 0m c4meo.i: r/kpu;m.xnay8 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 componxen:cmokip/4ra0. . ma8ym ;u ent of linear acceleration change?

Could a nonrigid bodynu6z8y947,w27 yso 7v cg iacisaj rh1 be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a -scms zgt5w +ao6ug..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 e65ngyv461 z zzxtj;z to do a sit-up with your hands behind your head than when your arms are stretcv6z 5z;y zjxg14tze6n hed out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear w2dlvkw)q ;l6z3 uu 8efheel 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 g26fq)w l z 3lkv8ud;ueear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bc )bl6 *,j 2yjc xe5inplx2r(leing 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 dynamics2),jr jpilc*nl5x6y x (bce l2, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nxrl. :/c f, y+tkpdhxx(k8 wzo84 pj)oarrow beam?

If the net force on a systemctcpi-4q 1o2/q fx apuq)-f p( is zero, is the net torque also zero? If the net torque on a system is ze-fi qc oppaqx(t p4c qfu)21-/ro, is the net force zero?

Two inclines have the same hsb6j96 l*j/ygxms/n/ fh .ylreight but make different angles with the horizontal. The same steel sm9r y /jn.*yjxhl6f / s6l/gbball 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 an incline. Onegw- h(*to a(he sphere has twice the- hgw*ot(ea(h radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They 4,xp eta.*tdvje,wpklaoc 39z 0.q+ gstart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinedl,pwg.xt ep eq3oz9, j0 a.+a ck4v*ttic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving m5-0f 6emcok )efmmy7or rotating objects eveemmekc0 6yfmf ) o7m-5ntually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, howl1- j,5tbf:2k -;bg(luyzqiv; f w j,y oz/zrz would thtyrk-zl-z z vb/yz;5j,quw ( bl2j,ff: g; 1iois affect the length of the day?

Can the diver of Fig. 8–29 do a somg 9.9fp :dzcdhersault without having any initial rotation when she leaves :gpd .9cf9dzh the board?

The moment of inertia of a rotating solid disk about an axis through its9 ig; xoch9z, pdu 5zuki(g**s center of mass isu u9 d *x ioi5gp,z(ghcszk9;* $\frac{1}{2}WR^2$ (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+ 9mmj71yxizz g4n 1luk4l3m o a 2-kg mass in each outstretched hand. If you suddenly dr m4ml7i gz9 ljuk43m1+zxy1onop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identea2yc;0nx jx-ical and have the same mass. However, one is hollow and the other is solid. Describe an experimx2xaje0 n-;cy ent to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle p)wur f 5es/g.doints west. In what direction is the linear velocity of a point on the top of the ef/sw5rg)u .dwheel? 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 decreasing?

Suppose you are standing on the edge of a large freter tqg,vs. n.h ;kl7z 8162bfe(bo yqi7(pllely rotating turntable. What happens if you walk7klq b.(efs.8ztl (hv1 p;obtle7r i6qg,2 yn toward the center?

A shortstop may leap into the air to catch a ball and throw it quickld:m z:y 21,-sia-h+vk2uzygfz*r gf, p-ypqhy. As he throws the ball, the upperq zy:aph*2z-2fsyydm-, fkgph+ rzv-1:,g ui 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 conservation of angular momentum, d k.dpb7igodxk 4: )a:ttg:sk +iscuss why a helicopter must have more than one r:7:)bgtktxdsko+ gika p. 4:d otor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30l;xrbqh if0h.t +z.6 r $^{\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 becas mf2z2rbmqzv*g 2 vg-s+j, ;ause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of,fass*g+mj2 m;2 - v rzbgqz2v 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 -(l8.3bqp x8 myn ;ahb-p*ob1fvjaah beam diverges atp--v j.8(hh*3 x ;aya8p bob1anb flqm an 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 p ::z*7c zpotu u-*( pvpud4zerotate at a rate of 6500 rpm. When the motor is turned off during pp7p(uetpd u z*-vzoc*:z u:4 operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makekciu k(bx(6 6ecn .+ ,nnj8v df7wmelkku *,.qs 15.0 revolutions, what is its diame, .c wc ek8b.ni(7 n,q x+ju6efvk6n dkul*m(kter?    m

  A bicycle with tires 68 cm i c 8dv7bcqa-zsu/ s-7vn diameter travels 8.0 km. How many revolutions do the wheelsav-bd/v zcus8-c7 q s7 make?    $rev$

  (a) A grinding wheel 0.35 m in dia en7jlog4oh6cu(.;a h -) ovyzmeter rotates at 2500 rpm. Calculate its angular velocity in vg6(.h oj lzho- eca4)ou7y;n$rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8bth/c;rc (lf 1fwd) u4–38). (a) What is the linear speed of a child seated 1.2 m fromu; cfr4cb) tdl/h(w 1f the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in *1 sqgzlw8m3sits orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spesbwi6 pp euec*,iv-;, +ecbv5 ed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm bo;r/- lsnn1- i dglj1h(ek5c from the axis of rotationned--1/ ( shr cbl5gl;i1 jnok is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its 68o nuqco*8-uy) )aershw; l zcenter from 130 rpm to 28*h))-uoqy88nsc ;o zaul w6re0 rpm 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 radiuhvf)s2uqe1m 6 h:qmw7s $R_1$ 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 spacecraft put bu4 xc9z9 7lfq:4rkx rthemselves 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 cqurx4ck:r99l 4 bf7xz an be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thro9 b21g9-q pqyvh*uck /gs wu2dugh how many u9q *h 1-kgyvwps22dq/9cbgu revolutions did it turn in this time?    $rev$

  An automobile engine slo,r*xcy+p ea wwj70p 9cws down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirl7v8tpq+x7f mc;lo6it ing “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching it m c87;t6ovt 7xfq+lpis final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 w of,r9pc vu7w+:g:phckcw 2*fy w)2w rpm to 360 rpm in 6.5 s. How far will a point on the er22gwc p+o7: ,kcwvwwy c: pf*w)9fuhdge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min I *lmg7jhl -y8wt turns 1500 revolutions before it cm7hy wljgl*8-omes 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 rd/8j lll+pu*ceduces its speed uniformly from 95km/h to 45km/h The tires hadul +p8lcj/ l*ve 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 uniformlyvd if1 gdak4en3/58nf from 95kdin/8f nfa5gevd134 km/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 puty o 7cj yjv0;rwo7(4xwt4wxx0p 8amn*mi*h o5 s all her weight on each pedal when climbing a hill. The pedals rotate in a circle of o;7w ov j(joaxr*y h5p 08xt4*m ym i4c7xw0nwradius 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 adz(q-j*y(q0m e;rb ed door 74 cm wide. What is the magnitude of the torque if the f yd-qb(zm r;qe e0(dj*orce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–31,:qta2gg+ vn qtxlf)9. Assume that a friction torque of 0.)g1l ,2q+vtfng xt aq:4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a masslesc2dmoiik:i+rptk.fm8a0 g4 w6h pv t8:ej8)d s rod which pivots as shown in Fig. 8–40. Initially thep8 682oe:)tcv aiw p+hd0id m8i.kkfrjm:gt 4 rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38n8jcf p6u*/any;08mel,g vmoj 1we9 etqq8 i: e8 8 ipqoj qjw*m86 cev,mft09yu :;/genl1na $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m whdcir)3hbww:vjz 8t )0en the axis of rotatic)vbt0wrdh )j 3 zi:w8on is through its center.    $kg \cdot m^2$

  Calculate the moment of ine7jpecu21 bv5bw2y 7bi rtia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of jb 72eubyi 7b5c w12vp1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of ade c4w p51bn; dqh:/hw thin, light rod is rotated in a horizontal circle ofhd5q w pd/n;:h cb1e4w radius 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 al 2- wia )v8 l(ib/d:) :wyxfr*ddozcc bowl on a potter’s wheel rotating at constant angular speed (Fdxl)d(*a:2 bw/ -o8f wzclirv:) cdyiig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of t-a(q (wm7;por krzpi0he array of point objects shown in Fig. 8–43 k p(i(wmr -rpaq7 0o;zabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms jv8/ ykbf 2ztr*t -pb4whose 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 cy0al8fhqltu i-;m p,n4lindrical satellite 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 if the satellite is to reaml; i,0-qahuf8n lp 4tch 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifor6 qvl ,le;,hzdt1/ydn m cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcul n6zhvtd1 l q;/d,y,leate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from l geum eq+/:d ysl7a+2rest 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 h+99cgew6gml2g(vt f small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Asfew+ gv6g(29 tch9 glmsume 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 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 eventually bro g;;d3cp,clvax(c h; -k9eqtiught u;a;c (qglc9pev ;kd h3ict-x ,niformly to rest by 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 accelerates a 3.6-kggxzx p1s6/yt3 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 zia;6y. 5t uihforearm, which rotai;h6tiau. z5ytes 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 bexf2 ojh,(txswag bn- 4;*gzt. considered a long thin rod, as shown in Fig. 8–46.o;twsft-*haxnzg j4g .x( b ,2
(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 tk+m .-.x i+zg whmqqitc x6: kn7p+zj.wo 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 wi2 g/**: :muawj*xcruiitu w.f thin four full turns (revolutions) and releia*u:*fc2r */ u.um:g txwjwi ases 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 mom xy(;bfo.a 6sjrp,,7vi i -jlrent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a 9 zdndby()al* 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 o ty3ncdm ca w,fln-)ao.mxpy 3)02t(radius 9.0 cm rolls without slipping down a lane at 3)n)2o (l otca3ca f. p mtnym0x3wy-,d.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun aug u:y 2.kl-mgs the sum of two.:u uml gy2-kg 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 and a radius ofmgwe+ww 6 *22goss2 ep 7.50 m. How much net work is required to accelerate it from rest to a rotation ratems6o2pw2we+2e*g s w g of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls w(m e7 ,-hwnf s thbz25gui64ojithout slippingbh,2j gtw7 (zhns 4io6-eu5mf 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 ver/v:1frrntemc mk1z/2y2 u nt h odeb5+a/k)c7tically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the p+//moe7k1zdhu: fvat2 2 ren1b 5r)y /cntmk cole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a b ph*dp1o,w b i4r*w8b0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 1 bphbbi*w r*1w po,d480.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2/wixc zd+) v.k.8-kg uniform cylindrical grinding wheel of radius 18.+cv kwix/)zd cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform thmod 1f u3 gl 5w0+ 0pmsd1in;qu20umtqat is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Figlpgu nftwq1 m3sdu0 u;520odm qi0+1m. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia b/ hcg36,i e4ixqd z2. swey0wyy a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in wd 20zci3yi 6.wsgy,/qh4e ex 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 incrrlpci- ig; g2q(n5 t 9b3tv;bgease her spin rotation rate from an initial rate of 1.0 rev;t;b( gtnl q gvb5i i-g29r3cp every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating aroundc js6l*75edld a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of cla7ced6 5sd ll*jy, 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 momenx;fid *:q cndz l5ws(:.sprb8tum 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 :qpoe- n,wg*k 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 momenta7 *y-km/f b;y3i yvidzfi1/ r: tzim/ of inertia I is dropped onto an identical m:73 fzymt/f/b/yi; ivizda-i r*yk1disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2eam58fyqn56vw tu 1w) .4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the cenhzwoa,e-5.fl,qw1co c0.5 z/r flty, bpw0yh ter of a rotating merry-go-round platform of radius 3.0 m and moment of ifo -ll owtyr yez,hca.bqz1,p50f.5whc, w0/ nertia 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 anguym7c)h zkn*9elar velocity of 0.8 )kzhym*9ce7n $rad/s$ 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 event(ey/)81rq cpxrhol03itz 6 apually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of q3 ayz08p cp t l1ohxe/(i6)rrits 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 12g d9pt97iry 6rx 17fza0 $km/h$ 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 dvjug; 3gy2 yjj( rzfc.3*3y t$ 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 fra )s 4qlauija,zu b264eely without friction. The moment of ine l)u,i q26ubaa j4a4zsrtia 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 diameter merry-a,9lizdly:h0 o:r r;i go-round turntable that is mounted lroh,yai; 9rzd0:: li 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 the rope lying on2/xb( nb f vhn/q/0ckn the top edge of the spool. A person grabs the end of thbq 0nkx 2 //(/nfcnbhve rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that thepc :m,g dkj5)vyu iqj0n.)/ ie same side always faces the Earth. Determine the ratio of the Moon’s spin v)m/u:in 0cjqydjeikp),.g 5angular 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 a8;5h:g(wj e du* jwr7t)ikma from rest 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 acl +9 b,gz1jujtn c/.zxquires a rotational rate of from rest over a 6.0-s inter,g ub.jzl tj+z xn/19cval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 esc2m4 4y,n5ldlkt/im- zr q)uvl 2 y/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 to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long 2lyy2c4qzm45 /iv/m lkr )n, d-sletustring, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular spk/5 z:6airh syeed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as grea-kpzr/at2rd qqq s(u di8moh3iw71( 6t, were rotating at a speed of 1.0 revolution every 12 days. If it were t3 u8/( zwmqqadp(r s2h7 ikrd6 o1q-tio 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-pollutia5ud 7urn5v8g8st fm3on 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 flywheel of diameter 1.50 m and mass 240 a8gvnuur8 md35t7 f5skg, 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 l0ioy.1fn0e. zgfntx ; lk hmy86b0r +$H_2O$ 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 s,u p :fdm0osv.urface 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 piv b yv,3u5j2yq--yesk mot 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 shown. [Hint: See Fig. 8–21g-vm3 y, kje2qys5uy-b.]

A wheel of mass M has radius R. It is standing vertically onu 7l0uki if(8m the floor, and we want to exert a horizontal force F at its axle so that it will climb0kl(f8 muui i7 a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wo;ry5 xuqrh0q 7t.l +uith a radius The forces acting rqyh7q x5tl. + ;ou0ruon 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 begls0k f:s8yw;m ins 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. What is the torque required to achieve this feat, and where does the08slyfs mw;k: torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin ruy7* ewab 8cahc-vx. ,ods, making reasonable estimates for the dimensions. Then calculacy,b7-wcuaxv8ae* h. te the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cy 2ku jhxg.*db,lindrical plates, of mas*guk d bjx2,h.s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 andvhn d +)xkm-l: radius r rolls along the looped rough track of Fig. 8–58. What is the minimu+d -n):mhklxv m value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do nd 9h jj.:rwb9lot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolution v;ed)rbak a3l( )31us8iapdqs 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 angular vdq p8a)k3i1l rsau)deba( 3;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