A bicycle odometer (whicx,+( )or muflx(zjhbq t8 -p/ n(1a gxu,rupr7h measures 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-inch wheeluxr, t-1(+pnq 8 rrjul(pao (ux/mg ),f7zbxhs?

Suppose a disk rotates at constant angular velocity. Does a point on thekmhs*9wz7:o.w d nu+ fstezp)(r ff p5.ay71i 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 magnitude of either smfd5 +pe. a:yr7siwnh) ff1k9zuz op7.w*t(component of linear acceleration change?

Could a nonrigid body bou;.0ch * z2 8.g mbgihvgefx(e described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thanry c,0hl:lw/ o 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 weq-dsosd:ro.8i ( b4dith your hands behind your head than when your arms are stretched out in fdbe:do84 i-(.d rss oqront of you? A diagram may help you to answer this.

A 21-speed bicycle has d0,*ket5 t9kjfo-fi;8p:xl) izzbk -pg; q gy seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or ap jbt)dq;g 0g,ek;:pko -9kxii*- ft8zfzly 5 large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able9l1 py7 j4ze nrx3yqd: to run fast have slender lower legs with flesh and muscle concentrated high, close to the bzr3nx1yjeyp9d 4 q 7l:ody (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carfa4 z)j.d.lg zqv lzk+p++qf *ry a long, narrow beam?

If the net force on a system is zero, is the ne0(/rxfee 9sznu5 ow 0rt torque also zero? If the net torque on a system is zero, is the net fexwnez rf su/5(0r0o9orce zero?

Two inclines have the same height but make dic ym;-ut5n+qj6r. dfsfferent angles with the horizontal. The c-rs;fq. nytd+uj5m 6 same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (f.4 inidfv/1v;ww2 i hsrom rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which hwi/4;2nvdi1h w i.fsv as the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They starz0t:y 94 aa7ysnmgdv*t from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which had 4 y9y7:zt sv*m0nagas the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or rojtd*v qnp) 5r*tating objects eventually slow dr*)dvn*5pq tj own and stop. Explain.

If there were a great migra*: zu+oca st3stion of people toward the Earth’s equator, how would this affect the lentsaozu+ *c s3:gth of the day?

Can the diver of Fig. 8–29 do a somersault without having any q6c-q7oqk 21t )mvyayf cf;(g initial rotation wh gc7qq)o1 2yq 6atc y-mfkvf;(en she leaves the board?

The moment of inertia of ypg w 6x so,t7psfcff24)g2no53ex;h a rotating solid disk about an axis through its center of mass isxnt xogco y f7)f4phe g2s3p s25fw;6, $\frac{1}{2}WR^2$ (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 each outstrsfzhl;iz6 /vxvgf c2lml*5 .)etc6/5xll*l .h mfvc)vi;f2 zzsghed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same massir+yi -c3 t+osyry,: :wf7 hfr. However, one is hollow and the other is solid. Des:tiy wch7fror +:+-y y r,3fiscribe an experiment to determine which is which.

The angular velocity o nw.ck+(c p,yutq u7;,c fn1ytf a wheel rotating on a horizontal axle points west. In what direcccutpf;.7 1(k y, nq+ntuyc ,wtion is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge nj0f 1hgc h2s5s 89rmhof a large freely rotating turntable. What happens if you walk toward the center? mh shr20sgcn89jhf 15

A shortstop may leap into the air to catch a ball and txo*pkm e 7+c4fbvwa 6zr317abhrow it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will not k *ze67x a3p m+fcv14o7wbrabice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentumvcxwbm0m fscm)( o z5 **pc-+k, discuss why a helicopter must have 0s*c mwk5c+zfp co-)m(mb*vx more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rat( v8efcmgjo 8:3u /mjdians: (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 becaus ,:fq:8xzhql ve of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun an,f:z: qqvlh x8d the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam dezv5iho7g-t oygej:es . .jpx2 11j4wiverges at an swze2ej:-ji 7y4 xej 5.1p1goth.ogv 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 the motor is t*(hrg 6a 9n og0h5ms yvs24gvuokl8v)urned off during operation, the blades slow to rest in 3.0 s. What is the a*vglu9( kn6a40o5gm 8)hysh or2g vsvngular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the balciq .*s:v*fp5ksa o 4pl makes 15.0 revolutio4q pis :*skocpf .a5*vns, what is its diameter?    m

  A bicycle with tires 68 cm inq ,v hbknx er8w84 .js+fgr8t7+9*o vj kp7ysw diameter travels 8.0 km. How many revolutions do the wej 7rsf*rhgx,788kqy nw+p.t o9s4vj 8wb k+vheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250ughy:tb: +xu+s+o- js0 rpm. Calculate its angular velocity iny:gs hxs-j u+o t:+bu+ $rad/s$ $\omega$ =    $rad/sec$
(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 revoluh8c .8 mb 0xenvu-.0 qujwaf6btion in 4.0 s (Fig. 8–38). (a) What is the linequhe-6uw. .0a8cnbvx0j m bf8 ar 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 Earthfxtrqc tvy)d6lyxi3y,n0xs7v(/60 y (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear splap///( f.wr(mjf h apeed 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 centrifugmgk 4ilj.ou.o8dym l8+f* 6p j*fzc,qe rotate if a particle 7.0 cm from the axis of rotation is to experience an acgc.+4dqy ff,zjlk*88p j ml*ou im.6oceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its 3*x9dyt rm zo/i5l.el center from 130 rpm to 280 rpm in 4.0 s. De*9tdxrez o3imyl .l5/termine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiushsb+i ucjp gs 9nrw8u05x54 z; $R_1$ 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, ast ek) * dls0c+obm6hx9cronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they c *m xelbd)0ko96c+hsaccelerated 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 15,02o rvu tx6v2-e00 rpm in 220 s. Through how many revolutions did it tue2 x6vot -ruv2rn in this time?    $rev$

  An automobile engine slows down from 4500 rmaq;q-pgyldm1s 0y;h 13e7o( de5w ffpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a r1ikgs9cj/4) u h hns1whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final spun 1ri41/s9hgc sjkh )eed.
(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 uniforx3ws)s ,epz )kmly from 240 rpm to 360 rpm in 6.5 s. How far will a point ))sxs wpe,3zk on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off wphqxntte063c u:k9iow r) --fhen it is running at 850rev/min It turns 1500 revolutions wothk)0ncx q 93ie-rup6- tf:before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolut,) .ukd0(dgcchp f :amzs)h* uions as the car reduces its speed uniformly from 95km/h to 45km/h Thdh)*)ap:gc d 0u k uh,(msf.cze tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed u.b nl)iqo)z l8 mtza28 rczt.2r9 jlr7niformly from 9tj8 btza2r.r )mq7rz 9 cln28.l i)lzo5km/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 puts all her weight on each pedal when climbing a7zw o31h3ebizvaiz81 hill. The pedals rotate in a circle of radius 17z33 1iiaz w81 o7bvezh 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 d)f f200hp;aflcq1l xthe end of a door 74 cm wide. What is the magnitude of the torque if the force is hff10cql)lxa 20 ;fpdexerted
(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 iu+yw ,wgl.w7ln .uct 0n Fig. 8–39. Assume that a cgt7 ww+ ully..n0 u,wfriction 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 x,bv9+-iosh 2s33obljamao2tw o 5b,of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posaobth3b-l3,2mwj 5o,vbos i xa 2+ so9ition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of *cvv f08evcw.h0af 1w38 0w* f8wfeh.1vvv0cac $m \cdot N$ 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.s: m)ry,jy;+9k(tscw gwgfu08-kg sphere of radius 0.648 m when the axiw:g9uyyg+m sfk,ts(j rw ;0c) s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamphnl:qd4wfhj*1 0x4f eter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (hjx4qfpfw0*nh4 d l: 1why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a wcty8w;d mwb4w1g39x thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculatewwd3;w4w1b c tgxy9m8
(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(phzex 3haf0 xgeg51f3ex4 /n at constant angular speed (Fig. 8–42). The friction force between her han4ng aph0/efe5h f31z3xex(xg ds 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 inertf +b1(01ck ea7pvxyy wia of the array of point objects shown in Fig. 8–43 a0+pk1e 7 y(f yw1xabvcbout
(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 whos2pts ;del /l;ze 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 correct rate,1jnf dmn x-c+bc v3*l- p:2/wply mmx: engineers fire four tange/*xmcxbl-mlm-n:pwcp d: 2n1 f jv+y3ntial 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 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with610jzq8c w qi ,ne,n,kswm-ut a radius of 8.50 cm and a mass of 0.5-i1qwwmn8u q,kn,jt0c s e,6z80 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 swingsq q,nn2b(fy iy 5: gmv5ska fk6*u-7oe a 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-round and icrl2jmco b11 -s able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round moj1c1-cl b2r 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 off and is eventuaiy.l0 8b v2c(aryi a9tlly brought uniformly to rest by a frictio(cyr l 9a8bv 0tay2.iinal 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- ;wxz/fe)ic3yrshyv vb;) u3( 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 nqj(vl.njk/npzwi 8 ctx2 hd8n9+z1-+kh u9oforearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformlzj+lnkno.j p981 v( n2 d/qitz n-ku8+hw9hcxy 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 showjz v6 mi)fgm/8n in Fig. 8–46vm8igmzjf) 6/ .
(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 matwp h +do65nu2mv9hy 4sses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turnsv054c kbvb xih- .e9fi9fp5vx (revolutions) and rel c9bx ebpi-vvx 550hkf. 9vif4eases 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 2i p:cw:yj9os 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 develops a 5e8i-w a8br,gf1 4jvtof 25vgzlc l /gtorque 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 a e2f;(nes/8f/umymm rolls without slipping down a lane atfymsea2( ;8/e/m nmfu 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earthxg5 b qtef- *y0y.re6j with respect to the Sun as the sum of two tqe g 6y5by*0etxfr-.jerms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much3y,(lxc/gnd5 o p-gvuo4 e3a l net work is required to accelerate it from rest to a rotation rate of 1.00 revolution p -g vdl(uo,nx3 l3yp5cga/eo4er 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg ; 0h+rou d m2wqic:7iostarts from rest and rolls without slipping down a 30.2 +0do r; :chw7mouiqi0 $^{\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 baza)t ;qrr8ct7f ro3 em*o1wt lib).0slanced vertically on its tip. It starts to fall and its lower end does not*)qrz e cftw a 8lmo1rr7;to3b0.st i) slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the endf2 g )l56tl)v0.sr wdqjvubd: gy8ai 1 of a thin strindbt l6avj0fi)s l5q.)8guv wgr: 2y d1g in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whee1t) q1)yylvitir,2z*ml7 a ydb /a r8bl of rav ,7dzyl)r*1rt/bya8 yma) qiitlb1 2dius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his w+wd d2vj+l,eog r,,g side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decregrg vd2edl,woj,w, ++ases 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 l o lvxx-*3go,oktax 5n87 a;aher moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotationxnko,*o3a 7xa -vogx8tla5 l;s 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 l:5+hkdoc : tikz,w2c spin rotation rate from an initial rate of 1.0 rev every 2.0 s2c:+k :tld ic,z khow5 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 arounmlj(19tcq3jrqhe1iq;h io q8 +w .mh( d a 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 andiih1(w.rh3jqh m+ ;m(1teqjqclo8q9 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 momentu nf6k:d38dnqim 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 of the Ear4mb6gri 2u +jath
(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 I is dropped onto an i()r6g zzp blb:e q/b(tdentical disk rotating at angulazbptb/ br(zlgq( 6 :)er speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at wr nu5 d;y (gu7ucd: qymxs2/22.4 $rev/s$ 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 ve)l*/e . ffigstands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertivgf.ef )*ei/l a 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 angularvwlm6q6v1xbrk )i s ). velocity of 061 vw)q m. )bilv6rskx.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half it v iav(/ j 36/qy12j0vdijvnfgs mass in the p02idvjfjnq/j (v36 1g v/ vayirocess, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve vgr 5dy4g.ia j6keh32winds 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 tteq4 . jhrvba1rh3*- $ 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 platf(f;szubbqulp)2 c8 6form, initially at rest, that can rotate freely without friction. The moment obu;8)(c 6p2u fqfslbzf 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 a 6.5-m diameter merry-go-rouq3e4fz e,n) ;typy9 ;oqsaw4jnd turntable that is mounted on frictionless bearings and has a mot yyeq4fpo9szja;4) qen3 ;, wment 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 thec t6pab:px -b/ r7dq+u rope lying on the top edge of the spool. A person grabs+u a-pp bt:bdr cq7/6x 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. Detex0dl mt sz;gyy 4 9xe 2uf6xs iz(+x5r0,v*yairmine the ratio of the Moon’s spin angular momentum (about its own axis) to its orb;xi(mxlgzezsd tr,*yy 00+ 9uyfisva4 5 62xxital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest,hal6 skdmn*;u 2z2c h at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radi21cxstjxh ;fx rk.i5*je8u*5 ,ezkg eus 0.20 m acquires a rotational rate of from rest g;,c5zjxt.* 2ufsie5xrke1kjx he * 8over 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 twoo2q)t6 d8 v h:jii:esq solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear e:j vtho d6:82q) qiisspeed 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 tjt jq9; uu2adis*+szwb-l7r5.p wz(khe rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 tixetbt7b*n i 19t4g fl6mes 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, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off n69x b*n4ttlfebg1 7ito angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy store,(3v19-dmqcz+mz6 /aqen xsz3w9xcktxc hm1d in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylinez3 61 hcdvm +nzstc9- 9xmkqwx 1c3/,aq m(zxdrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates 7aya xsoo(7r7 +k vhqf.d gf)hmr04:yan $H_2O$ 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 g6p+evf9h:pt - k0og 7yup)gispeed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we inx)1) gmji,n*a + meb, ji0aeggnore 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, determin jnab+ien1emj*xa) ,)m0i g, ge (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 ra ba15ro ny :iz gb3l+0-ukaq7gdius R. It is 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 which it rests (Fig. 8–55). Thek7g3 ga 0:b bun-z q1yli+o5ra step has height h, where h

  A bicyclist traveling wiibbp0)52 qzk:v hg5iuth speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the norma0z2u) bhg b:k p55iqivl 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 begins whirl.vx /m3ebkuk .4zxci za( u/v0ing the rock in a nearly horizontal circle above his head, accelerating.u/.eb0v/u k4cz(xvzi 3 kamx it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thint3epa szp v03x-v +)rt rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with ov z+v-r at3t0e)pp 3sxutstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of twohpir)(amk 3fx )pt 5w dm5vs4(-vm5be cylindrical plates, of vd3555(h p)x iam fb)-rtevpw(m4 kmsmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiusl*ra1i hpoqv4 + (l.ls r rolls along the looped rough track of Fig. 8–58. What is the minimum valuel as rqlo4*p 1(vl.+ih of the vertical 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 q)i7t4 9jactke/ ql2 hassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car e) k(trwxy,,tja;d6treduces its speed uniformly from 90km/h to 60km/h Thetwjttrea6d y(,x ;, )k 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