A bicycle odometer (which measures distance traveled dvqm 5o7xb )0xd)cf0n) is attached near the wheel )ob fdx5q 70mdnxc0)vhub 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. Does a point on7f(-c.j t 7:okdbqyq s 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 eithec-fyb(s t:q7.d 7o jqkr component of linear acceleration change?

Could a nonrigid body be described by a single value of the angularw629;mujlk7t9i qzg h velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaipd,t9li j6y d2n.

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 your head thdg2g1 i;u)eod ,hdj:ja) r/hvan when your arms are stretched out in front of yv gd :;o,gei) dhud/rh1j j2)aou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rea1j4i:.k v btfc(n(k;ezwx t,kvt97uk 1r9xymr wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which :x,f (kct z9ybk jt v4(m;tvxe u1ink9r7wk 1.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 lower legs with -;7tnrrcvb9y5zuie: 9 qm3e* g n3ajeflesh and muqb3nu a c3*7rv5r9z j emegt;n:9ei y-scle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. ufx m9x/,-tnv 8–35) carry a long, narrow beam?

If the net force on a system is zero mrr y2x(k,x7ak0h,a g, is the net torque also zero? If the net torque on a systeka h0x(,7kmr 2xr yga,m is zero, is the net force zero?

Two inclines have the sa.ulih:/ti wjc7vvq5 ( me height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bott:l(i c75vutvwi q/hj.om be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down * soa(2o9yf:zg urydq/e.:olan incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which hal2y*y orgd :9eqzso.(ou:f/ as the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start 4884 eob4-ri tdkll9wxg10 dihmzqa) from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greaterdim8br)g4k498t -oiadqlezlw x4 1 0h total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angn14*ji22y:nlu j v+uo nq )llld8dx/oular momentum are conserved. Yet most moving or rotating objects eventually s)xol li*n n +u1ldnd/ 4j22olv qy8ju:low down and stop. Explain.

If there were a great migration of people toward b 5r)wzv2 sd1ythe Earth’s equator, how would this affect the length of the dzy 5w2bsrdv )1ay?

Can the diver of Fig. 8–29 do a somersault without ha jfsc,vb5h6phf/1 r6s ving any initial rotation when she leaves the boav6/f1p fhr5j c,ss6bhrd?

The moment of inertia of a rotating solid disk about an axis 8 cg-;rveh0rj6)c1wik x. ltwthrough its center of masv) e0tch6j8klc 1r rwg;w.xi -s is $\frac{1}{2}WR^2$ (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 q(ogzj6mw*)o jh7.i erotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will yourj qh)6jw* z( moeoig7. angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicaldx fg0,5kjaco9u e-y. and have the same mass. However, one is hollow and the other is solid. Describe an experiment to9k5du cjf.y g-e0x ao, determine which is which.

The angular velocity of a wheel rotatingk rg3qjif( z1.4abv/e 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 pov(.43 qg azkei1bjfr/ints 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 turntab8f5dxe wq; p4ole. What happens if you waleqd o8w x5;4pfk toward the center?

A shortstop may leap into the air to v9ohi;u t/aeds)+ nt1pou6g *catch 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 that his hips and legs rotate in the opposite direction;t*tp6dos1/9uvgh inu)oae + (Fig. 8–36). Explain.

On the basis of the ob;)l9 bkotr +lzk- nd4ya/ua -,v f(rlaw of conservation of angular momentum, 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 soruoblv-+dl bn k) ; ayztar4-9/,(fktable.

  Express the following angles in radww x 2+vjty;t p.y .,g: owo0o61nxjyk4( nvqfians: (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 oftov(cvqkj/y:.wl+i m23l9 np an amazing coincidence. Calculate, using the informa moi +lc:wl 2kpvtq.v3( /jn9ytion inside the Front 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 from Earth. The beam diz *fn0.8tpcu zverges at az.n *ufc8tz0p n 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 rotate at a rate of 6500 rpm. u;rgw 0wz-u3m un.v,cWhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? mn.wcg rw;3,uu -z vu0    $rad/s^2$

  A child rolls a ball on a level ftca.gkz o d4+rl , t-0ecchzfzml,(3;loor 3.5 m to another child. If the ball makes 15.0;kl0 a ,f zmg-dtzhecc+4, (zrcol.3 t revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutip ),xst pm hf)z3,md+7evka5twx(u 8tons do the whee,mp, p53t8) 7uttdm h()kv w+sxex zafls make?    $rev$

  (a) A grinding wheel 0.35 m in diameteczqh yy*:6,: fc;whf sr rotates at 2500 rpm. Calculate its angular vel,:cc hh z6; yfsw*:yfqocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolutioeblxu axfvx3l;7,,7gn in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m froml;xa x,f,b7l3ux vg7e 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 its orl vo-2 ttht s:/r,n+tj3uid9bo-qx2w bit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a xf)vp 1b+v.s0ga bgt: 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 0f nr njt-1eqxij tl1/p0+ew/rotate if a particle 7.0 cm from the axis of rotation is to experience an acce/1l /ie+q 0jnnxrt10p e-t wfjleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centome,mf47ng)rb w g b1 c3-ud/per from 130 rpm to 280 )r4gef/ nc13d-bmmp 7 g ou,wbrpm 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 radiusq i fmi2ig03g9)p7;4u qwjqq u $R_1$ 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 a od(km t35:vixboard the Apollo spacecraft put themselves into a i5v dk: (mox3tslow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be 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 +zqovjg2u u2*from rest to 15,000 rpm in 220 s. Through how many revolution2u juzv*go +2qs did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 r+0taw)a+ikum1x44ts n xz)vd pm 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 whirling “hy d y ;yb8+:7put)byjeuman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final spe+7 j ydyp):uty 8by;ebed.
(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 in 6.5 pr7mmvv1, k-zs. How far will a point on the edge of the wheel have traveled in this vp, -17mrmkvz time?    m

  A cooling fan is turned off when it is running at 850rev/min Itqyhua9h*yj66 turns 1500 revolutions before it comes t9 ujqhyya6 *6ho 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 revg4;vutdw (/s 5ox6 sxiolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of g(ux5tsx; 6d vo wi/4s0.80 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 the1od 5b) 36fsyfy7 luv7or, a .owopbn7 car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 6rl,d7oa1.7 v3y 7bs)bw f no5uf pyoom.
(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 peau.7dcjtk j6, dal when climbing a hill. The pedals rotate in a circle of radius 17 cmc6a.u7j, dktj .
(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 Nku uf(b)kl 6,r on the end of a door 74 cm wide. What is the magnitude of the torque if tfrlu u) kk,(b6he force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Ass 7rro8+1hf ejx7u buo3ume that a friction torq ef+borruo78x37hj 1 uue 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 ends of a massless rod whikxk53w ,;g erych pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Cal wk3ge 5k;,ryxculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighte chn) (/g:q ue1)xjrxaning to a torque of 38a:r g) x/jnexc)q1 u(h $m \cdot N$ 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 moment4 s+va3 d2ulsq of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of2d3av sq4ul+s rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.y)krl;rvv j+1w c4sd 5wjk1c) 7 cm in diameter. The rim and tire have a com)cjk14w rv)c k v+s1l5yd;wjrbined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a ho;/mt vr. m hwxv.nf01tqwoo 3-rizontal circle of radius rwn.f to -o v/.0xhtmv3m;q1w1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant ang+ vxun9moxz1t j+mwf:8b 3 vv,umm1 i.ular speed (Fig. 8–42). The friction force betwez9m:xmtn wimo81.++v, v3v1 bujumfxen 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 Fig. 8–44bc :2m oix0lo3 abcxi2omb4 0:loout
(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 whosh8 iq au -sm9hyf8u*(he 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, engin8qrc9n 33aw zh0m,pq ieers 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 theah9mncr zpw, 0q833qi 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 a30lq6z/ m noh5:q jebynd a mass of 0.5zhq 6l q:jm 3nb50/yeo80 kg. Calculate
(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, acceleraf t:ozv1syd*)d:8yx mting 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 smwd5wz3 v9 9 kjr-fspxuyc 8q6,all hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of 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 reqszf9ucdkp5vy6wj r9,wq -8x 3 uired 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 brougbrid;tu, q / g4sj;l.kht uniformly to rest by a frictiona;grbj ,qdliust.4 k/;l 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 nw:u ba *dekjo) cybo*1+ia47accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is th6. ,na e1zbln)i-k ttprown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball isenbkp-z t) ,at6i1.nl accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as u2- sk+s75o tmhhwz0l zkr h+5shown in Fig. 8–4l 5 mts+zu5h0khs zro 27-kw+h6.
(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 4y guohx i)10t.tkux.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 fromhyf8p8dxtpn +) qz/11 duw.uj;u-s dp rest within four full turns (revolutions) and releases it at a sp8w. xuq1 p-hd)nppu/d jy8tzdu1 f +;seed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia ofpog5+ie71e rp $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine45 j3l fkvha*c develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls witho2c ; 3v4vwwqncut slipping down a lane at cvq 4; w2cvw3n3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the 8n r vh(cfv)p0sum of two termh) fr0n(cvp8v s,
(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 y .(kpoefgpi)x 2i m)2a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cyliximg2i ek(oy).2f p )pnder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls w )8i)aa jn4qubithout sun) ba i)8jaq4lipping 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 to fall andcny h j 5)o3f-4i: aw,viun/ln its lower end does not slip. What will be the speed of the upper end of t/h3l :n)iw-yon u4n5fjc, iva he pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball cg0;jn;rf0p g rotating on the end of a thin string in a circle of radius 1.10 m at an angular 0j;;c rggp0fn speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindim/ hgaqdo*5 )tng wheel of radius 18 cm d*hg5taqm) o/ 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 siden; cjk)+or .f89gsqbe, 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 speed of ro je) of cb.q+ngkr89;station 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 inxc*um(pa8 1 tzlehuiu3 9zq 0;ertia by a factor of about 3.5 when changing from the straighu1x he a* t;z(93 muizuc0l8pqt position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate o9ezde *.ii;xfa o 5f47mrkmb)f 1.0 rev every 2.0 s to a finaleo;ifdaeki5.b4*)7 m 9zf xmr rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its c*.hnzwkx1 fr.ibuq9, enter at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 iu 1n.hfqb,.kxw *rz 9cm, 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 angulart u+iuk-fld8.mv ea8 +7fmc)s momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentumiysk;, uh1/a e2dpe 6 4vh/3pijnzk;k 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 cylindricaj ut.c(n0syxc3osgo --sfo,4l disk of moment of inertia I is dropped onto an identical d4c 3tg 0onsf- xyu -oss(cj,o.isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at58lc vi p/h6bhe;n2ot 2.4 $rev/s$ 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 center of a//uq ozo 3wc0i rotating merry-go-round platform oco o/ 3/wzu0iqf radius 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 frdk(cn.jw8* gadn6 oy :tn 70gieely with an angular velocity of 0.8 :yn(wa8jdo tdicg0*7 n k6 gn.$rad/s$ 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 i1 5 zmx:2eqccsnto 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 z c5xe:s2 cmq1rotation 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 excessza y 8bnp02pc 0b ys,52ql:vgv 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 1 qehlx.gd8soi .d*v;$ 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 plat4t(43wr1 gj 4giv qruoform, initially at rest, that can rotate freely without friction. The moment of inertia of the peoiw4gq4r4 (j r3vgu1trson 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 s;y2o7yk:;rsv;r w(r ai cbp6 hvk)g* the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a m;p;ovyr 7ikah2vcr:wks6gb sy );*( roment 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 thgrrao .( uwgn :sck;s.fxzg0 7- 1bfk:e end of the rope lying on the top edge of the spool. A person grabs the e:g0fz kcs.ux(ob1fg .k wr- rg7:;as nnd 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 side always faces the Eaxgt q)g1dt 0ml6w;hp2rth. Determine the ratio of the Mdwl g; p0)x16t hm2gqtoon’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 fbd w9re)s2y z3 gx )apq .o8uw bze*2h/kxn0,drom 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 vl8 *n -xj-zublg59 hdof radius 0.20 m acquires a rotational ratbzl89l hjnv-dx*-5 uge of from rest over a 6.0-s interval 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 kg an*hsqzj6q.x(mt:diu8 d 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 whensdq:jt qzhm6 i.u x*(8 it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel (.9vqwk *kk0bz2x3w b jc2 /ask$\omega_R$) 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, were rot,re/gg.r7hq 9jq;ig zating 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 itsz.jg;r/q rgqgh9 7ie, 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 mvz30)x.(tsq8 gf sg4na0gmsa 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 flywheel of diameter 1.50 m and mass 240 kg, and should be asn s0t .gas3qx )f g80gz(vmm4ble 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 p0xd-5wxos )6y(a qmm$H_2O$ 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 horizont1 k7lk r8llgo) ee;:rhal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and lengtifnu71z1 y ujt72oerm d;.*c ah L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center rc 1fa7mndtz*ueui71 ;2y oj.of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticalk ciqy;gl,epu* w76o,8uk ljc 93wis2 ly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has heigh239; kcup7ey6j*c, qlgisoilw 8k ,u wt h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makiz3nz xp,j+x 3rng a turn with a radiusr3n3 z,p+jzx x The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of lengthg6wtsrd -85m o 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. What is the torque required to achieve this feat, and where does the torque come from?g 5dwotm-r86s    $m \cdot N$

  Model a figure skater’s bopf zp05x:naq- x(vky-7bw9 *n/mfmqqdy as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds avq f- (:kfwm-pxbqnz7m x/ynq* 90p5for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindriagc)y 2uhk73e cal plates, of masygaku73c)2e 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 and radius r rolls along th/bs3o (rmno .p0e;etv e looped rough track of Fig. 8–58. What is t(me 0 b.prvsoote/3n; he minimum 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 nm1kx0jh(+ 2yofa lg9c ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its spee)y bvm1vjuwz*3t +kc8pnr;(m d uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wamtvkp nr;wcbyz)mj+3(* u v1 8s 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