A bicycle odometer (which measures distance traveled) is attached (8 g xk2 lra0u99yegeinear the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-i8 ie(2k9gyaru09xg lench wheels?

Suppose a disk rotates at constant angular velocity. Doe.u)1 5b jx: s9vqvyxt 7 ,msh) vo3yi6zzb4euvs a point on the rim have radialm.yu:)uz v6 t41yx 3e,s 97sq v5 )hxbvjizobv 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 7 s7 zu2 -t1fihb- nnr-ikjfb+8zda6ksingle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a6 1yg 3fna5.j 1:ed i8mbroruy greater 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 twzv8a rtv:(9jo do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help yozrj( tv9w a8v:u to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pwof. d-ly 9iz1edal cranks. In which gear is it harder to pedal, a small rear sprocket or a .oyli19dz f-wlarge 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 abl3 ( z:c/i(;arz:bs xc o9zwyp8uvgh j;e to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageorg 9ivj c;z s3bx(:z uohw yp;ca8(:/zus.

Why do tightrope walkers (Fig. 8–35) carry a long, narro-.ylm n;9yr. o6y.abpw mfw;w w beam?

If the net force on a system is zero, is the net torqu i.wc)e8)04ktzns a ype also zero? If the net torque on a syksp. )nwz8a 04tci)eystem is zero, is the net force zero?

Two inclines have the same height but make different angles with the horihevl p/4q037r x*esoizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bo/erxqlv3h 47s 0ei*pottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. O isn*az.+90 ,xcy-ujjv4oc opne sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline fir04+*9 z vyoc jjpa,xnoc.i-sust? 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 tlc0ozc15.6b z ky *it3+s trtphe 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 kinetic energy at the bottom? Which has t 1t6ol b0skz5c.3i *trc+ypzthe greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most cqg1/lux7:9up q+p u emoving or rotating objects eventually slow down and stop. Excl p+7u:eqg9u1puxq/ plain.

If there were a great migration offorv s(/kez; m lkv1-8cwl 36s people toward the Earth’s equator, how would this affect t 1zm re6lk (wk83;svvco-l/fs he length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotat/qb46 kh+ pbqm k x-j+ahk*-btion when she leaves the bh+ akj+6 h-tq *pmkbq-kb4/b xoard?

The moment of inertia of a0:rjr/z. /i-aoec aja rotating solid disk about an axis through its center of mass is a.io0ec/- r/jj:aazr $\frac{1}{2}WR^2$ (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 hold-jp h uw0v/*mding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase/ 0vjuwph md*-, decrease, or stay the same? Explain.

Two spheres look identical anduuqx4 k)9cl4 x have the same mass. However, one is hollow and the other is solid. Describe an experiment tolq4kuc x 4xu9) determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poin z wcps4/f 4z4zc.0e;6ykvifum0mjj 1ts 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 thevc 0zkjcz .f;/zywus me4064 jm4 fp1i top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turn oz++b9 /bf/etgk3 beotable. What happens if yoeobozf 3+ //kbteg +9bu walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quic+nr a/q, kuba (*w;cjp2gfyb3kly. 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 directin +j(a bbprqu*a; gywkc,3f2/ on (Fig. 8–36). Explain.

On the basis of the law of conservation of angular m3 ec n dn0aa5opv8(bx-:c.)hcf skczgo9) s4 momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or mored8kops.h5czx 0ca))9n(o4vn 3mb c fgc se:-a ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30j-p6j a h p.bi42cnb 4qjc6zo1 $^{\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. Calculate, using th-)udjs75n3f7 )k notnmogr+oe information inside the Front 35rog-njukons))od m+nf 7 t7 Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380q hu:h g9 ne,4oodkj,itbi88;*k yh(l ,000 km from Earth. The beam diverges at an l8jkng 9hb4h :*htekii o;8,dqo y,u( 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. When pcgs +.v o 6vte 4ju0sm:i5-ldthe motor is turned off during operation, the blades slow to rest inv4ts0-d.+jgop m se:cv6 5ilu 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 an-bjicjy1twj x3 t, ::bother child. If the ball makes 15.0 revolutcj:jx,bt -w 1:j3b tyiions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do o0d46s4,: qn(l x f4ftvkm hpotxf,f :tv0lopds464o k 4mqn( hhe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. C i l-rac:(q2i3z gbcoh6v8.bo0 ep .ltmtq4b*alculate its angular velocityori.hq :i8btec3bmal og.0 2v6 -lt4*b(zcp q 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 oht4f ;fu qh8ko-oe 4i5ne complete revolution in 4.0 s (Fig. 8–38). (a) Wh;u4 4tfhk io- q58efohat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocd.i)xc d*xko 1ity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofru3i, i6f 1 9nwje/uye 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 centridyb zy4*r/fqq)s+,twfuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration o4 ,q yt/rsq bfd*+wyz)f 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cente)v0. .6rgkidf*xoqbz r from 130 rpm to 28g6d0rzvbq fi ok.. )x*0 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius,qyrfni(jye s; uy 1.4 $R_1$ 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 aboen55q8r:wo bn w jth5*ard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, t rqn 5eoj:ht8nbw55*whey accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with 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 u sv/2lsz23): rkoi udgniformly from rest to 15,000 rpm in 220 s. Through how many ulir2vo ):/zgs2d3ks revolutions did it turn in this time?    $rev$

  An automobile engine sl2/y72 yunktem vf:l3 rows 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 stresses of flying high zqhx1/y ds21aspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rea2ahqys1x 1/ zdching 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 3 :bl4bychmx-+ gsp* aaccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far wi l*4cs -a xb3y+pm:hbgll 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/.xdjs 1vu,f t/min It turns 1500 revolutions before it comstv1u /d.,f xjes 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 ca;xef * t;oswg8r reduces its speed uniformly from 95km/h to 45km/h The tires have8;s; o g*xfwte 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 f ,ohbv tdfdn82ve 1w1+gl5o34fpdg+ from 95km/h to 45 4fd wvd813lb2f podvfhot,+ +ngg15e km/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 av)n,4-bhacwbf* u;r p bike puts all her weight on each pedal when climbing a hill. The pedals rotnfrbb huap,v )*4w ;c-ate 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 od g,ijefjyr67 2h qe5;:wbre3n the end of a door 74 cm wide. What is the magnitude o q5 ,fw3b7 ;yrgeie:j 2ejd6rhf the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle ohj hzs(lfgqz80t 3,9b f the wheel shown in Fig. 8–39. Assume th(q3h ,tgj08sfhlzb9zat 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, are attachergv a5, aks7(ptl 0ye2 9usjl-d 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 ans07 yekjs g25v a r9pa-u(llt,d direction of the net torque on this system.

  The bolts on the cyli8:c3mux l yzgp4kk8c qgmv *3,nder head of an engine require tightening to a torque of 38 v x:3kq8 c,lm u3yg48kcg*mp z$m \cdot N$ 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 b ,w:fzjzya+lkn oes8 /)l)5aof radius 0.648 m when the axis of rotation is throua)lza zsw5l:) o, 8/ek jf+byngh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm ip1r)v0/cr2 gly;0 bz m vdt8ndn diameter. The rim and tire have a combined mass of 1.25 kg. The mass 0vmz2td1dp;8v/y)g 0nrb r lcof the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a ucc e4;g ,kv 2e4(cqrvgy0rt+ thin, light rod is rotated in a horizontal cr c er(;evtuyqvc04g42k+g ,circle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping az. biwu:.q+n g/*ej yi7a) i8x wxxdb8 bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N w.xiyw8 *7)njg:/ ibiba x+xdezuq .8total.
(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 objects shown in F itiy;spcccun9.r:/ 4 t qjef3e5qu;i:m9a-dig. 8–43 ab9ir.5 :/tds:c3q9 iujupc;ma; 4i f-yete cnqout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass isc d.:k ley1w2n $5.3 \times10^{ -26}$ 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 spinningtra y5k0 ;-7u5yo jt;iws79suomovs / 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 /v5-s;ik5t7 ot7j; s9m sryy0o wauou 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 cqr+a k5 n-y5kzr:*c bdg h4e5tm and a mass of 0.5r:bh4z tr*qd5n+5-c y5 ka keg80 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accvtrg eit8v0t.kmy9 u+s,( n/celerating 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 onsqo5:z n-h;. ksrs8fenzt0tcxdd6 + 4 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 ndxz0 .6kh:+tdo t4z sr8 f;nsc-qs5ea 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)ep7gnb5g -pnhk9a o3 off and is eventually brought unb n7h) gk-aegn395pop iformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. dc.(x d*uia 4k8–45 accelerates 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 a5 zq3.u mgrk2dction of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fqukg5mzd3 r. 2ig. 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 Figyyyhloat1 40xhdh7j ;k* .e ,q. 8–46.k1t0xy* jyhe7 h4ahl.qo, dy ;
(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 masseb k l*j./xji4xs, $m_1$ 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 u7 wsk;vf*ir3 from rest within four full turns (revolutions) and releases it at a r*vwk3u ;sf7 ispeed 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 o12 +sossmy-yp f $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine dev+6 k dt0zvkplqhpcs9m*2 rnx3(zg6 i*elops 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 radius 9.0 cm (:o a,pqljpf2uq3m) *cuj.ke rolls without slipping down a la2lpkp ,am.)qceou u3f(jq :* jne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as ;f ;z.; ugyi2rq nu(zbthe sum of two t;; q2zb(.zuiru g;nyferms,
(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 164br/nvt kalwg 4q 2rfb*95 6*6fj kpo,f0 kg and a radius of 7.50 m. How much net work is require6ptor fgqv5rl/bk n*9a 6 fw4, 2f*kbjd to accelerate it from 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 wi u* beu oibgp9.9*a8ef -el5euthout slipping down a 959 eubo-fe 8l*pu*.aubeig e 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 ik:2lhv nm w/*deh0 ry4m 3vo:05kqll xts lower end does not slip. What will be the speed of the upper end of the pole just befy 2d5v:*hllx0k/ mq 03olnv4m :kewhrore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentnh(zq7/6kp k w2unw4dum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 2 67(qwknw4h pnu/zkd10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8 wh9jon1mg3w +-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? 3j9n gowhw+m1   $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hca6, tibz9u jtrxa/5 7is side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–4 6zt /ju7i,t ba9c5rax8, 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. 8–29) can reduce her m+)vghw 5y8g3dojbl x2oment of inertia by a factor of about 3.5 when changing from the w2xjg 58 vy)h+gl3obdstraight 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 incre tffzjgnqps06g qpn/ :7rh-n8yf ) 0i7ase her spin rotation rate from an initial rate of 1.0 rev every 2.:s6fg )i/f07yfjr-p0n8 qp h zn7qtg n0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical sop z42 h9d6x g9fkv6suj9q(t 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 of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rota xz6qfpt 2s9hvk6s u9j g9(o4dting wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angu ys)avty7 dsgo8 +rf;lvxk7 20lar momentum of a figure skater spinning 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 92ima:s l;3mjc 2lbf isi.elw)ru:a 8of 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 inertia p69rod rwald4mi10g g2a-c6 s3 uo*pnI is dropped onto an identical diskndo4m6w -p9g dar2 61 0arsgop*l3iuc rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at dnbl25r s )s;)2g h2rnln*ucn2.4 $rev/s$ 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 4t zm*;kzsbx9rotating merry-go-round platform of radius 3.0 4z9 tk mb;zs*xm and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular v+tdhp kq2,y n* 8taie*w6.xzqelocity of 0.8k pa6iw+qx*tn yhdt.* e2z8 q, $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing q- fcvqu)cy9)sej - e5hv iny,;a14mos7*yaaabout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass car-mj;1nha v) acyoc- 94qyeqe ay*) ,vi57su fsries 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 involve winds in exg kpv d-5+tx1r+ght q-cess 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 if ohs g03x.w2v 3psu,$ 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 freo a) cu+ qte,,;pe*c6a0rs ki1 ma+qheely without61cahs)q em;e+ * otpcar0ukaq+,ei , friction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of ahr6 du1wv3h kxa*1wc(xx;o ucw04 x+y 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inercx vwd(xxychh; u kw+a1rxw46 30u*1otia 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 ropbq/4 vqhj33pte 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 b v3htp4qj/q 3person 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 (h8c*7mvqsgh)2 y5e +u n,ultp l2fjpf*iwp )faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In thecpf + fhwm npi 5u)(qstp2lh8,ul*v2y7g)* ej latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fro qhy5rd70zp9 hm rest 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 i3jen ;s1)wlnyn the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 3wn;n y1)ljse6.0-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 cylind;e: 1 xtzs)vqygd);emrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub od:m)e1evq ) ts;z ygx;f 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 rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel mz +ex +ye7d,8wv 8ctb($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rot7w+)jyygte w8 ating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of yw8e)yjw tg+7radius 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 t v(0wu6ryz*gf9a;nado 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 travel0zna *wu(r;g6y9 vdaf 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 oczsqx4mz9h.dt h4* n(k a-2n $H_2O$ 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 su x0fl5 ufjba80rface 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.,wy9m d v2x n:kvn.xt*: 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 acceleration of the tip of the rod. Assume that the force of xv9nkd 2*:v.:ymw nt xgravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiuswwdftg60tr 4; R. It is standing vertically on the floor, and we want to exert a horizontal force F at itgdtww0r6t 4f ;s axle so that it 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 m,pil j -rzfx*-;r bbkn4dt4 -xaking a turn with a lr*fidtb4 brjx zk;4-n--px,radius The forces acting on 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 of length 1.5 m and begij6m w.5zhzx,zns whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 1 . wz,j5xhzzm620 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 armrx rcyjltue 790 4vv+:s as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, andverjr49tlx0 uvc7 :+y with arms held tightly against her body.   

  You are designing a clutch assembly w(ma rd*6icf+bv d 2h7khich consists of two cylindrical plates, of d( rd2c*k6bm fi+a 7hvmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rad azsttm-.q r qw1jbr,dr9 8-:mius r rolls 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 witho,jbaq-mt .twz9msq 81rd:r-r ut leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume fz8w)mq w:.hk$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as tw:vw5bu ,5bg the car reduces its speed uniformly from 90kmbbu: 55w,t wgv/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s