A bicycle odometer (which measures distance travelahj z2neg.x4+i+1d5 kv tjve2ed) is attached near the wheel hub and is designed for 27-inch wheels. What happens if e1gt +a2+k5 xd ehvvj.ni 42jzyou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim haq 4 8(/8zu pgzo8ohw a1-(khrb7bos6wia 2uuave(8kuau2 or68hb( b7g-4zuq p1 i8zhasoa wo/w radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocity d 8m*s/ w/ kwdxr +mc7yic(-zw$\omega$ Explain.

Can a small force ever exert a greater torque th2 vdxv7yyo( 5qe9e d,aan 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-up witht*,wn6/+sx5y ep raab your hands behind your head than when your arms are stretched out in front of you? A diagram may hp5estw,/b+y6 *xa nraelp you to answer this.

A 21-speed bicycle has seven sprockets at thk5lfb +:jqsom z l/.h)e 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 hardjm):fq lk5/sl hb.o+zer 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 fleshs/s0i+j;y 3 zwi/zt x0sv+t jw an/ 0sst;z viiw +/+j3 xwtj0zysd muscle 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. 8–3w-1z(uf hu38.+faub f 5qng cl5) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net l:og hbvx- w o7;wb,*j, m;niutorque on a system is zero, is h,o;: juw,mv; lowinb7-gx* b the net force zero?

Two inclines have the same height but make different angles with thodu ,0; olva(ld8hg(ge horizontal. Td;u,0gv l(dh o8 galo(he same 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 res)osx:n*kt ,c st) down an incline. One sphere has twice the radiu*tcs, )xs:kno s and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They starha6kv6q-n drsiaqba 27 j :g25t from rest at2a6j-b: 7qh 5d kvignsaq62ar 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 rotational KE?

We claim that momentum and angular momentu1 +dn2g,sn f*xz*ygedm are conserved. Yet most moving or rotating objects eventuaz+xgegsy*,1d d2 *nn flly slow down and stop. Explain.

If there were a great migrati3mmr .p. wuxd*on of people toward the Earth’s equator, how would this affect the led. *.wpmrum3xngth of the day?

Can the diver of Fig. 8–2bof. )kdy)e 9q0l90sndr ,twe9 do a somersault without having any initial rotation when she l0 r ey)n,kwol0tde dsf9.9q)beaves the board?

The moment of inertia of a rotating solid disk about an axis through f24o,xe3 lv l noq:9yqits center ol nlqo,fo4v y9x q23e: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 e z-n /gu)y8y.8kbqr jdach outstretched hand. If you q8gd.yj 8 /n)-uyrkbz suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical,n/gu.p1 u szegr .7qd1 o)dql and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whicn1lq, gsq7/ue.g ur )dp dzo1.h is which.

The angular velocity of a wheel r.x:2z, bcyzxss+h f9 d6.pmm7fr1z nu otating on a horizontal axle points west. In what direction is the linear velocity of a point on the tons6ms 71d92:p yf+z uhrz,x cf z.x.bmp 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 f z(f3t2n c89s(us(xwhvteu6)klp/ i the edge of a large freely rotating turntable. What happens)n6 2p/i stewvs9(uf 3(cftuklz8( hx if you walk toward the center?

A shortstop may leap into the air to catch a ball and throwwxu: o3o d*7jq it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly *7: xo jdwqo3uyou will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation ov9jc5 1enxrwu-tss +6 f 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 helicox9ej-t n65 cus sr1w+vpter stable.

  Express the following angles in radians: (a) vd;x- ukk 09lo30 $^{\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 cor+t*t,c 87u8)h e ijfq.g 1kgtnpbw 0uincidence. Calculate, using the information inside the Front Cover, the arj t .g0n+1 uqcwgp7hb*t8utk,8i ef)ngular 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 b:4b)s qjq) ce+ vj-ouseam diverges at an anglejvjb ) se:4oquq s-c)+ $\theta$ (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. klijb:+ nw1 12fftj61x)okb wWhen the motor is turned off during operation, the blades slow tbjn1o:fb 6 klw1x1) wji+f2k to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anotb;yk:web.-,t xm ee;pher child. If the ball makes 15.0 revolutions, what is its diamt bew m-bp;e.e,x: yk;eter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How f :3 ; vu(iawlnlg9w6nan dp6;many revolutions do the wh696f;: ;la wi n3vnwngp(dlaueels make?    $rev$

  (a) A grinding wheel 0.35 m in diamete elal/yoe0 :l.r rotates at 2500 rpm. Calculate its angulaleol:a e./0l yr 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-round makes one complete revolution in 4.0 sv c*wwu:eec5iwl5 (m vn +yf52 (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center? cw:+vm 5i5(*vw ywu 2e5nlfec   $m/s$
(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 orbit around (a cb77 qa1suethe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear set94bnpt-)pkv7l my4u ;a m j;,f3c bmpeed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm f4 p/5-s z(l8ujzepivxrom the axis of rotation is tiues l-(/zvp5p4z8jx o experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to:5:+bx wdgd; we*pk-lk s8oise;1z sq 280 rpm in 4.0 s. Determineopk:swsk+e15eg s;i8x w- bq;d:*d zl
(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 radiu g/6x:uf3v xuw 3ylzd- ji:d3ds $R_1$ 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 Moojd*7a hj502:seaj tinn, astronauts aboard 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 can be thought ojitdnjj seh* a720 a:5f as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Througlf dbo:a7 zm/,a ly16u ccs:,th how many revolutions did it turn ,z ud/7asboy,mc6 ct::f1all in this time?    $rev$

  An automobile engine slxsq/9 e6 2(h9qedfppgows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the:)yk7(-m1hmq akqnqj stresses of flying highspeed jets in a whirling “humanh :qqnyj -1)k q(mk7am centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerate4s((pypgqiug2pg 4 w 4 h;ddl3s uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in tgpud gh24i4s d pl3gp;(4y (wqhis time?    m

  A cooling fan is turned off wx)5tahr w5a05vuvoo ;hen it is running at 850rev/min It turns 1500 revolu;r5o ha v5t0v5w a)uoxtions 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 f p6f; ijx/ih8revolutions as the car reduces its speed uniformly from x ;pi ij/f6fh895km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car red: 7v;dlnmsv9jxos;m(w 4 c5qguces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0lmvgdqm5cxvno: (jsw s7; 4 9;.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 pedalo6ls3. pro/6b *od ke:a,hkrr when climbing a hill. The pedals rotate in a circle of rar. s h6la 6kbo,e rr*dk/o3:opdius 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 dd9fh(h4 gkr/tc.es a) oor 74 cm wide. What is the magnitude of the torque if the force is ex 9ecgd4(kf)rah/.s hterted
(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 a2ht 3wirz c;6fm .8faexle of the wheel shown in Fig. 8–39. Assume that a friction torquefc8rze3 hi twa2m6;f . 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 whhst(hc(ap*ify5s8;2zsd ) gwich pivots a(5z d*pis ; c(f hswtay2gh)8ss shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38780o(sl mck+g0sgdxi7 pc i m;c smogcmk0xlip0is; 87+7d( g $m \cdot N$ 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 ofa3tluwkmm kcfe1y3 +d,9 l- 5k a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through+,k uecltm13 dkm 39lak wyf-5 its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wi(eypp4u+/4t z.gge sheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. p4g/e+ 4gs t uizpye(.The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the ja 4 ./y isxl1+ndalm4end of a thin, light rod is rotated in a horizontal circle of radius 1lna4ajs+1.4l/y imx d.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.sq*bxe h k4:j3 brlm(’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay4rb3. *xjh mqske(lb: 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/;8 ; mu al(.p4mgfe zqtt1vmu objects shown in Fig. 8–43 abou u/ zq;;(1me mf.aglt u4t8mpvt
(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 consirk u3 sf +0c5l.evmn8zsts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinnz*oyuy+98zu htr 1j, 5wdcdm0 ing at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satelli*w 8tcyz j +,u9hdr z5d1myo0ute has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius j4mpb01 vd6pa( 3dhsdof 8.50 cm and a mass of 0.580 kg. Calpm pvd hdd03b16a j4(sculate
(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 4,- ynhuag1fkbat, 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 tangenti.rde g;ph;t* ef k-:jially on a small 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 required to produce the -e:;ji ehk.*;gtfrdp 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 r; c stgv7nt(7is eventually brought uniformly to rest by a frictional torque of rct;gvn7 7ts (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 acctk u)w ;pp p2mguq9-/nelerates 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 thrown solely by the action of the forearm985t9 lr0o)bkeb0 klz ihj0dd , which rotates about the elbow joint under the action of the triceps m d hlek0brj09ob tl)5i 89kdz0uscle, 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 *ib+(bc cton/ can be considered a long thin rod, as shown in Fig. 8–46 boc(b +tni*c/.
(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 two mhak znt7 sxow6-/*1 odasses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turo lhq286 qlcq(p;wd6gns (revolutions) and releases it at a speed of 28 (cq l ;gp lhodq6w2q68$m/s$ 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 of ;4vsaq,)e2mykajsr 1)qem / j $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develop( d: i3bb,jyufu5 iw,ks 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 radix)hj goj)a9 f *;gn(kius 9.0 cm rolls without slipping down a lane at n f(*o9h;gig)j)xajk 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earkruhr2l(:1e3zbxz 3z th with respect to the Sun as the sum of twouxbrl :(1z 3zre3zh2k 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 netw-j+62i , uzukd0jj 98fzgrp o work is required to accelerate it k,pd80r 6zj+oiz9ujg- w j 2uffrom rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wmwq-uo7chr e.p9;k,m7vy w/ dithout s m o.ph9u dk m,7y;/rvc-wew7qlipping 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 balance3r5l yec -mh)z+g3 p6(a fj,lqmtf; eud vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it56 fg,lc mfzylm+hj3;-(t qeurp)e3a hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on tiky72 9v*y kh+(jyuhh he end of a thin stri2hyku9yi h hj*(ky7+vng in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu:z,jaed +x:ag/0dw 75s:a b x .mpyrellar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm whew:beza7 m5,r:dxa d +0ap gy:lex/j.sn rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of* 0.oe.bbnq22nps z6t. k hvpq 1.3rev/s If he raises his arms to a hn s2v.p 0o.6q. e*ntq2zhbbkporizontal 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 o7+1u3mpg jato u-b75ma a6da sne shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight pu6gutoaa17m5a3 +mbds-p j7 a osition 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 incre5ah,ljgn:c 9 h8w j8sxase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final n:jl95,8jxacw hshg 8rate 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 thfhi fk .cewt8-5kz2;b:weivp9 o; xg/sko+g/ rough its center at a frequency of 1.5ree i- ib9.pk2ok/5 zw/hv:ffok + c8xgw;;st egv/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 3.5 )ka) +fs6mf(ce xdb 3v$rev/s$ 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 then z je(trrrkt 8-1otym6)pk,8 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 fmi2a:*59- co* fdt tjsgv)eh of inertia I is dropped onto an identical disk rotating at ag)j* femitt2av5dcf:s*o-h 9ngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a(qx ss-5 s.,pm(lqni/*ex djlt 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 merv/e1 ousohf:jgjk q-f ,:um+eb* 2 .qiry-go-round platform of radius 3.0 m and moment of + v-2u:ee*o:fjbukfh.m,/jg 1qoi sqinertia 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 rotatintt0q -a crysd/xe:2xt-b8c-n rl* lzmj;t9k6g freely with an angular velocity of 0.l 2;ttezsycj0xlmr6a x ck8--:9t*qbrn dt /-8 $rad/s$ 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 in f-qpk+8- x5: zlfseys,8dyaq to a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assum-f5s8qsdqfka-8+zyx : lp,ye ing the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involvpt h7n:mp38r9 )r qyepe winds 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 zj5-5blgpk f4$ 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,zpku/st7 z.l3 z kz:vb*.x)v fecd2z7 that can rotate freely without friction. The moment of inertia of the person plus thekvl 7 vbz.zuxs/p: t czz .2d*f)k3ez7 platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter me3:gxn0+kex y-avux8 xrry-go-round turntable that is mou:v xa -+eynxk8gu x03xnted 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 lqia.othg j6(p )t :xotu5 :m733ye reoying on the top edge 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 d grut 5:mto6y: 3j7 t3x)hp o(qeoiea.oes the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwtbe++cfu;sww2: v+g;acp f h.ays faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to ithc++pf2.ua :v ;bctfe g+;swws orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1jz.3 rjtgiwhm2g:f 4 : m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m gr r: cee(/z+iacquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate thirce/r :gez +(e torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, ea37z7vm- e)zs xb 8cploch 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 w -)e83 z7locvpz mb7sxhen 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 angi02ketdq 5pz .o4/2gtn *go uxular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the s-qtm (zd99kkc ize of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12tqk9z( md9 -ck days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the 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 for it to use energy2rveo3 la,jjbo07c+n stored in a heavy rotating flywheel. S0vle najj7, oc3b r2+ouppose 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 p(gb/2yh kx 6u$H_2O$ 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 surface azhihzo,so+ 3;a/u.yadm 9tv+t 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 (i.e., we ignore frict-k4b ns9hx57k zd gz u)awvl0)ion) 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 (anlah kv0 )745-g9bkuzz)sdwx ) 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 floor, and welzc pjh j60 3d)jo1l/p want to exert a horizontal force F at its axle so that it will climb a step against which l3 djhjj/c p61lzp0)oit rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with mib4uxq/p:7 k m p+;h 0phczh(a radius The forces acting on the cyclist and7upkp ih:b /z;4+phh0x q(cm m cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m ans5kielogd3 nsb98u 2(d begins whirling the rock in a nearn ku9(32g8 s ebol5dsily 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?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin ro2p dj lgtxsre46u9s8i:syz40* sme , kds, making reasonable estimates for the dimensions.r6mg sujt4kyesz 0sd4 : li 8pe,*2xs9 Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plateh677ppghz)+ a dg:lb hrv p-v7s, of mass zpppa h6: gh7vrh +7vgbld-7)$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rolmgemtdou00c7;2 y)6y of; neba5zu t5 ls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leavint u;050 etn6geyu7mzod 2fcm;y5ao ) bg the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n fc(100mac rhdot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed is1bfiuv d/hx z;h5*bp7n9)jhx9h8 ouniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acfvu jhhb)xhh5 nz9 ;sd*o 9i/i1 p87bxceleration 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