A bicycle odometer (which measut c +,a:tu6jvlres 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-iljvtcu 6: ,at+nch wheels?

Suppose a disk rotates at constant angular velocity. Doaw* x+2kf*bbr oq)fvfp 0de* 4es a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitudaw+d*xof2kp0) rvb4*bf*e qfe of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the anz1ogw.- :(,m:njgdt, knihfc* bby3ngular velocity $\omega$ Explain.

Can a small force ever exert a greaaa8y/z1-*vl - lqwiqeter torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head ;c ao4mz-gdbm0p9un: than when your arms are stretched out in front of you? A diagram may hgm m0pna;9u-boc4 z: delp you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and threejn-jpi5*j. 5 *cd v.f;nnvkuc q9o;qi at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprockej..5 *inu5p c9n-ink* dqcq;fvo;j jvt or a large front sprocket? Why?

Mammals that depend on being able to run fast have slende;6jzbk3c2ilxvf3 hclt; 2 se3r lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distrizh j 3tkbsc6c3vli 2;;fl2xe3bution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow be cvr*mod6zqjk2e *60 uam?

If the net force on a system is zero, is 8 np;quh:sytw 4sk2v-the net torque also zero? If the net torque on a system is zero, is the net force zero?;4sqh 8-ykwsv2 npu:t

Two inclines have the same height but make different angles with the horizont.zcfv )xu8h km3tqa// al. The s uvhak)zt/fx 8m/qc3. ame 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 rolline*qvg do rtypx.wtj5ax1/-xa*d8 y5 )g (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 thtge*8*1ydqd/ vx.xwjay ao 5 x-rp5t)ere? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same masset wzo-x*gt2ww c43y* . 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 kinetic energy at the bottom? Which has the greater rotationaxw3yote* g2-ww*t c4 zl KE?

We claim that momentum and angular momentum are conserved. Yet most moving ora-) m1qmhxs 2k rotating objects eventu) kh2mmaxsq1 -ally slow down and stop. Explain.

If there were a great migrati4 mmvh8bh /)etomg9xcy- ..dvon of people toward the Earth’s equator, how would this affect the lhh4)m -y8tv mxm.d. obvge9/cength of the day?

Can the diver of Fig. 8–29 do a somersault withouxkv+eyqd6;jt4bt xcuk lj8-i4+ 7. svt having any initial rotation whe k+7je4qtsjvtkydlci vx6b+. x84u -;n she leaves the board?

The moment of inertia of a rotatv) p4lmd)n3 x+hc ye:p/hzxb c/,y) mning solid disk about an axis through its center of mass ic :,h b+)/c 3pmxnezl/yp)v4)ymd nhxs $\frac{1}{2}WR^2$ (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 ea074 9-jgmfl cztph3pw*cdl /ach outstretched hand. If you suddenly drop the masses, will your angularmp t zj/h g3l90afpl*dc w-c47 velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hol g-f*)vp huiny-5sd;tlow and the other ist-i5 un *pv;y)g- shdf solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points weqgyh;- 0o1bbm: siu)b st. In what direction is the linear velocity of a point on the top of the wheel? If the angular accel 1g) moq0iby:ubbh ;-seration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Whatu 5vh2-q, y uyzut*. jo;l;(6phr clvb happens if you walk toward the centeruvhp6zvh yo(r-;qj ,u 2lybtu;5c* l.?

A shortstop may leap into the air61ya)6 anq /ewq xssr/ to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–3666a 1 e sx/naysqrq/w)). Explain.

On the basis of the law of conservation of angular momentum, discujxbutpq 285 8bss why a helicopter must have more than one rotor (or propeller). Discussjp2u8x8 q5tbb one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (avy :vcwi wd5j0s81/uln- o/x9brsu a1) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. 0qp2er-t jc93 o xs/kxgqe(l1Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as se qcgq 1e l9xkjoxrt2e/0s-p(3en on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges aj2mhp8q kbcz z8i96v)t an p86mzj9zc bk2h8)vqi 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 blend ft72gd;kmj(z pk19mt er rotate at a rate of 6500 rpm. When 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 dpkm(9gm zk;1jf2t t7down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the b1m77hf d 4+xqy bqziy0all makes 15.0 revoqxfh4 77+dbzymy10 i qlutions, what is its diameter?    m

  A bicycle with tiresheyeq ft/ ce:dg 3160xh*1u9(o gqqoc 68 cm in diameter travels 8.0 km. How many revolutions do the wheels mac 1qg ( 9uoe/0qo:eqfydhe hgc 6*tx31ke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itsg0xwue 51r04/z qscva4i r(gs angular vcgq4e vix4aswr 1r/0u0g(5zselocity 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-ro i29md: t6iu)kq o;ovvund makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is 9vdkuvoi m )2 6ot;q:ithe 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 of the Earth (a) in its or8 6ql nsx 85kp/k3szjgj hy)i8bit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed:n3w,c vjllt3p mygperq 4/m25j h8e + 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 from ,9z a ptykjv2bg/ -3htthe axis of rotation is to e3t,vzapyght /b-kj92 xperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abeuzyh59a+ sa r9vztdt h)4ny ),m*ub3out its center from 130 rpm to 280 rpm in 4.0 s. Deteur5abthamduv+h zz3tn,y9y4 )9* e)s rmine
(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 4h 4ln5ua opd+g/ov(q$R_1$ 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 Moohwx gwx3n+uhu ,uvph,m s49*d9x h98j n, 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 accelvsxhn,x49uj ,muhgh *pd99 +wwhux 83erated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throuq 3no:.ozwvv -cl/zr /qn7x3 igh how many revolutions did it tqo/ q3r .c :v 3x-l7/niovwznzurn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rmq w x:hu9/b+*w6x*cqmet ef2pm 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 high :ba)mzkgmd9 /speed jets in a whirling “human centrifuge,” which takes 1.0 min to b/ kzdgm9am:)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 accelerates uniformly from 240 rpm to1/w) m-xxsw kk6hkhr :wq5z s3 360 rpm in 6.5 s. How far k3k:wx 51h /zwrss6hmkw xq)-will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 oa z7wf57ro n4revolutions before i5rfz7 oonwa74t comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car r / a1qr(hd)ilkeduces its speed uniformly from 95km/h to 45km/h The tiqa) /hk1l rid(res 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 reduces its speed uniformly fr4:+: a nroxsooinqbe-(j2k (tom 95km/h to 45km/h The tires have r2o:n (-jabno xkt4:e(qis +oa diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on ee95+omzr9g r.8 hck nmach pedal when climbing a hill. Thee.9c n+ro9hzm 5 8grkm pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on thee(3 mu:uozf) v end of a door 74 cm wide. What is the magnitude of the torque if the force is fo(em) uu3zv:exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown iyk (w4cljha 2k9 f)yd71lapy 7n Fig. 8–39. Assume that a fjah(pk9k)ca47yylyd1 l7w2 friction torque 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 mas.5 yt yf9e k:(brgi.eml p9w ,*(yn*ifckeuo4 sless rod which pivots as s 9ulk5r:y f p9 ee.g4yib mnko(.(*wetif*cy,hown 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 pu;sfx k j1l.4udbgj-ax;2-:hfx (tj a torque of 38 xs1pdfg(tx-;fax.;2: hj 4l jk juub- $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radiu/rj4;jbb( nmm 3z+yots 0.648 m when the axis jjn3rb(+ z/;oymb4mtof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameteriglf,j) hj2fd7(l8rzq u x* o;. The rim and tire have a combined mass of 1.25 r)xiuq7 zjo 82jfl;*h ld( ,gfkg. 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 i ip1kvjb f(1)/dw-ezl s rotated in a horizontal circle of radius 1.2 m. Calculat/ke jd )1( bpw-flviz1e
(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 hb :;9mdgir- qconstant angular speed (Fig. 8–42). The fric mihq g;:-r9dbtion 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 in *.z )dmatbnos uribhr5,*/k3ertia of the array of point objects shown in Fig. 8–43 abouz5 m/oh3ktr*,d una s r*i)bb.t
(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 consisj) o;hmr6m wa+ts 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 cylindricalmu;j2xnfr jz ;2wix3-r i0 j2y satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0f2yr 2jwz2xn ju-i30 ijx;;mr min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a massv bkh*n.bn-9* r(g04pnxw m px 78jhgt of 0.580 kg.pg*0wn*p4 r7x ( hhtbmk .jngx vn9b-8 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, a qzgx0:,d :dfcccelerating 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 hano yra )1*nkih.xd67af d-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-roa d *korxai7 .y1)h6nfund 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 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 of;d 98ul ;9/yib5lw/da-d5rjvtzc ax vf and is eventually brought uniformly to r8b 5 -l dj9dtv az 5/aydv;x/9wlir;ucest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg bci5 i3pxbpzd5d, za2o6m5hm7all 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 t5wnruk(t :h1phrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45pukwn:1( th5r . 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 rodf5+szmuvq* 1-arz :u0 fbcit /, as shown in Fig. 8–46. -c ff 5/0asu*+vb1:mz qtruzi
(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 masseszw)n91pul1 zr 3 uhha ,r;r-gm, $m_1$ 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 fourd, a+xb*uti0 n,6v nl.zkb0ar full turns (revolutions) and r +t,6ubn.dl0k v *,zbaanxr0ieleases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia of; :z:+4+wh xcxd nojdae)e/bh $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 28qgg v3;uei/4s6 5t:gg sbthe) 0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls wibxa104myl vob 9 j1f7:mcczg ,thout slipping down a lane at 3.3 jyb x9bf cv104a7,z:mgml 1oc$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic15.wvi xvh 0 tq7ygx7i energy of the Earth with respect to the Sun as the sum of t ig.tv507xq 7 y1vhiwxwo 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 yzakeu46 75 ick(ar00 6dtynpradius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 re ay iya6nrd57et(kc4p0ku 6z 0volution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1t1hdpo;u sc ( dw:z6y8.80 kg starts from rest and rolls without slipping :8wy16( od ct;zhupdsdown 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 starkyh;w-sp : 9vfts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint;hk:fp 9wysv -: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg bali, xgx.qh p*.ql8t i7xl rotating on the end of a thin string in a circle of radius 1.*q8x h.xxp 7 .i,qilgt10 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 unig-n nwfmf 3zf:k53 j5mform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? fkwm3nfm:5g5z3- fn j   $kg \cdot m^2$
(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 isthrked 5;9 .cza3j hap ;)i5mj rotating at a rate of 1.3rev/s Imh9t5;a3ijp .aj r 5dzck;e )hf he raises his arms 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 in Fig. 8–(ju7)wh1lyar41kf h r 29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight positionlfj ru1w471k ry()hah 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 ;lsyevd 0j.56 vtns4lspin rotation rate from an initial rate of 1.0 rev ev .6syt4l;vj0v es5dnlery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around ao w;w2d hayj6 r0b;pe; vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk o;ajwp wd 62rehy0b; ;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 atve/)mqm5e xx5/-np/by peoi, 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 momentu0 x5j 5w nu6rpq2e:r(e zl jsjf4v(p5qm of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

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

A nonrotating cylindrical disk of moment of inizza/fgada,;d es-j9(+8iso ertia I is dropped onto an identical disk rotating at angular sz-9ia ez j;,o/is(f+aa8ddg s peed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ay;fhpok5i, g+ p6bz r(snz9/it 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 ,h :risg4ps s(fi rg,5r6aa- , /nucomrotating merry-go-round platform of6nm, gr( iui,o -crsf prgas:hsa5/,4 radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-govea2vx 2pf+p2-round is rotating freely with an angular velocity of 0x2p2e2vfpv+ a.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 collapse2+dq zy,o(rrm0 4l ljns 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 angular momentulr0dl jn4y or, m(2+qzm, 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 8a9eqoi-u6sq ta c-/vwinds 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 e9uqdnfp82 z(ax cf30rh41 i d$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely witho3hai1 /3v+zy v d8rr5el) ane7y-rpki ut friction. The moment of inertia of the person plus the platfor/5 an+1l3i3ery-rzpdyihkr) a7vve8 m 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 th/lxns+nc brk i 5mx83/e edge of a 6.5-m diameter merry-go-round turntable that is mrxs8kxni 3ln+m/5 b/c ounted 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 withfp2x jej:irj/ 2( weak9;3mhd the end of the rope lying on the top edge of the spool. A9m j2rxe jwije;2p3dfh(k/ 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 Earth.v ma4-jyx ;v+b Determine the ratio of the Moon’s spin angular ymvv-4j+ab ;x momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of ,ctqfu8wx 2 zx ln:41g1 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 uniformtt( gm/bw/-va 6yb9x;hga. xw cylinder of radius 0.20 m acquires a rotational rate of x t/ -ma6vtb9/ x wb.w;aggh(y from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and jumtmxt/m+3 6ovd yj81 .wqse -b3fp6diameter 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 when it reaches the end of its 1.0-m-long string, if it is released from r buvpm63qx/ .j1yjemw- ts tdo m+38f6est.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the 7c j+2fn(vtz)b/dm fyangular 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 size of our Sun, but with mass 8m9lpr 7*q1rnoz qr+f8nq.bb8.0 times 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 of its mass in the process, wqfql *+bpq rz8n9o1.7r brnm 8hat 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 a 4us+jq6 hckm0o50 fbeutomobile 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 b k006+h ofjq5 4muecskm 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 illustra+j,:lxdo d 3cetes an $H_2O$ 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 rq otaao 3a1s6s*58a;u b(uytg olling on 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 length L can pivot freely (i.e., 5 mb /9deuwww20h;czywe 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 acceleratiod/5w2czu wwmeyh;0 b 9n 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+oyudcdjbl8y :s;wi)d*b cmm 3) 6bb+, and we want to exert a horizontal force F at its axle so that it will clim3ybj6mi cdmob+ sc lu db;)y)+dwb:8*b 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 with +znr ;ji 2 +64.au*eoyrsxmd ma radius The forces acting on + uax n4 rory*m2zi+mj;es6 d.the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling 6 *ioyux-v r 4d35bgtoof length 1.5 m and begins whirling the rock in a nearly hoodri3txo 5-y4bgv 6 u*rizontal 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 ar5v7t+ un0po -ahj6tq zms as thin rods, making reasonable esho5 0t p7+tu6z-v naqjtimates for the dimensions. 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 consi y v2taphrbo9,j;req(v9:im1sts of two cylindrical plates, of ma(:e; qv9bvi r9ao,rmht pj12y ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of 3a cbf (yl(8mbmee5.j Fig. 8–58. What is the minimum value of the vertibc 5e83(f(myjl . baemcal height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as1m 0 l)-vwkbkasume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its sp 2nkf- l8k*vtueed uniformly from 90km/h to 60km/h The tir k 2nkl8vfut-*es 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