A bicycle odometer (which measures distance-,)j +gxv;t8 p e2jnrw1qdjq i traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use)n 2,j1-8p viqwjt+gqxd r;e j it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim hav ,/a7ztf g hpc8xjfr;609adkse radial and/or tangential acceleration? If the disk’s angular velocity incrhtr7c9j, a;z6 f k8axp0sf/ gdeases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the anguh6erdou+ :(3skhz n 3ilar velocity $\omega$ Explain.

Can a small force ever exert a greater torquewt 4utop;z ;jywrn7*ed5 i 8-i than a larger force? Explain.

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

Why is it more difficult to do a sit-upze btgw7h:n g()1 vbn+ with your hands behind your head than when your arms are stretched out in front ve+h7g ( )1bnt bwzgn:of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at thw1- xw;dk8we th9:lx )sii +qhe rear 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 gear is it harder to pedal, a small front sprocket or a ) 9-whiw1hqd w; e+kitl:xx 8slarge front sprocket? Why?

Mammals that depend on being able to ru6iz -g5y/uzql-c pa5c n fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the bauaqp-- zg6c/iy5cl z5sis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 4hk;mmu jz i51-m6z5c gt6yap8–35) carry a long, narrow beam?

If the net force on a systemzht:hrk4pryyw0t / 26 is zero, is the net torque also zero? If the net torque on a system is zero, iytry6rwpzkt/h0:2 4h s the net force zero?

Two inclines have the same height but make different angle3v koew .vhd )un918nzs with the horizontal. The samh.9)w k8env1 v un3odze steel ball 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) do -k ro6g2;3aiz7lzpso wn an incline. One sphere has twice the rspk7i3- ;ga6zl rozo2adius 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. The452z-xcgvm e qy start 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 kinetic energy at the bottom? Which has the greater rotationgm 25ez- cqxv4al KE?

We claim that momentum and lyn-at2t up q;3zs21 dangular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explainynz2t23a; ltudqsp1 -.

If there were a great migration of people toward the Earth’s equator, how s;ogp bv 7r)a9would this affect the length of th;avrobpgs 7)9e day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation ta/h)s5 s43pi6 o/jvw) uf 9 -dykgfhtwhen she leaves the )k/f/vogi) j5ypfd9t4 a-ush6s h 3t wboard?

The moment of inertia of a rotating solid disk abou4kyz2z wik sqye2g 94)t an axis through its center of m4 k9s2w)ezyg iy4qzk2ass 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 rotating stool holding a 2-kg mass in each ow) s8 n9darl9outstretched hand. If you suddenly drop the masses, will your angular ve899ldasw n o)rlocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one 36cujvw;u p+o is hollow and the other is solid. Describe an experiment to determine which is whicp3vcj wo6u+u ;h.

The angular velocity of a wheel rotating on a horizontal axle points west.)qjvqc l*skz3* it0t( In what direction is the linear jkqsizv*0 c)l qt (t3*velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating tu8 *fx gmdp-k*-nczn7hrntable. What happens if you walk toward the cente-n 7*zdxnfh pg-m 8c*kr?

A shortstop may leap into the air to catch a ball and throw it quickly.t;45yhhx/ icrk-ta+ k 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 d5aih+-cxtk; hty/ k4r irection (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentu: 6t 7a;;kpn.(b nfavbbndz h3m, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can opaba:hn6k d3 ft .(; nn;zbv7bperate to keep the helicopter stable.

  Express the following angles in radians:o5zyo(m: gl 9ty1 x mx+*cxcb- (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 c jorg9ejl +y,f:s 1 p8s7f6hpjoincidence. Calculate, using the information inside the Frrj69l7:+ p ,yfhejos 1fg 8spjont 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 M3 m21over pjyxa3c)7e oon, 380,000 km from Earth. The beam diverges at an angle p)j 1y 2rcxe3e7vm3oa$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turnemr ,r0gs 1bsc4d off during operation, the blades slow to rest in 3.0 s. What is the angular accel,bss1rm cgr40 eration as the blades slow down?    $rad/s^2$

  A child rolls a ball on 29ym4fo1ky-ux j4jg qa level floor 3.5 m to another child. If the ball makes 15. 2fyu yj1xgjm 49kq-o40 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travemh 8ej.(3ivjo ls 8.0 km. How many revolutions do the whmi.o hj (je83veels make?    $rev$

  (a) A grinding wheel 0.35 m m e,lom mrh8/y3y (a1min diameter rotates at 2500 rpm. Calculate its angular ,ymm1l3h(/o ayme8mr velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-roun f0- gs))uok6bsqqcs5o q d)0od makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a childfsd kq squo)0c5)q-0so6)o bg seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its o hlrb,crfm5):3 2grtdrbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point d2:vqw8wj1vw31 tmv o6 8mrfo
(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 roti x,pa()r r7f,zgy2 mjate if a particle 7.0 cm from the axis of rotatio)fim( rzxj gr2a7,py ,n is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates un. g/gc 3 sggmlqc4y57piformly about its center from 130 rpm to 280 glmpg g7qs5/ y c4c3g.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 radiusl9d .y7 sqn8zs $R_1$ 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 th 2a(zorr *qoi( .4t(aa2qxcwae Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with 2rciaa2x r. o* 4qtzaw(a( qo(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 acceleruvne (-utv nts6 irq;v; w3:w/ates uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this t6-n(rv v/u:;qenvwi3t s;uw time?    $rev$

  An automobile engine slows down from:jk (p7u:v 9 hdhsv(un yfu;f: 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 jetpdt f9ua c)wl/4)wr 5is in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions tdwpua w)/4 ir)cl9f5 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 uniformnl qfh uack/,la52c -*ly from 240 rpm to 360 rpm in 6.5 s. How far will *5cf, 2kclq/u h-l anaa point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 , 5v;i;d,*z s+l57n b)skjyyhxm yg jwrevolutions ,n ;j, *)xvy hz 5ws5;y7ykgmdjbis+l before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as thvif,q: f si(wepa 5,h7e car reduces its speed uniformly from 95km/h to 45km/h The tires havevhqei i (w7f,,5:sfpa 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 spee)q m4a*un3m :jf uivi,j l7xa3d uniformly from 95km/h to 45knju3f m a, vliqu)*x 37a:imj4m/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each paqt ecx7 b(k+ c-sbh./edal when climbing a hill. The pedals rotate in a cireb 7-cat(bk s/x.q+h ccle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 c7i59s xzqfza8ui 8m :lm wide. What is the magnitude of the torqu8 8zsqfl9 mai xiz:75ue if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of j h0v(oxiqd;3f; qub gf *f7d+8 qv3qbvd+jl/the wheel shown in Fig. 8–39. Assume that a friction torquebq d3vf+* d+v0hfl/jqf;o7dijvubq8 q(3 g;x 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 e:gj kpyl l(e)q 7evvm/a0;we7 .hz9cdnds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position vcae( :y9lhvdewjl q)mp;ze/07. gk 7 and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylind eup-a9jz62 xeiq3bg.er head of an engine require tightening to a torque of 38 z.q- biap6eu 9e23jxg$m \cdot N$ 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 radi7 oea* nghiy+8igp3a)us 0.648 m when the axis of rotation is throug eh*naggi+o3y )ip78ah its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wh.dj2h8t *-eh n;s3ouu hpv a.yeel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can b*-s 3 u8.2pthadjoh .vuynhe;e ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hofqcc 9k4b j*6u;fq-t wrizontal circle of radius 1.2 m. Calcu6t9 b;ff4k-u cqw*jqclate
(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 p,(4ttw gb9;gng clvb, otter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her gg g 9tbn;,b4c, (twvlhands 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 amu9wj /k* k+c rrz7gr5rray of point objects shown in Fig. 8–43 about + gw5r /7krmrk9cjz*u
(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 ox3 x kw(,:h ki npsth6 j36v:vmfm94fgiygen 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 cylw0fd3ct6ht u3jo u:qwc;b +9n indrical 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 s3n:dwu b6cu0oc 3w ;qth j+tf9atellite 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 gw3 :ceit i5q089umqga mass of 0.580 kg. Ceu m8q3tq9w0 g i5gi: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:2 , wpybg1vhya3+ck q, 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-driven merryp nm kkh)i.p xs16*8ze-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 t)1.kp8*nkx6p h ezmishe 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 av/,r3fdi vd2ou ku13ind is eventually brought uniformlyfviukou ,32vi d3/ r1d 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-kg ball a1 p2s6w-fvy swt 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 sogvnp b) ov ;1w9*c)kyxlely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly ov91 ;vw*nbp )cy)gx kfrom 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 consided5e m, epjs5k 8tvs+g2red a long thin rod, as shown in Fig. 8–46.+v g5ss,m tpj52d 8eke
(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 consl-6jg/ ;goopk- ya 8bqhy .ei;ists of 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 foudi wo+*+xozl- vci .8m5b+fnp 8d)u lfr full turns (revolutions) and releases it at a si+l)uldzi dv+o8 w+ c.pb *m8f-ofn5xpeed 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 inertkpcz,pu7wk 7xq*o e9) ia 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 developog .bj+ i.rj8h akja,5s 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 r0 by (qt fz7e5pg2qd4oadius 9.0 cm rolls without slipping down a lane at 3 z gqd4yf(op e02tq75b.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of hk-jo4aof; k z cdl/,*two,j dc/;koh*4-lo fzak 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 of 7.50 m. How much net wog .y .i-penmhkem4x +6du, hz2rk is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? n +d.iy2 4mxm-hze, p6g huke.Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fromk59lcuys e6(m rest and rolls without slipping down ek5s(l 96uy cma 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 and itsw/fbvlee2p;/. ; 1s2zn jefpu lower end does not slip. What will be the speed of the upper end of the pole just beu; ps 1/;zfje.eb nl/2ve fpw2fore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rota;v:xr n r(ohgiz;+q c7jvdd* .ting on the end of a thin string in a circle of radius 1.10 m at an angular speed of 1g;rn joc dv.i;+*dvr(:xz7q h0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifobuw)cz:roowf8 * 4 jq5rm cylindrical grinding wheel of radius 18 cm when rotating at :5o8o)4c b w qfzjrwu*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 p4 w16l 2a 2neye)f s:iduatsd h6.tjh5 9-ytkolatform 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 1w: 24t-ol.th2kd9yy ae) eisa5thsd fnj6u6 rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the8/ sm f -kb7+ nu -3cbfi2:dqglevrl8p one shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straigirkp fvmbng8-ul2 l3/f bc8dq se7-:+ ht 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 of 1.0c9+c yn1 x5*hfexmr0pku2p5 r rev everhf9* 5cyum102x5 necp+ kprrxy 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 around a ver jj1t8j01zdc gtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform01djtj g8 zj1c disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spi 7*b6s4ihjrb lnning 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 w,(j ze kz68wr*2bp- gbcdo9athe Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto anflrt pv1 j2u15ykz)4d identical disk rotating at angutz 1ypd4) 2f5 l1vjkrular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at o(y+6oce272om dv wpb 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 rotating merry-go-roulngsf 70w7 1oo(;0hnvj xs1v vm7 )vtznd platform of radius 3.0 xjs w)ohl 7o7g; v7vvs1fv(t0 nzn0m1m 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 rotatid0hpa2:r 4nt+ gvy q0nng freely with an angular velocity of 0.8:atv qrh00p+dn4ng2y $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwawew)grgdhw ky1* o,i: k9w9/.p(xmxp rf, 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 nwgyr g9 (k.hdk/wo iw9*p1ep) m:wxx,o 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 126xjr2hj 6 6vro0 $km/h$ 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 it08lxw ) /w0y xpn .yuw/x7iq1d3 sbv$ 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 freely wit zc nuecpeq;j, fv:i1 ip:63i;hout friction. The moment of inertia of the personzuc1c : 6;pjni;3vi:,epiqef 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-m4nkid q5a+ew6gfub 32z)n) fb diameter merry-go-round turnt)k+z3b54 )gd enuw n6bq ffi2aable 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 the rope lying on t0t3om k.-yz hfhe top edge of the spool. A person grabs the end of the rope and walks a distanmyt.3 z ko0hf-ce 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 +)geluara (hx)rk;n 2the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earga) rae+khx;u2(n) lrth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate rargfj1evk 2 mn.*c.0iuv) 2tof 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 the2/3piwkl hfon 8g2rj2 shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of fromkn3rg l2wj8i phf2 2o/ 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 ands00 p s. 8vxwc:k0jydw diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.0d0:x.c 0ws0y8j wvksp 10 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of thaj- q1g9r8ade(ltu9f - lmm,we 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 gre ipu(xc*x:q c s-i 1sf) :p(/onc1fkqmat, were rotating at a speed of 1.0 revolution every 12 daysq -1cupxsk cf/1fsi:(: px*mo )iqc n(. 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 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 automobile is b c:x9fdjfw-+ for it to use energy stored in a heavy rc-b:f xd9f jw+otating 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 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 anu2o9 nz2*kq9nt vz,am:mfl1p qhl)0h $H_2O$ 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 surfqp86pqufi3su7zh8- g2:w k cgace 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 pivot freely (e(*x *ie i noq4:y9urli.e., we ignore friction) a:ru*e9n4yi xe( qil* obout 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.]

A wheel of mass M has radius R. It is standing vertically on the er +27cy2qgjy uwy+n* hi+nc 8floor, 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 nh jr 2iyy*n 7eg+c8qw+ycu2+ has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn witht 8rtbyg,nk/ 8hyy,, l.+mh ud a radius The forces acting on the cyclist and cycle are ther +/8uy8bhym t,kldhygt, n., 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 -i*r.h4blpvox x1 hmpft-+o, 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 px4 .p,-v+ilbt hfoh1 rox-m*this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her a zodeegx .4dk h1r1*mbxg7k .:rms as thin rods, making reasonable estimates for the dimensions. Then calculate the rk17 xbeheo. d* :.kz r4dg1mgxatio 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 cylindric08ldi 2h6no sjal plates, of mass ons2li 0dh6j8 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough(+6ff5p*d h;eeimir v track of Fig. 8–58. What is the6 e5mh+v d(rffip* ei; 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 n3rh8bdw73 l l7 d+kwfxc3l 5qqot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolra5 ip; r0 7ks*t r+ope,lfax;* 3dtkiutions as the car reduces its speed uniformly from 90km/h to k, p;al*5*ired+siot3pfrx; 0atkr 7 60km/h The tires have 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