A bicycle odometer (which measures distance traveled) is attached near the wheqx 6q2p3tbyc 8rp)g,;ji ngs) el hub and is designed for 27-inch wheels. Whayxcq)q 2s; ib83j)pg rn pg6t,t happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. D w32xwmgfx,3rlo7p e1 oes a point on the rim have radial and/or tangential acceleration? If fxwe7 lpgm3 or,w12x3the 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 single value of the angular velocicvp09 2p/kni1ynq vih lab831ty $\omega$ Explain.

Can a small force ever exert a greater torque thanhn 2hf4eljc25s y39+y r7oh ms;kbqt0 a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than whevvoxuxz- y2/ij 2d4)en your arms )ey2u jv x/ zv4di2-oxare stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seveor3n.-zfwt-o*w, b /lf3 szmnn sprockets at the rear wheel 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 ,lw *fms.3-ntzw/ooz-nr3f b a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower i1lxw * fyiahd0mce6f3(x80d wzc7z7 legs with flesh and muscle concentrated high, close to the body (Fig.iw7hc dmlazwf0eid3xf1(6 y7x*c0 8 z 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope wal*lad nxx (evy6r 96qm5kers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque amx l04flcbo-m--a m x.z1r3ejlso zero? If the net torque on a system m mmflzxrb x34c.-0e1ol j--a is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizotikyg n1 t. xwuh z3bgm5ytr2; .q.)4xntal. The same steel ball is bg m;k ttyyxizq.)w2.u54tr.x1n3 ghrolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously star;9oiupu tpg 7)t rolling (from rest) down an incline. One sphere has twice the radius p7g9 p to;iu)uand twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass.8p 6rij l dzf1 -v8nt*s)h5vop They start from rest at the top of an incline. Which r1 pn z8tf6v8is5* )vrojhdp l-eaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yey0x +-k8cp8xay hl mc3t most moving or rotating objects eventually slow down and s+ a3c y0-mk8x8pl yhxctop. Explain.

If there were a great mi ;ft -jk+2v(cr9t svxpgration of people toward the Earth’s equator, how would this affect the length of the ;9t(-s2vkvtxfc jp r+ day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation jwj5)bsrc u/coded4yz;v p k(r 9w139 wkrcb( vod;)w3jzs95dwyjr/ e 94 1 cpuhen she leaves the board?

The moment of inertia of a rotating solid disk about an axis tho+8s 5hbf*pbo 5 c4uapm9ddo+rough its center of masp*ddc +5o uhaopm9b 84+b5fos s is $\frac{1}{2}WR^2$ (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 ccsc32.xl s ;ba 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? ; 2sslbc cx.3cExplain.

Two spheres look identical and have the fxp6ul+6hst yh0w al *c32;vg same mass. However, one is hollow and the other is sol*6 xphv+l3su w6a ;h2lc0gy ftid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west)h6uj rbq8n3 ls7*y lw. 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 ty *q86l j3uw7nrsbh )lhe top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on tl oeo17i+ rq9ghe edge of a large freely rotating turntable. What happens if you walkgil q7eo +o1r9 toward the center?

A shortstop may leap into t rl3g803 dhnwahe air 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 grw8 30lahn3d and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momf; wva),/bto f3;vek bentum, discuss why a helicopter must have more than one3fvbtev;/k ; w)obf,a rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in z+6: y,dgesa bradians: (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 an amazing coin0 .)g gtiub;nbfn8m g7cidence. Calculate, using the information inside the Front Cover, the angula 0nbgmggu; .bti)f87nr 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, 380,000 km from Earth. T(.-v6sr9w bgymv *zbdhe beam diverges atbv9r (vd*-zsg.m6wby an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a ra,s oc8ufykkv)a mbs25ln n655 te of 6500 rpm. When the motor is turned off during operatiol6us5yo 5,bvm nsn5)k 82 afckn, 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 level floor 3.5 m to another child. If v v7cxojsz;6q4m9 osr:75c) .yrjhxf the ball makes 15.0 revolutions, what is its diameter?;9q7 4h5j v)fy:csj omxzxvors7.6cr    m

  A bicycle with tires 68 cm ivjcv3p 8n1 vnm4 ;oo1yn diameter travels 8.0 km. How many revolutions do the wheels jv184yonm pn13vvo ;c make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atfv,9ugi1p 7 3qlk tgk. 2500 rpm. Calculate its angular vi1gt9k.quvk,gl37p felocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–382) )wgfjqz-s8 ks2u zz). (a) What iswju2-2 ) f8g zzssk)zq 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 velocity of the Earth 9d(,hryot(vx sztcqtt;+c ) :s/ 8 upj(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 poc ;kzlb1kp k 0s rsny(61-uw: h5a+ngmint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the a7z(no xak)dc/vi/ ; agxis of rotation is to experience an accele(nz /akg ;7co/ai)xvdration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly ab p5kok)5k)1 4 fxoeqeaout its center from 130 rpm to 280 rpm in 4 q kkk5fx))ooae1 4ep5.0 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 cx z+b pwu,d1 g+hf6dz7yr+az *-n x3b$R_1$ 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 i9t;7g 3ep/wxczu6xp (qdpo8 aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accu ip7(p9top ;3 /86dxqczwgxe elerated 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 unifmfk 1e z(rkdceed 60+(ormly from rest to 15,000 rpm in 220 s. Through how manykd zkf m (e+cer(d06e1 revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 r-x+c m;:oe5wxmco mkjx57ig ) ktv(7apm 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 j3s: /d2pupxonk p6 q8uets in a whirling “human centrifuge,” which takes 1.0 min to turnp/:po6s d32u8kq unpx through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly frosq, /- lfu4zldm 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in d/q-sul f,4zl this time?    m

  A cooling fan is turned off when it is running at 850rev/migj ; bstw00o6an It turns 1500 revolutions before it co; w00t g6ajbosmes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniform )nc7ds dlnvar /q(00ely from 95km/h to 45km/h T vd0cal7s )d nnerq0(/he 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 revoluts)c+2p:yf qmd(d( 2 3pdyk6zx-en 7 prsrlhv/ions as the car reduces its speed uniformly from 95km/h to 45km/h The t6dy2rp d/3 ns ydx+rmp:k- (qzf7)svl (h pe2cires 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 o sxvbuye8k0co4 +m x)c+m;nde2pu o(6n each pedal when climbing a hillo2m64e(bnsvu;0ypoc +muc+)e d xx k 8. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on)qw*u +v ek2d y*7oxaf zlpb: gjj39z: the end of a door 74 cm wide. What is the magnitude of the torque if t:w93 )ba+u:xjvyfz pg jkole**27zd qhe 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 of the wheel shown in Fig. 8–39 qqxm )a 6 rlph;dld;s6-d/r)e. Assume that a friction torque olr6;de);qapq6r- mh /l) s ddxf 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of masm3bur 0yh0w) aguaxsq;0n 8*nimu;o : s m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is nab* :umxau;y ;uhgo)w8nm 0r00 3qi sheld in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head ofyyrhv e7l8g5 pw4g 3x;x.nox(nxi4;a an engine require tightening to a torque of 38 .xg8hvn 3xa 57 g4;4ewpx(iorn yyx;l$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 mvoiq:dc1 jbid 7 /u+u1 when the axis of rotation ocq/u:1 diu1bv+ ji7dis through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamet9uyr*o9 b82fwwz8 d,0nb dqegv3/ dmrer. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignoredvdy e3dw2/u* o r808w9b 9qzrdfg,nbm (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end o0 +r/vmojd zr+pmmr ).f a thin, light rod is rotated in a horizontal circle of radius 1.2 mmojp rrd +/z0+m). rmv. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant anguluspo.)u6rtdom1x x*2ii 5, ie ar speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.u)di,p.sri2t*e6 x m5oo iu1 x
(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 obj, cmbkd w07u-k9u.l hjects shown in Fig. 8–43 abu,. kb9k 7cmwulh0-jdout
(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 m0 ao; :q dm g -dj.z9t1yu0kekdk0a1oftr3.ifass 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 z6;o*:sl scrm*qg y iewc5w- cbo8x *1satellite spinning at the correct rate, engineers fire four tangential rock-;o5wmyw* :e oz *6bs 1lqsc8gci*rxc ets 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 with a radius of 8.50 cm . ofq:aza2-h0i krm fv.. i:sqand a mass of 0.580 kgmri.vo:f2.a hi0k :.q-fqasz . 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, acced 4 ,gvthf7e 4me(gaq2lerating 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 is abl3yu*f6g8o ncr9-enjo7qvn z0k znny5p/ 8.h oe to accelerate it from resthn 5yno p6-r/0.y98qez 8uz*jnncn37ogvok f 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 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 uarso4s y(v41mf -nzu8xtk ;)and is eventually brought uf )k srn1 4x8uvsz; m(tayo-u4niformly 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. 8yc(op83u:g,y 028q g 3rc7 dwin figkt–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 action of the forearm, slc8ihm86q 19 9uyratg(, ntrwhich rotates about the elbow joint under the action of the tricepsugth9 8, nst1q9rma86ycir l( muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin ro emy av38+uf*bd, as shown in Fig. 8–4mfa+ vb u8ye*36.
(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 twobkhl ;(;:0nchqs2 jry masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammkb)2 :b:6 kg xisg;qedja a)w4er from rest within four full turns (revolutions) and re2g aq:6aw:bxbi; )e)gkd k4s jleases 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 momepoo7 u.+xr/ vknt of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a to1m67n0xu9n qpj 0/y v4xfn nmirque 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 kgi( ,r 8jtmdff: and radius 9.0 cm rolls without slipping down a lane at 3.it:rdj8( fmf ,3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to thf07kg 2-hfy fue Sun as the sum of two terms,k-f0yghfu7f2
(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 much net wocrc8;i,.ixh: rgd (okrk is required to acigirr8(k : dhxc,;o.ccelerate 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 rol3.v fdpbbh2ncsdw/hq972 s)2 jt uz/qls without slipfh2bqj tdn7zps/v .cq2 9bu/3d)2wsh ping 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 andnrg.)fj bd;j2 its lower end does not slip. What wnb;2) jfgj.r dill 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 0.21pbk( ol7a h,flqfy80 10-kg ball rotating on the end of a thin string in a1ylo ,( k80 falhpb7fq 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.*ap t6y4p m4fux+hat uniform cylindrical grinding wheel of radius 18 cm when rotatxpap4tu m+4yh*.a 6ft ing 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 hld b*vt*0( ti(m4kcgsis 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 det lmig dv( t(bc*0s4*kcreases 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.q,.n2l nta)wzgi) d/p 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck posi.lg d))t q2,/apzwin ntion. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial ra2 sccv3d-qm2,gz8y j lte of 1.0 rev every 32v,dczlcqg8y s mj2-2.0 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 6f- -auewpn i9around 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 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-9faw6n pui e- clay sticks to it?    $rev/s$

  (a) What is the angular momezlbsu j46yjk36:w wz2ntum 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 E nz q-t ahain9x6dj6ce.q17j )arth
(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 dropy ay nxubgq *yt)*3pi5m12fv4x1 m1cz ped onto an identical disk rotating *4p yig3x 1z52uyva mxmf yc11tnqb)*at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 /k6. pcnl2yvu $rev/s$ 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 crc3vo ax1h-w r82 mw1kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920cw1rac r138 hv2wom x- $kg \cdot m^2$ 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 freenws 1;9*0szks g;vza yly with an angular velocity of 09g;1zs*;0vskzsw n ay .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 iu8kw bioo(v 9)x15txmts mass in the process, and winding up with a radius 1.0% of its existing radius. Assuwi9k u(1ob)8vxx m5toming 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 involv0ib8oh,p(blktb;h aw 0 vk:v+ e winds 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 x a )rty/bw/n;$ 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,sptx404;6 midv ebt bg3sa8i ) that can rotate freely without friction. The moment of inertia of the person plus the plap6t;s v g3d bs i8b0a4)ixmet4tform 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-md p;8iyy/m *pa ubtn.- diameter merry-go-round turntabdb .i/ -nm put;pyy*8ale that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the ecen9lduvma q+18v +d/nd of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distancd8 vm9 n1+/ a+vqucedle 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 the6pr8ok/f g h/s Earth. Determigrfho /p6s /8kne the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from,h9 gh 2mgh0l ua7xo)8)h hszw 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 in the shape of a uniform cylinder of rada1wx5pepjjr cc) v7,e4*z3y wius 0.20 m acquires a rotational rate of w xcpw jee31*74 ,p)cj zyv5rafrom rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solidj at at-g;.:fz cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, g- atfa.:zj ;tjoined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed2 pru.6vf6avro .vp t1 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 si;f zr (c6layar )g,.ecze of 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 process, what would itcz (rge;,c.6yl)r af as 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 to use energy storedp oh*c fy5okiqpr3 :*h25a8i s in a heavy rotating flywheel. Suppose such a car has a total mass of 140 ohip8 5 3a:k sy5**iqhfcp2ro0 kg, uses a uniform cylindrical 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 an cw nv --r:zk4qf3-+zp oc i,pk$H_2O$ 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 (hoo8at. r-)g 8usgpzfaa.4ob d7b/vj-p mp) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fricu ik88yx7nl.vph. x4gq ;h,j ltion) 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 tixq4x jg.lyv,;n h.7h ul8k8p he tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertica)6 gpkkmqqqv4 x9g 04p.hpyl(fg(-khlly 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). x.m- kvph)( py49hpg q0qf6gl4k(gqk The step has height h, where h

  A bicyclist traveling with speed v=4.n h.hk+z ru7ilue:i 4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the nilu:4.kr hhz +7 ie.unormal 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-k + da/it pxmkm 8.cf,2lamr(i7g rock into 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 thi rl+d7am ma8ci./mxf (, pti2ks 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 t4wrrax/ xm omf+4h c*i;52ixohin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speed;x c5 mi4/oi+4 fhrx2w moxa*rs for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch asse1zlhkqs +6o3p z )rvmdf tj)94mbly which consists of two cylindrical plates, of m+9m)vz f1th34rop ) zjls6qkd ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 1djj7p:kzvqv(r u 1+z 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 withouk(71vudjqr1 + jvzpz :t leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do .s lz-mwavynazm0ut9b0u/* q. h *2vg not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 r3rxv7/ae ui 3d6uetx67 2ochtevolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of ed ae7c rvxhitt23766 3u xuoe/ach 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