A bicycle odometer (which measures distance traveled) is attached near the whez *r*60:g-bg vjmanxoel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-iobg*:0g- z*xj an 6vrmnch wheels?

Suppose a disk rotatesx a4m 5hsnf5b7u.qsd1 at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If 4m71n as xh.qs5ufbd5the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angulfaf g;fm(vm;a sb v7cx/lv538ar velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explair-ie(qim/oc 9 n.

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 than wh pnnrnl,tizlp1+4w 0pl16 bg -en your arms are stretched out in front of you? A diag1 4 rlnlp+nt,0p6n pg1-zbl wiram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at tcu +j,:sr q2h9hc .ynwhe 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, ,.9 hqc uh+:r2w yscnja small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slent t9zoz :;q /u4kyi9as3/aaoader lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantag9 u9ktsa;/:t qa /y zoa4o3zaieous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beampanyps-6obz(su3 r x/9/h3yg?

If the net force on a system is zero, is the net6 *o4f yfps*fpxx i3cs1a6q4u torque also zero? If the net torque on a si4*p1x u fs6ocaq3 fyf6s4px*ystem is zero, is the net force zero?

Two inclines have the same height but make different angles with tgbci d5l+is .wy:a)9k he horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be gr+a d :yik)sbgl5.9ciweater? Explain.

Two solid spheres simultaneously start rolling (+mbvgf44vc g,jjac4,4bpn p:bc+ym96j 2kbq from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which hc +pb2g 6 ,4kqccj4,b9n+vyb4 afv4jb:mj gmpas 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. Therq sz:eerqr9;/ *dta,y start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottoms:dareer r*qq; 9t/z,? 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. Yet most movw y7-w/ht bw0aing or rotating objects7w-t wyhwb/0 a eventually slow down and stop. Explain.

If there were a great mcg6. +xfkgui )igration of people toward the Earth’s equator, how would this affect u.k 6g +c)ifxgthe length of the day?

Can the diver of Fig. m c:lyrc2o5.w1 f6e fj8–29 do a somersault without having any initial rotation when shw coj5cfe 1m.2r:y6lfe leaves the board?

The moment of inertia of a rotating solid disk aboutb0sbkl+r* yxx (g*t/ v an axis through its center okb/gtb(r+ vy 0xx* ls*f mass 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 outstre.0zkuweea,jidc1 92jtched hand. If you suddenly drop the masses, will your angular velocity increase, decreas0ie. j w2k9d,jac1uze e, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and +)2 oehyc5qj 5 zrr.qbasl;5dthe ao ry5b; lr+)cqe z5 52djqh.sother is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel ro. bz p1a qa-j49tyn9tugb6.nztating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is jtz b4.nyn9-t6 b1zau9g qpa. the angular speed increasing or decreasing?

Suppose you are standing on the edge o 5gahz 4(oz/txet 0+lof a large freely rotating turntable. What happens ifg ha05l (e ztxoo/zt4+ you walk toward the center?

A shortstop may leap into the air to catch a bal-fi*ha8bh 5mfn,lb o0l and throw it quickly. As he throws the ball, the upper part lbifbh 58- *n,0fmhao of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law ofd*x6he w)sq wc 7ui-2e 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 operatuwe cxdwsh -* )2e76qie to keep the helicopter stable.

  Express the following angles in radian;iqjpp)c( 0qfe b/ /j-mtra:a 9ds 9or o0hjo;s: (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 Earcsa3l )+u*pajth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and 3s+ )caa*ujplthe Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mooncu l(y 9a 38dkutz-j+w +cwpt2, 380,000 km from Earth. The beam diverges at a y d+tzkwlau( jpcw-+3c2ut98 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. :po2l isy5d r-t6; pxtWhen 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 dowd 6t; sr:t po-2lp5ixyn?    $rad/s^2$

  A child rolls a ball7b7t;ps h 0n+i krs79vzhc -eb on a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamt;v7k+-s 7peh rbz7bc h0ns 9ieter?    m

  A bicycle with tires 68 cm in diameter travels-p otbwgswt k1.7o*;f 8.0 km. How many revolutions do the wheet .g p;ft71-wks ow*bols make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpmd2nn.t9 g68fn hq f7wv ykx0f/. Calculate its angular velwfn d2nx/hf096 g kqvntf .8y7ocity 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 revolutied dksk.a+ 6y-on in 4.0 s (Fig. 8–38). (a) W- s+6kydk.de ahat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity ( w sm b4bm2hl.zl0d:dof the 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 pp;*lgjex8n kjg*e mx()j j 19boint
(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) muy;z g 5df.xiui/m5 /a20ld i;rkhpg-ust a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to expeiif ud0rz;xgdpk/2ia/ 5lm;y . hu-g 5rience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abow3ym.s6 x xpw/)aa/hq ut its center from 130 rpm to 280 rpm .maxqpahsw3 /6 yw/x)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 radius w*6h m.e6copw)rv jk/$R_1$ 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 ab7hq3bkwzclg0 ts t+pu+-kav.p (.igx0el. .voard the 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 canq..il0v h7x0 +zv+3gkwst lk(cea.u ptb .gp- be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest qp i d3wl +ok++s*mc;ag3prv1 to 15,000 rpm in 220 s. Through how many revolutions did it turn in this;p3 +c+ vm3o1li*dk+qrwp sag time?    $rev$

  An automobile engine slows down from q2bmqhnltaw* 3 9yd+k,3 i0qr4500 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 whirling jix a9zmm9l +:“human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befm azm l9:j9+xiore 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 fkz/(s h;v2mvy(1hrxv+ 5 uxy ;wy8enjrom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have hhzvv5ey n(us8;m;kx/+j yw1( xr2 vytraveled in this time?    m

  A cooling fan is turned off when it is running at 8506;c2s jtrjhk1rev/min It turns 1500 revolutions before it comes to a strj6ksc;hj t21 op.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduct9 -z 3:lozin7ohve;yes its speed uniformly from 95km/h to 45km/h nzo ve9y lit-:;3hoz7The 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

  The tires of a car make 65 revolutions as the car reduce4r/ /dy3f znp*zcw5qos its speed uniformly from 95km/h to 45km/h The tiredf5n z*r/4 w/qzc oyp3s 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 puts1 nl)du wwmxejz0p( 80 all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 1(0lx1wd znu80j p em)w7 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 doo-d1wrs,3- f4:jfq* bg79bvs m filhgcr 74 cm wide. What is the magnitude 4s3*s dmcf jg-wfl:-i 1gfb9qvrhb7,of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel showne/yj bb)s ec.+ in Fig. 8–39. Assume that a friction c yb/)+.eejbstorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachem ; demkbd+. 0vs exdo()mm33pd 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 ma0okb3md 3sd)e;v. m dpm+m(xegnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engit(te2x+lf h -vne require tightening to a torque of 38+elh txf 2(-tv $m \cdot N$ 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 is6x t4sx *5lnjnertia of a 10.8-kg sphere of radius 0.648 m when the axis of ronx56tls j*4s xtation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rimzdp* /4pu k lk4u 3/o(a7)m*rnqvx bpt and tire have a cokt 3l74*na zdpub 4mux /)kpp(/oqvr *mbined 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 rotf. k,8g/1*xkn khaghjea 60m4yw1ee tated in a horizontal circ61k4fj gae8xhkayge0.kw /,n *mht1 ele of 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 9 z/-xjs t,udcon a potter’s wheel rotating at constant angular speed (Fig. 8–42). Ttsz jx/9-u, dche friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of poftwbesh7y)j yv0;h- *int objects shown in Fig. 8–43 abosv*7 )b0yhw;j f- teyhut
(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 whose total m p(.6sdq/0nsgy o)z nuu//heo0*nzfcass 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 y3fv3: 6jrhtsvkz.ks)xd8p; rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a massysj )vrkvk :h8.f; t6 zd3pxs3 of 3600 kg and a radius of 4.0 m, what is the required 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 withzpl8y/- tvme4 a radius of 8.50 cm and a mass of 0.580 4t-p8zv eym/l 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, accelhjj dtc+( ,k 6olib6tyzo c0t2o*ezs;k a5.8aerating 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 merry-go-round ao-rhh a en5t;-nd 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 required to peohha- ;t5 -nrroduce 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,j -9:a.wphp0mav x pc eventually brought uniformly to rest by a frictional torque of 1.2 -mapxapch 9.:,j wvp0$m \cdot N$ 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 balljiehu h)bt2m*p5,a;nl t)jmji)9fb p ,; o wp5 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 so )6rdcq84b.q uu5bkp7s 0wydnlely by the action of the forearm, which rotates about theuub). 7 0scq 6q8kyndr4 d5bpw elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig7ue4.r gxm70 rttugv6. 8–4ruxrvt7.u4 6eg 0 mt7g6.
(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 conw bd*;ro*1a1z5xzh z csists 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 th-y++ww w0pavq4w5 kx 8xm6ztu e hammer from rest within four full turns (revolutions) and releases it at a speed of p kxzw5 a4w txw6 q8-+0ymwv+u28 $m/s$ 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 ou)2qaa:qm8l . y0 fglhb z*8vej0td5bf 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 280 18h qmqf9mikf.;p3fusm. ms 4$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radifvks2/ l)bf d1us 9.0 cm rolls without slipping down a lane a l12f dk/bs)vft 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to t/l6 muze6 kqr6he Sun as the sum of two 66e rlz/kmq6u 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 radifo2wm**qex 9ius 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 cylindemi*2 ex wo9*qfr.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from .la0s jj t74m 3x-9iptm maq(nrest and rolls without slipping do4is7 nt ajm 0lx9ta3pq(mmj. -wn 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 star p9qalg*y g5kxt5,0srrzd7 -ets to fall and its lower end does not slip. What will ba7-9kgl 5zp,erg dyrtsx5 *q0e the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of /i7;w1lb y wcm2+hgoga 0.210-kg ball rotating on the end of a thin string in a circle o iocgw+/2h 7b;yg l1wmf 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 bz7a-3 tkck0dlkp3 0duniform cylindrical grinding wheel of radius 18 cm when rotating at 1 k-0a0 3d pdlk7ck3tbz500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that iszkoxph+ny9un/p. *;ssw 6ct+dgq 5p / rotating at a rate of 1.3rev/s If he raises his armw+pp/*yshut9 g5ns;nzpdx /o6.c + qks 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-*y)t t)rpd wfsm hw61 in Fig. 8–29) can reduce her moment of inertia by a fa6s wrpw f-t hd))*1tymctor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rail:jt i(w u88ate of 1.0 rev every 2.( uaw i8:lj8it0 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 vertical axis through its center at a frhwd95wj a4f g oj;3f*qequency ow* jjgdw;4 fhoa3q95ff 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 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 at 3.ub9++pgz f g5i5uo.*e t f8yib5 $rev/s$ 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 Earjn4n * a(j z.0 r:c6s-vujrvaoth
(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 cylindrfez24bf2qq6 zf:i msrrro4 70ical disk of moment of inertia I is dropped onto an identical disk rotating at angular spe062 mrq2rzbsfrze fq7f4 o: 4ied $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns oeeu k.,.n0;oix8ola at 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 centf)vigysvrhpx9 zb56 f+gx0 1. er of a rotating merry-go-round platform of radius 3.0 m and momenth) yx1095bg.+g i fvxvfpzsr6 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 f,qlnc*3: fp ecreely with an angular velocity of 0.8 f:n c*l,qe3pc$rad/s$ 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 fk aiju0*5ls*f l-w )j*r:tbwcux :h/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 :*)5lx ju0*c j*t/kfai-ubrfwh:lsw 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 wi+kl2 (ge ym-w/1amc wwnds 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 )/h8 tqh0vgz66 2ylwvqyop 2 i$ 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 tad) 2 mm scj)p/+zcmdb+ku -.rotate freely without friction. The moment of inertia o)dmdms+ -/ujk)mp+bc.c tz 2af the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diamet0 kq 1-x.psxh(:ckzsfhh0 5ak6m qvd0er merry-go-round turntable that is mounted on frictionless bearqxchkasfhxs vq 06 (: h01md0 z.-5pkkings 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 grounamt zg g45ox;;d with the end of the rope lying on the top edge5;;ax 4ztgomg of the spool. 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 side always faces the Eart6xz*7k2sxy a6vmqq 2gh. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular mom va66xgk* y 7zx2smq2qentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from wxdj w(j31thex/ z1sj i;5w3+ dggqc1rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0c (46gnhsd)jy .20 m acquires a rotational rate of from rest over a 6.0-s interval at 4y njc6() dshgconstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of;vfe,id36qip huz3 2z two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo whee6 ,fz3pu3iqiz;hv 2dn 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 w zglx9or8 4d.zxro(b )heel ($\omega_R$) 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 masgxlry /ys2f u-)t 9ui4s 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If ir-t)y g 4 ulxi9sy2/uft 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 l c l xa7+q70j2.mjmierow-pollution automobile is for it to use energy stored in a heavy rotating.m 7 mq2jaec7il0+ rjx 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 an teb( )ktw-5ui $H_2O$ 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 surf1zc4/(6 jidizi6tdtn ace 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 frle)un yg yi*0 /zd8m+deely (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 accelldm d*gy yi8eu)n+z0/ eration 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 vertical/+4og)eqdr n kv9 ae*xly 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. 8–549e) ek+ adrqo*xgv n/5). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a0orm 3t0cdnq,e s+bu/ turn with a radius The forces acting on the cyclist and es30q,0/bmnuo rd c+ tcycle 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 begincis 1 +z(:b, ;feesofqs 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 th ,qzcfse+i1( ;fso :ebe torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms 9q9j7fc /ovxqas thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spi vfcxoq99q/j7 nning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutd zwj)e*u67/y8z tvn mch assembly which consists of two cylindrical plates, of mass)w6je d8*n7/vtzyzum $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roy.p4gylrrkaglc;ts 5gm ( 0k1 tq.; ,zugh 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 point of the loop without leaving the track? l.pl ygqg5cg k,tr t01(z4my;skar;. Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not;2oa7 (t(dv 2c xwmieaej,eu. assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thynhwx;4j3;ggpo/+ct1g y3,lwqyh:me car reduces its speed uniformly from 90km/h to :xhoy w;h,gnm 4+jy;t3y p lq gw1g3/c60km/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