A bicycle odometer (which measures distance traveled) is attached nwk jzo7kss,r 00,hb hw*p6ugv;(kq1j ear the wheel hub and is designed for 27-inch whks k ,*zjvbk17s qw j0ph6rgo;(,u0wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vunow.f(5 e *qxe)y9 fvelocity. 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 whic e)(wquf*xnv o 59yfe.h cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be de+ fq5csnc/.quhp;+ kjscribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a 9aeom*v3 i51 aubfn 2egreater 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 riud:6dyx zr 63) c9owdo a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagra:r6uy9 dczr d3o6iwx)m may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at th* nc4 q-8m;qf vsvjh- g+qhar6e pedal cranks. In which gear is it harder to pedal, a small r*a+6 -;-jg fcqh svqqr v48mhnear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender low hv: z95zm 5tkc +p+mdgtzi8(eer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of5+t dzkm5zm9 i+gh z( c:tpev8 mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a,o pch:8+0:bi5yxxstjf* o mx long, narrow beam?

If the net force on a system im.dxx.rze/ mo)u.s a+p4tf6ks zero, is the net torque also zero? If the net torque on a system is zero, is tmx).ps me.ur+ 4.ko txf 6d/zahe net force zero?

Two inclines have the same height but make diffe8;w,m *d0d m gckmw6(r vtdv5urent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the bal5vu;6tw,k8rm ( d gmvc wmd0*dl at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (b+x ;9qww 4kztfrom rest) down an incline. One sphere has twice the radius and twice the mass of the otherx ;wbz+qtk9w4 . 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 r 9lco/+/,cq g: qlk:94 irqzdn)o eh:cbwor1eadius and the same mass. They start from rest at the top of an incline. g1 qo k)rq/oew9/lo9 cqli+::4zcncbr:d , e hWhich 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. Ye kzprud d/r;54t most moving or rotating objects eventudzdr5k;4ur /pally slow down and stop. Explain.

If there were a great migration of people toward t8c5rfvq8v uo.k jq5h8he Earth’s equator, how would this affect the length of t8.8v qck5u5hr8j qfvohe day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotati1sc4n6d (dqod5k h+lx on whelqch51dk + xo 6sd4d(nn she leaves the board?

The moment of inertia of a rotating solid disk about an axis through i9 oa+v1rnc9fpuv2pp7 ts center of mass isa cpfu7nvv 219+p9rop $\frac{1}{2}WR^2$ (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 p-k987kjse(j c o5ur w-z.lpsholding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increkj prsewk-lp-co9zu (58.j7s ase, decrease, or stay the same? Explain.

Two spheres look identicrn4.w dy9w)m4e vw )dtal and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whic 4n4rt wy v.)m9wd)ewdh is which.

The angular velocity of a wheel rotating on a horizontal axu :fawf sx,*;dle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular fx:a;w*, fdusacceleration 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 freely rotating vvmlk.4fk4 2aturntable. What happens if you walk towardvakk.m4vl24 f the center?

A shortstop may leap into the air to ; v24d5ki am d2si2rwgcatch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice thisvm2 iw gkrd5;242d aat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momt,.a3aj6. 1hwh wk2hnaqew- yentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter s,..3w 2awtwakj h ehna 6qh1-ytable.

  Express the following oulx5i-k qyw2 g ;fpm/c9;.ppangles 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 Earth because of an amazing coincidence. Calculy 3zlwge: o04wate, using the information inside the Frontyow03we4 zlg: Cover, 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 .pwf4adhtcvl+( ( kz8 from Earth. The beam diverges at an wlh.fza(vpc4k d t( +8angle $\theta$ (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 qg*l vjcq7r32i*je4 f motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular a*e gj4rjiq7 3q2cv*lfcceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anoict f)-:mc2,enzhp(zther child. If the ball makes 15.0 revolutions, what is its diamet(ic)z2mfz,-cpn the: er?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutz,cly3kqweo smc7 vp4y 92 tm3h*f, 7fions do the wheels makvy47kofc9f7* w mmszle,t 2,q y33hpc e?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its aaq3k z8. qvp- d1oonn3ngular velocity oozqd3q31-nv 8a.pk nin $rad/s$ $\omega$ =    $rad/sec$
(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 /1qsava1a/zaayhq+ (makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cq y+qva 1a/as1a(za/henter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of gd rd/ nsnup,rq-e/f b/ ,)8ohrr:cf 1the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a,j lhcw38yco,; sen h n7s(q0)p lvah8 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 vasf- u( rdvak s70otz (162rffrom the axis of rotation is to experience an acceleration of 1o krfu(t as fvr6d17-s0z va2(00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its(36f +wcdig pa center from 130 rpm to 280 rpm in 4.0 s. Detegf (a+6c dwp3irmine
(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 m4nqg0u* r( qwla(6wj$R_1$ 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 spacecff92sk( gobqx p -1l *2iqahu:4n;szy raft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the s21 oqk y9 4-g2su*sx:falbfzqhni; (ptart of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniforml0 ,tt6bwfahm;) 1 zbyhy from rest to 15,000 rpm in 220 s. Through how many retfyt6; h b1,) 0wmhazbvolutions did it turn in this time?    $rev$

  An automobile engine slows d-bf1yjn *-qvhx6pue7own 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 wh5vt fii vbte61zs9d8,irling “human centrifuge,,bevi f81titv5sz6 d9 ” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm adv vk8no:;m(in 6.5 s. How fa vm;o:8kvnad(r will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at ,q . /hpbyb ljd4rfw4ju; +kegs9kv +;850rev/min It turns 1500 revolutions before it co dkp /;s+;hy9u wv j4fgbb,e.4lrq jk+mes 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 ma9v.f, 8p/ 7*xjuedb2 jndfu qvke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diad7qf* ,edn.2j 8uvj vxf/pu9bmeter 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 reducet0 f:d0b;e1x gqrxrk5 s its speed uniformly from 95km/h to 45km/h The tires have a diameg0tf1d ;5q:x krrb0e xter 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 on5iu pr)b7 gq7u each pedal when climbing a hill. The pedals rotat5q7bu) gr pi7ue 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 5t+knvb. pr38(k* z. 1akczbxd 5 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exer3 +.* (zb1pzk rc.kxn8tdabkvted
(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 qdl nd 4ifd)j00 zol45Fig. 8–39. Assume o4i4d l 0dqf0 jd)znl5that 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, aadxb 0nh)7r l.re 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 then released. Calculate the magnitude and direction of the net torq 0)rn.dxl hba7ue on this system.

  The bolts on the cylinder head of an engine require tightening to a torquib. pj7y0vzmh- vy 1n0e of 38yp-0yb z1j0i7vhmvn . $m \cdot N$ 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 ;wb*jx0yq; 1l6fhe srof radius 0.648 m when the axis of rotation is through q 1l0ejrsyx*;hf;6 wb its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm inogv ufies-1,; wl i(b0w (5qwg diameter. The rim and tire have a combined massgf5biw1gwoi 0s(uv lew( , ;-q 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 is rotated in a horia q1v; laa8qp1y /4a , r*b8hoqychc.tzontal circle oaaa81 yhlqcqq y, a*/ tbo41;phrc8.vf 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 a bowl on a potter’s wheel ;8ugv rq +5cma;y9hugrotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.59a 8vurq;c5u gy+m gh; 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 ahqme6ga 4*xm3bk :0hc y+8svsrray of point objects shown in Fig. 8–43 about3h cvkmgs q8*64 syhb0aem+ x:
(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 conc 1kc9n2(-/ hnxym 5q4tezclb sists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ra ; c ta(yb8+7fbkcl8cjcy77of r*x0 tlte, 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 requirf ;tc xfy j olcktcbcb*+(78r77yla08ed steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a maq5afcba 9 yw38ss of 0.580 kg. Calculate b3aq yca589wf
(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 batm*plmxsa yor99*-s1h 6a8oqh , 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 tangentially on a small hand-drivg-,/eck fva v(n /u uz*zj+y znkc93f8en merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg,jv(fz+-u8 ,f/cngv3 e*z ycka/u 9znk 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 b.w:lu (kxn r+is eventually brought uniformly to rest by a frictiobx+kr : (.wunlnal 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 8.t7rd5s er25ox pzcm 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 fiari ti6 2-q:of the forearm, which rotates about the elbow joint under the action of the tric2ri6-tfa ii q:eps 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 l /xn0tb6 ;a:6rpuep tcong thin rod, as shown in Fig. 8–466 tcrx;6:e0anpptu b /.
(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 o+zat 1lx8 1hcp- 41uycue a0alf 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 the hammer from rest within four full turns (re*r88joqo 7j oxvolutions) and releases it at a spe8j7j rq*8o ooxed 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 ug :x4fado0 x: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 of 28 t6vyrntoo5 mbi89b(5au+k6 v0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of ma*dc 24tjvus; oss 7.3 kg and radius 9.0 cm rolls without slipping down a lavs;utcd j 2*o4ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth withgm1 1kspg vk,zc: / qg29nsjq; respect to the Sun as the sum of two terms,/j 1s,ksvpg: ;k q2cm19ngzgq
(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 av+)23a orz roia7jx- h mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from )xj 3irav 72roz-+hao rest to a rotation rate 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)gb *rk8 p+bjr)f5dpaen uo7( starts from rest and rolls without slig abrnj *b u)7df8ek5 rpp(+o)pping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to3lrf2reode ey/hz trydrj p8 4*a1o;) xl-0/g fall and its lower end does not slip. What will be the sp) 82 *3r/hozxr rlol;pdd-ea e1y/t4rfjeyg0eed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mq -uzzm*goo8*.p7gby fu .tw4omentum of a 0.210-kg ball rotating on the end of a thin string in ag.u-fzp* ty *mgoubw8 4o .7zq 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 cyly 6fb ;fbqt5l*m6fn)t indrical grinding wheel of radbf 6tf;6f 5n*y) qmtlbius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands xe3tsc)q)-gvye7qi h e-x-fk 5g iv34at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his armsq)e 7eiskve-4 hx3-cgx5q)g t- v3yfi 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 i -)8pmsbjzjk d-. (spfnertia by a factor of about 3.5 when changing from the straight positionszjk-jfm -p (8bdp )s. 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 hme0g/jio cne, k5u jo+1yl.6m er spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate gi,ej ku.om5n c6o1y0e +l/mj 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 vre; *x)a .ltid) di5ffertical 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 txe)ial;.if 5*d td rf)hrows 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 skater spinning, i fml/ ,u6:c/oet.cf yl8hp.pqo z.l 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 Ea8li 6k/; advzzrth
(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 dropped onto an i p g.z9 .1,a)zisojgrkdentical disk rotating at anrjp g.kg. so9,azi)z1gular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2*v/h y9 (efjce.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 1*th 5wc.p zkiat the center of a rotating merry-go-round platform of radius 3.0 m k zt i1*cw.5hpand 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 rotatia8v :hh7ios -ing freely with an angular velocity of 0.8isv -8a:h hoi7 $rad/s$ 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 col biqx50 :cq*ke9jtzfwn jp-(8lapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integerj bzepi 9(q0 -nwq5*kx:cft 8j)$\approx$    $rad/s$ (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 wiudu x)tep6n8i; e+ qd0nds 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 w2f0lu3(mjux/f ytv9 $ 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 m d+tod1gp q:+ymb09u)s+www ,tweg/at rest, that can rotate freely without friction. The moment of inertia of the person plus the platf mw e1p 0w/+9yt+uswq:wm,)tddogg b+orm 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-go rb60g27 8mp,g-ye6yfsv;0hmbo4 tlrijd 8md-round turnt,m0 vo8 b2gh-sd6y6l0yrm;8itj rgfdemp47 b able 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 l81 nki,98 4 w nqdqh:vhrs)erthe rope lying on the top edge of the spool. e49h8 hnnq rk1)sdw 8il,rqv :A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sideuoke +d 246gjb always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis)k6obu 4+d2 gej 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 at a rate of 1 m/ e+p )*s nkevbc7 j,d1mc9;sfu$s^2$ 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/*nytc3 u;jsc uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the tos;j3 *uct ycn/rque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of tne5+ sz) 3qj:.atqznnwo solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.005nztqj3an5s. ) nqze:+0 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, howpoay bx.* :9w/ g3nyls (j)lsy is the angular speed 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 Suwp1tc )m+kss9n, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the t1) +pcsswm9k 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 authc2fse c 4-hiy1*bc0 jomobile is for it to use energy stored in a heavy rotating f0-sc 2fiyh b*cc jeh41lywheel. 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 illustrat /fr lae3-4pvyfe1(nyes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on/pea5tv 1h woj 6l2:irc ;wv(o 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 an3jyo*kyb k8t ;d length 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 rele3;8b*kjykoy t ased. 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 isttsqpb(e1pa;m. /q 71n/ aod(cxnuy* standing vertically on the floor, and we want to exert a horizontal force F at its axle so thaoxq /s( *;1yubtne(dp caq.tn1/am7p t it will climb 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 r7f6mvqi8-ii x84v c twith a radius The forivr74civmf8-x6i8t q ces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begl y6p z:qu4ljt/y4hr mcys8hpc(p*3 ,ins whirling the rock in a nearly horizontal circle above his head, accelerating itqc8yzpyl/ : h6 p, r*myp4t(sjuhc3 4l 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 andzkho:xz);8wm*8 w 4 zov(ggib her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular s z 8v zx*gbh):o;w4 (kmgzow8ipeeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clu1ia6);i ygeyz tch assembly which consists of two cylindrical plates, ofzy)ii1y6 a ;eg mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 aly-rs:vg ud;6x bwfy96mf8 p/p *h9sso ong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest poin 6s-;9hvud6/fgsfwm 9 osyy :ppb*8xrt of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asnfm*e4 1lxl8q4po s0p sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 rev7 z;;7lojr)j+m8sv irwtt 5kwolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires hjw;l ot7jzk;)rwsm 5+i tv7r8 ave 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