A bicycle odometer (which measures xbcvy1bk 1f1u25pg6e a9,yca distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-in5f bkye96ay cbvua,1x1 21pgcch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim hr46u a cp)x;f5oyq 2ptave radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? xf;)4uqpy62 toarc p5 For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single valurd:rqo. p+3uce of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaeg3ab(q9nkaoc17 5,3qj clx kv c 1x2xff/mz- in.

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,by2- b.u- ke,njbms+3, qkkoajic 8f sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram maynjk,,.bfo3mybe uqi82k ac b-,k-+js help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel andl3on fj6g4t n3 three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket ontf 3 3lg6no4jr a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs; a, uh4r hkijzom+l/5 euci3lj0n4,h with flesh and muscle concentrated high, close to the body (Fig. 8–34). jo a/uunl4m;jk z h+4 er3hl,i,ci h50On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkej98v c 4z.vvld);v ok- pwwntu+-c- nmrs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If*1,voc,k kdxwe6.o pn the net torque on a system is zero, is the net force c6,kv x1.*ono kwepd,zero?

Two inclines have the same height but make differen1l q+il y;pcqx w-m5(kt angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom(lx 1ik;c pqmq+l -5yw be greater? Explain.

Two solid spheres simultaneously start ro- y 38uhg3:bia*hm( c5az0wy4k z2bqfgbz tm6lling (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 has the greater speed there? Which has the greater total kinetic enerbkym :bm*hzz 8-bfc6t(yz323 uhwa g 4q0a 5gigy at the bottom?

A sphere and a cylinder have the same radiuszki od)ey4-c ) and the same mass. They 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 y)-idz4k)eco kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mos(tq sco kv:s8b j,svs(i;(78x, 7fnrqw : wdxzt moving or rotating objects eventually slow down and stoqv;w8:t( s , dr(oxk,w:i7(qns cv z7bsfjs8x p. Explain.

If there were a great migraw,2bbb7,fwmqv; hx5ksm d+;z ;p3wqm e*f1 oation of people toward the Earth’s equator, how would this affect the l7a1p5ebwwq ,k h fo;s 2x+m;bw*v3z ;mqf md,bength of the day?

Can the diver of Fig. 8–29 do a somersault without having any ing-z 8 uyz;;ms.pa xjq1itial rotation when ju;-g.8;1pzzq axysm she leaves the board?

The moment of inertia of a rotating solid disk about an axis thrqi6m:y4 sm9 xig3tx7cjn0re2ough its center of mass nts243j 6q 9g0mi:ier7x xm ycis $\frac{1}{2}WR^2$ (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 rotatj,trhs 0h64n4cgv6lf;f v,ged. s tk 7ing stool holding a 2-kg mass in each outstretche sv6s.;,er6tgflh4tjv7 g4ckh ,d0 nfd hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hol 0 qfd- *otp,qdjc4ldp*/r0gelow and the other is solid. Describe an experiment to determine which is whicrl-*d4*poe d /0q tqg,jc pfd0h.

The angular velocity of afnvpwli+ ,++ u wheel rotating 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 ui+++nfp, wlv the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatin7luhp c-2 bme)g turntable. What happens if youhblp)eum27 c- walk toward the center?

A shortstop may leap into the air to catch a ball and throwktd 52i+o xut- it quickly. As he throws u+t-x25tik o dthe ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helim *7nk(qvl4e mcopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the heliv(q7 *mme4lnk copter stable.

  Express the following angles in radians: (a) ka t7u/z -8* aoxsu)nh30 $^{\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 b 8oh xwaguhdk1zq+9,4 ecause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen ogxkq 91u, o8hh wzd+a4n Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed k2t/,ncw3t.k e+iwj z at the Moon, 380,000 km from Earth. The beam diverges awtn,zekt i/ 3j.wk2 +ct an 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. Whel02 (fdm6tsxon the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the 0so(dl 6t mfx2angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level wnhj dc6y);4wfloor 3.5 m to another child. If the ball makes 15.0 revolu jch yn6)wd;w4tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions d4-cbr nei*ia(co. c/0fs p1nzczib .4 :sec *qo the wheecqc i.cb1 *oc( ss*frp 0n/.ea:izzi4b- 4necls make?    $rev$

  (a) A grinding wheel 0.35 m in dig zu9-z4 x 6jahkk9h8omkt inwf od:- /7bd5w6ameter rotates at 2500 rpm. Calculate its angul9iz 5 h wof-k6muzh6/ :w87aoj-k9dxtgb d nk4ar 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 i-4vva/ybas( yn 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from tsav(abyy4v /-he 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 orbit k6 8/q v8+mowf7r uhnyxexq-)around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed wi/m+f05k ;gndw5nmz 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 ce8tg(v k,0e ut)5u.nm au3yrlantrifuge rotate if a particle 7.0 cm from the axis of rotation is to exp8uuu, n0(5ga mt t.y3)arevkl erience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fn:87b,hhs /hm tu (tu/knxk6urom 130 rpm to 280 rpm in 4.0 s. Dets/6:7/u n(uubnx8hhm hkt,k termine
(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 radiuss 5.-9g;zdp6;tqf.fx ksbn ha $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put zw+nrj75t-c q7xxme.qh+d 7ithemselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from nox -qm i7n r7d7xj+th+z.e cw5q rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through ho0v(cw-fd8y- + /tckqmehx.y g w many revolutionw tfcm -vyy ./80c+eqh(dkg- xs did it turn in this time?    $rev$

  An automobile engine slows down from 4500 :m xi2-5/ ytlompel/f 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 whirlac8k++gow +xi ing “human centrifuge,” which takes 1.0 min to turn through 20 complet okx+8+ciw ga+e revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpj--h2g v:8:3ztf9-aq zbt djj ab0bn sm to 360 rpm in 6.5 s. How far will a po -a-bvnf2sqt 38z :b tdhgjjz0b-:aj9int on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned of o;gmzyehggr *j7v (a2,,: n5x nul+yyf when it is running at 850rev/min It turns 1500 revolutions be(y+rxyn g7mj5gn,z e2ha:; o v*uyg,lfore 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 ax meg9jseop3*a:n0him8z-)vz:az g4s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of zg09gi a3:v8hmeap:s ejz -x) z*mon40.80 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 eb4sw22 lsck v;8p 2ftcar reduces its speed uniformly from2pf4svls282ek t bwc; 95km/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

  A 55-kg person riding a bike pu7 ijm*ka l5jdcu6z.eg4t 9c,1+pqnb p ts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 duzjt b.mkplgp ae,n cjc*i149+567q 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 cm wide. What is the mszx j k el,27iq)d9zk-agnitude of the torque if the force is k,d)e2 sl7k 9izj-zxqexerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. As(j1j;x eza2 qssume thj2(jze 1qa; sxat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arov4g); 9z1m umty)1on f+ynylm uhg:/g-np p9e attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and directunlp4gy ug-9 9mm yt):oo+mf1h/g;v y 1 )nnpzion of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteningkpwc ,; du.esk6)j8 hz to a torque of 38 ,6 zkdc.8uwk;hpe) s j$m \cdot N$ 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 /rb5tz4 yfgdc,odu82 10.8-kg sphere of radius 0.648 m when the axis of rotationfcytd ,52zogr48ub d/ is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in din .1 ,c -e/grht( xp0xap:obmtameter. The rim and tire have a combined mass of 1.25 kg. The mass of t0e -ab x h1tgp.mp,r/:to ncx(he hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on thzlv4 t)oyll;/ e end of a thin, light rod is rotated in a horizontal circle o;4)yl ltlozv/ f radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angat mm+82htg5 sular speed (Fig. 8–42). The friction sam 5mth28+ tgforce 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 6;: ) d+bqsom v obbbg+q9ld+ter,1w hthe array of point objects shown in Fig. 8–43 aboutbt whd+ +eg;m d soo) 6br+qv9l,:b1qb
(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 oxy4ep*lf :sgnso oun16 +gen 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 satelliter2/avgvuk80gqh1 w l4t cu f.oj58p5s 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 ste 5k0owu./ 2gr1t58q ucvsp fvgah8lj4ady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of;s ig,k8sq u :z1agsx6 0.580 kg. Calculaszq ,;81asg:s6g k iuxte
(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, accelerati/o9wmiw 1p4vy ng 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 onf g4vi;dosj/lsr )k pz(*-5 thwg0a5j 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 c5o5ir* tjs4 -vd fw(l/hgpk )g0;z jashildren (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually broogq.7zpo,l8zt)szf g*vh+ 74 mq(su sught uniformly tooplz*t )(h.s7o ,sfgu84zgqq +7szvm 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 aivt-,yz /4vdrccelerates 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 is8se7 ,;u ;xj igfn7 mgp emajbxj2179a thrown solely by the action of the forearm, which rotap2gnxm;j xea1j;8i7ef9 7bjum7g s , ates about the 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 Fig. 8–4ho 4k 5bg0(hvm6. gb4v(5m0kh oh
(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 consists1oib6y) x m*ar 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 thhk. vw. ox9fs6e hammer from rest within four full turns (revolutions) and releases it at a speed o6h fxv. ko9.wsf 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 mo3/ .j sj.og wjn1jxoe8ment of 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 v9p fevsbuf.(jlf m z37k 17)kdevelops 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 a1srqn/w - zg9yyr.q;i. n7zcand radius 9.0 cm rolls without slipping down a laneasg1yrz qyn./79w ;r-.ni cqz at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with reg0 dknp3e 5nrtpwc.zq/7j)3c.p /or ispect to the Sun as the sum of twn/3 0 kqd) c7.ec.3itgp/5zrpopj wnro 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 s wm4y9pg n-j :fe5ec.jr) (umand a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rafjp:jy(ms. wue- m nrce9)5 g4te of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20hpndv; ; :0mxa.0 cm and mass 1.80 kg starts from rest and rolls without slippinp0: ; ma;nvxhdg down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and :g/kt70 1xyda46 lrrp0 a jax0ftlut3its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Ut 6au p00 xg:47ayrktx aft0j/l 31rldse conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end 7;xc/ fo.lqt * abxd ye,o5vr0of a thin string dba;vr7o, xo0tx c q5yefl*/.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 angular momentum of a 2.8-kg uniform cylindrical grinding wheel w 34xihqd:l*g of radius 18 cm whghl:3xw*4i dqen 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 s (emfnox4wwr(7x27jc ide, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation de 7je7wcf 24 xwnxor((mcreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inert 47lz70h ruqgdia by a factor of about 3lu7hq 7zr4gd 0.5 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 d e ml;eax+;ldva)k2)from an initial rate of 1.0 rev every 2.0 s to a final rate d ;dlk; ame+le x)a2v)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 centevgx.ziuca6.r k/ri )* r at a frequency of 1.5rev/s The wheel can be considered a uniform diu i /.c6krva.g* x)rzisk 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 spinnin:i5gj* :ubjq hg at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the n. b.zwn)b24arh3f wo l7 saa4Earth
(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 disk5g- uj +2dke ;d hi6rg.+(gekkjsr fgdbna)/7 of moment of inertia I is dropped onto an identical disk rotating at angular sprghsekn7 5)d+gb 6+ekd;j-g . / dgfr2a(juik eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 0 kd9ac+1 ,rzvgtfpv ippy7/0 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 q brbfxw*wn-o;z b)54*u hhlc n1w2+k at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 9wn;lwb*krbnbqw+zu )14c-f hx52o*h 20 $kg \cdot m^2$ 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 ish oiqg4/ih+4 h rotating freely with an angular velocity of 0.8 gih/qi4h h+o4 $rad/s$ 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 ( zhobyts6c;eaq;*v4 into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angu zoyt;v4; c b(*q6hsealar 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 win1yaq pihj;x1z8 ipm/-ds 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 evm-svl u)7*11j rftx$ 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 platformbly+bxb7o;9bm)hn do ,;qzj- , initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform i z b9b-;bdyh7nqb )+j mx,;olos $I_P$ 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 standsnedk+yih+( :a at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bear+: ink(day+heings 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 o* a.dxj lxtpm1 y+3mj0n the ground with the end of the rope lying 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.ldj0 y*j+x 3axp t.m1m 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 E 6phrz8 -l c)dzulh 01aqgk:)tarth. 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: zk6q1gh0c)u)t rdl zlpa-8 h as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from restv9-n s3h q;t k0 41,rezafknhq 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 radiuv,iv9y+e j )dus 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcu+dujv 9,evy i)late the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disk *q .b eo6dm *p1dofqqr8fk-x*s, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 pfr *obf*6mq*xed d8q.oq1k - kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed s,bm-k i28dk cof the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 timev*: :.bp+9o px qaadxss as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of9sax *bvpdpq+.:x :ao 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 aut1tmn t7w5q .k2rf zb,urf p:e.omobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 kfrmn, w rkqtt:f 27z.1 5.uebpm 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 ak7fayuc t2-t6 c4)x ujc5gwo;mjuh+: n $H_2O$ 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 ojerrht:ff:audod2r ir+ /(,0n a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and ppaaq8c*q3ic wyz949h(7 peulength L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear 94pzappq u *yw389i7(hc acq eacceleration 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 pap9gxm)cc (k 1oca /,standing vertically on the floor, and we want to exert a horizontal force F at its axle so that cm/),cpga c ox ap9(k1it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wi-fpex :sbj r,vvo)m2 l d;b6)ep.;lkbth a radius The forces acting on the cyclist and cyc.jseme:lr,fp;o) v l-2dbxbk ;bp 6v) le 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 an01g,zx t)rk: 1wcst)g rt*sgmd begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and whexm,1zgrtg)gsctsr:wk *t ) 01re does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms rj az pe:f h w.fw8*83w2+qxegas thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a za: j3 2*wwe8egh.8xqf fw+rp spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical p yodi b.s ue8)v1t,:xgk51ulb lates, of mass )b gu,.o 15y1 l8sub:ivet dkx$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 aiwd8s 239kxbx5 je9aylong 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 reacikd8wey xx953 2abs j9h 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 not asqo5s r:+1um9 l clwbj.sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redu *0vx 2,e5n ;;komnz7trurcyq ces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a)konuxer*7cn;qr2m zv 5;y, t0 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