A bicycle odometer (tvqf vge,h x8..l)i8 zwhich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. Wha.e)ilh.gv v ,8 8xtqfzt happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates hc8 ls i )abq11u5bfh-at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the1aql)ucfb5shh-b1i 8 point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single +s4dsy mby2k/ +9d9gtlm,lsx value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explai/l3x okx b5cj(n.

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

Why is it more difficult to do a sit-up with youwhvj gzb9/m-08u v- eur hands behind your head than when your arms are uw8jv- uz 9hb0em-/gv stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wau(vm +p8.du8 z,2uur0gbh *d8u mi,m+ hchwrheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a larg.udp08rzhu+8,hm u*r2g(dahviw8b+ c,mu u m e front sprocket? Why?

Mammals that depend on being able k-0)g 8qq opg,uz-uaoto run fast have slender lower legs with flesh and muscle concentrated high, co-z, gpagququ)ko- 08 lose to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narroig *puq39wqc at1rf a7z8)ww5w beam?

If the net force on a system is zer uf*mfv6fz9wpnak,v / y*a:m7 v3b*), htw gilo, is the net torque also zero? If the net torque on a system,9a,vv7)*z w 6n3tu aym*/fkwh*m fvgbilp:f is zero, is the net force zero?

Two inclines have the same hei*0.sojziec.g7f1w+pk v zo:b ght but make different 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 be greater0:kfsbg+p oij1.z zw.c7*veo? Explain.

Two solid spheres simultaneously start rolling (from rest)kj51:v i;ff xyf .kid6 down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom ok:;1 f5. 6xkdyfvijfif the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start frzn 73vka8brb3om rest at the top of an incline. Which reaches the bottom first? Whicb8 ba 7kvz3rn3h has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved.tqy)j5a,bazs j):fw)y0 dh6y Yet most moving or rotating objects eventually slow down and stop.yjf5qt:)6 ywyahd 0,)a )zsjb Explain.

If there were a great migration of people toward j)mz,5n8 ofqgthe Earth’s equator, how would this affect the length of the day?z8 o5n,q gjfm)

Can the diver of Fig.m9 a e85q9,uhioqema r;r ./d ,jhjsp2 8–29 do a somersault without having any initial rotation when she learujj,sedhq8eaq2 m/ 9 ah,5 m;i.9o prves the board?

The moment of inertia of a rotating solid disk about an axis through its6 yga(br(i2 //p dxp/ mhzl2jc center of mass is d xi(/lg (rb6my/z pjchp22a/$\frac{1}{2}WR^2$ (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-k/ un;rs)amh3j(w. vfql0n r 8rg mass in each outstretched hand. If you suddenly drop the masses, wil 3v nwr;lrjrf08 qn.s /m(a)hul your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and hp 39as q/ ;bygm) n1q)yxrbp5wave the same mass. However, one is hollow and the other is solid. Describe an experiment to)gs;pr3wm bx5qy91p )qb n/ay determine which is which.

The angular velocity of a wheef* 8;vdng -trfl 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 linng* ;tfv8d r-fear 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 rotatig2dtqgw)by.; 7 ad8 s;k)xxkm ng turntable. What happens if you walk toward the center?g.)gk x2wkx7b) sq8 ;;tymadd

A shortstop may leap into the air.pmq6w8la)/a l0sppes h* wa: g0ho x, 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 ro8,a wwqss*gphe60 olmx) lha.: pp/0a tate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law m zdgcm-2 vurt.i7rd/ea00x,gasz q/hnl 308 of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propellrdt 20r-namuq.e8d/az 0s gh0cg,i 3zm7 v x/ler can operate to keep the helicopter stable.

  Express the following angles in radxcas..814 j gngti7 kqians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of x-t8srdep sp 776wbe-an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moesbrpew--7pd s8 x6t7on, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beamkbp1vvz ,qn.6 diverges at an anglqznk,v.b1v p6 e $\theta$ (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 6zpsb/ dy*.ktm33auz 1500 rpm. When the motor is turned off during operay mk/zsu1t.3d3bpaz *tion, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a lj:q0k+x 3 dtwqevel floor 3.5 m to another child. If the ball makes qkqdt3w0x:+ j 15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many9zx5bri,/ t q/ztwek,so jt-* revolutions do the wheelsx i,9j- rtq,b/ttswz/ 5z*eo k make?    $rev$

  (a) A grinding wheel 0.35 m in diametex )u9; btsf4skr rotates at 2500 rpm. Calculate its angular velocity inu)ss4;xkb 9 tf $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–3ygidjvp(i 0lk* v9r yk +9*s pfea0s9+8). (a) What is the linear speed of a child seated 1. +sp0iy9 lds9k0*vjy g+f(kper *i9a v2 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 veloc+ hsxe;-- pw:gtwf gp*ity 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 of a p* 5i-s**ulg,i zqsgkioint
(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 centrifuga2g6q+ rx2pn fe rotate if a particle 7.0 cm from the axis of rotatiq+ax22pfg6 r non is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformkzt16 f /h .gh)h.qmrvly about its center from 130 rpm to 28gr .vkm)hthq.z/h f610 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 radiusm2vv/ q.ybbf t2vnj2-:od1b /z i tf+o $R_1$ 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 aboardau:c m l.u;q*rpnera+ilt8/e -5mbe* the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft5+*p luc tabe:q/ r 8-aeml;.i r*nume 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 ) igfb,,xo+rc *pve .bvv82hwrpm in 220 s. Through how many revolutio+.vf2 gbrbowi8v,pvx* )ec,hns did it turn in this time?    $rev$

  An automobile engine slows d++n /w 71djpwd5)vssemxz-q rown 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 f4b r,4 bctodg+ddx,05kup9 j,/ vmp7pj4 mxhklying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through huo4j +4g / ddk,pmkxc,j9d0r5tbb pm74,pxv20 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 fvs 4,,pgyzh 0 hnpifc bb073c*rom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of thc,bg0 sn4p,i3 7b hzfphy* 0cve wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 kst ivu-4/ fuhd i3/m5revolutions befo tu/f d3sm -uk5vi/hi4re 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 as ths5tfmvf..p)2zm :l1i l-4ws0v3 i 4ncixlgrue car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter oflplfvv5s2 1mwti0).-c 44zg x :rluim.sfin 3 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 may( 66clpvc-28kwamgs ke 65 revolutions as the car reduces its speed uniformly from 95kw yps(2cg8-kv6a 6lcmm/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 p ow w(r*lmn4f amq+,5ay.:a pjuts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius j5m m o(wa+ywr.pla:f4,naq *17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is thgw-x,bwu*j4ny+tlr f9jf: ej4 t9jc0e magnitude of +l :90y,uct j-fre9jnt44f j g*wjwbx 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 aboq(um0hwkr g gc-5/z, yut the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4 yg(0kc m/hr,w ug5qz -$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached dmq0 g evp;c7q62v:tz1,skyp to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal 2gmqvk;ec s: ,vdt 7p0yq6p1zposition 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 tig+z utruf1sy+bgusjaumg.* u 5/x7 (a+htening to a torque of 38+ b uuuzsax+ry(5j gt u7am1+/g u*.fs $m \cdot N$ 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 oo )ji)9 vpt:wmf inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cetw:iv po )m9j)nter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7ijyarg05 o-0.p ccwd/ cm in diameter. The rim cc5r 0y0o/wd. gj-i paand tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horilgid4- t7ccb qfz*8g8 zontal cir 7if z-ccg*8dbt 48lgqcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a pottero1umx.ccpuk m f(gy.8s/rl mi/e; n 4,’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hagc k fpm8u/muli ;1.4e/,y rxcn.(smonds and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the a9t+hlkmcr/ *z.j c(g orray of point objects shown in Fig. 8–43 aboc* /ogtck .hl(+ zmrj9ut
(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 oxygensxn7c/g/ yjkg93 n 4md atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the e *lt yv1m.e)floifk7zf(f) 4.n agv3correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius o iftmfan*) v4e3vl ..z(7of y) egf1klf 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylindeekdq 7y1 34kp. y1tl3, vlrwwnr with a radius of 8.50 cm and a mass of 0.58v .7lr ,t43 kkl1 nywe3y1pqwd0 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:) br yj4nvbq.xbffm*+ eha)z9j4+ yf bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-rouhhj .t :t,1n:wt)rf; sdg svv/nd and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the a v:nd;wt1hsh:/ tg)st. vfr,jcceleration, 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 oq v9; jlnj:ipm8t+ 1deff and is eventually brought uniformly to rest by a f9+j 1n mt i;dqpjl:ve8rictional 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 k3:7qmdw cy 0 5qwlfvfe s94o7a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which ;r y9 3*i.m3ofhnzj( psf*jecrotatesyh*ce3o;n .jf3*zj r9 fp si(m 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 vc3v3kjfxm6 s/j( a :rbe considered a long thin rod, as shown in Fig. 8–4(ra 3/jscvv6j mkf3:x6.
(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 ofh h.,zpf4e1)myh 1kb b8 dyot77cx:fc 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 acceles4cwl7 sev46g rates the hammer from rest within four full turns (revolutions) a 4s6cg4s7wv elnd releases 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 inertity7n: :jwn 0bzl9u5hma 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 dev/nunzy1 9b zr 3em.wl)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 radjwzhqh7 6p/5vius 9.0 cm rolls without slipping down a lane at 3zvj7qhh 5/w6p.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eab/u 5 vhhb4xq2; i2zhtrth with respect to the Sun as the sum of two tehh2vtb 4xih;52 ubz q/rms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius ofdnay:)b ldl,5uj ez-;uam 3u/.rule 2 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.2: ,l3 ed/bejaal)nu udyz uumrl;. -500 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 6 b9j.knoy9ul3topo 6without slipping jyo963l oupkb 96.nto 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 its hcvq7n w42+xh lower end does not slip. What will be the speed of the upper end of the pole just before it hits the grov7qhcx4w +h2nund? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rf uw.n73rw o.botating on the end of a thin string in a circle of radiufnboru 7.ww3. s 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angula.uds ocx3dg4 w.8d1omr momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when r 3.dx4gcws81uo ddm.ootating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate cz9p7hq05te(e giph8 of 1.3rev/s If he raisesp g0 p7zqie5hh(e 9ct8 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. 8–29) can re dja6f 6,fikz5duce her moment of inertia by a factorj6 zf,kfa5i6d of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can incrhky ryszv0m5we/z; 3 6ease her spin rotation rate from an initial rate of 1.0 rev everyz;z3mykhsy 06 werv/5 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 a vertical axis through its center at a freqsh6g/n /+v :16abwug 4wrvlrh uency of 1.5rev/s The wheel can be considered a uniform 6 1/6r +whs:w4lg avrvhnug /bdisk 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 spinningraph:hbzf* fv 86ya.e )17/y tz mgjc- 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 moment jgu+ptn:x-)wum 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 inertia1ngvtu . 2 ei9fc,ww+s I is dropped onto an identical disk rotating at angul.+uwtfs ge n1cwi9 2v,ar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atieu (pnc2*8eav da f13 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 thcj66t69 an1bk ce9kc m7 jyn9je center of a rotating merry-go-round platforb96k mn6ckj6jy9 9ea 1ncj7 ctm of 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-go-round is r)mfmjlxxti4 +sv4z e9t s-78f otating freely with an angular velocity of 0.exf-m tfxjl+ t9s4v i)8m7s z48 $rad/s$ 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 about :-d4lde ykayn;yb bv: -(k0gxhalf its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming tyk - 0 :y;bdbngk- dy(lxae4:vhe lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds )r,c)imm3ot u41afbn) g0 ymgin 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 y1e l. 0ff, wk)hv7bby0wm/y a$ 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 a 8h2x -0m(mc*lvejimgj:jh 1kt rest, that can rotate freely without friction. The momenm ivj j1e8 mch:2xg0*m-(jklht 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 personmr0p d5,e.sgmqd c4mn *n+q0tonsg* 0 stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on fri*0mc .mrngd0q* 4pmn0 n5qe,+dsgsotctionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lc:ul fbq/+e0g 1y-vskying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding ontos1e/k y0q+-bg f:cuvl 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 s/2);m/0 fpzly-p0 ltglygw 8ygg1 qj duch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In f/tg)8ljy0gz g2l0wgy;/-1lym pqd pthe latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest atui 3al ,,*zbxoz6azf* a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquirx:5ep/xf 4 jes7 -0dhl(0 doeo(6uuy( s veziles a rotational rate of from rest over a 6.0-s interval at constant angular a hie5l/l-o(es( u70pu:x zd fyv6 (dsee4jo x0cceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two sol rc2*pzijolm8eitjhavo6r o.(80n: ) id cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050mp2 or0c8tv8ni o6olah *j.zj:()ier 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 yj-r/ /m4pyq xis the angular 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 okdoi v :s fhi)z,+nz3.f our Sun, but with mass 8.0 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 +:3 i if,.dzvnzhok s)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 khp (:isx 8;psis 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 ands ;pi8x(kshp: mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrate2:wa (ta/f-tlwvsub8uc gp-/s 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 rolling on a horizontal surface at sp0ox*nlg f qxstfz;lm2sur*.9 t(/3h qeed 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., werjqs :+t2hq 2 yxwhis 5,0tr+d ignore friction) about a hinge attached to a wall, as in Fig. 8–54qt: 0srh2d h ,w+i+rytjs2 xq5. 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 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 im+1 u,ggnc,lm7quvn9 s standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against 9u, m,gq 7gnmlncu+v1which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wi1lqff f/wlc4e/ss:rb go- l 9.th speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on-q/ os.bs9g:/rl c1l 4lfew ff 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 int2 y-il fdu4 co-we,)jbe0c .eco a sling of length 1.5 m and 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 where does the torque 0oi e bleefjc duc-y4c)w,.-2 come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, m. x qmo/--mwz)mov; qvaking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against hvomvqo .-xq)/w mz-; mer body.   

  You are designing a clutch asse*g wshgb 31ul)mbly which consists of two cylindrical plates, of mass)g1* uwlgs3bh $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 trackme)xaaqfs gg/ (me *r.4uws (, 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 high)e, esm(maqgr/ a (x.4*fwusgest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but fi:9 4 nor3mf1f;iqtj do not assume $r\ll R$ h =    (R-r)

  The tires of a car makt4u 1vi44c 9r hetkx5me 85 revolutions as the car reduces its speed uniformly from 90km/hix4ht k5c9 1 t4urev4m 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