A bicycle odometer (which measures distance traksuhfq 8- 8pov v;:z+qtk0d*kveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a hqpks8k vf-ut8d;0+ q*z vko:bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a z//d e yzgi,a9kpvn-7point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases nd9z//i7pz ,av ye-gkuniformly, 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 veloci gw9dq k*k)0mmty $\omega$ Explain.

Can a small force ever exert au fx o ylgifnyu51a+34l21f/ u greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind c, v7+2kvkn2iym bqh0b99 wusyour head than when n2 +,qk0 2ch7s9w9ymikuvvbb your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rearc.zj h usr;6v/ 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 a large front sprocket? Whrzc;vjh.6s / uy?

Mammals that depend on being azbtr h7u4m- i* ohvf5;vf.b3qble to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the5*fh4;tufr.q b7z vm ovbih 3- basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers zof/0lo -rf h.(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net t.+t*4m ws ei/-d ;buaixhet a,c05evoorque also zero? If the net torque on a system h;t/ ixwbaa0ev u,*m to.-s4 eed+ic 5is zero, is the net force zero?

Two inclines have the same height but make different angles with the horert k612)hlwq i, srah,-e3ahizontal. The same steel ball is rolled down each incline. O hrth3h,s-ia1e w,6q)ka2l ern which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frostc 2u0idfe y 0.efo*1jte yk,a,)j-om rest) down an incline. One sphere has ts0e*1 ey0ojt k-fo yji f ad)u,.,e2ctwice the radius and 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 sam(7q* c,bpg rrne radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Whr,nr*c(q bpg7ich has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are consebzs( svk)jm5b .f0sf*rved. Yet most moving or rotating objects eventually slo s*vs bfbsfz0.j)5 (mkw down and stop. Explain.

If there were a great migration of people t6adrqgx/wf ;5ay v9;l oward the Earth’s equator, how would this affe9walaf5x qr6 vy/;d ;gct the length of the day?

Can the diver of Fig. 8–29 do a somersaulte y g.;ega*;bxwcl f/- without having any initial rotation when she leavb;clee-x.;gywa f /g*es the board?

The moment of inertia of a rotatingwx 5o4j3rnv; f ;ofz/ hzf3+xi solid disk about an axis through its center of mass is /w fi jx35o+3v rof;;znfzxh4 $\frac{1}{2}WR^2$ (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 rotati,*( hw 7)hyvtq0owagvp 6fw;cng stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, de6wv a ,oh ;h(tc 0wg*7qpwfvy)crease, or stay the same? Explain.

Two spheres look identical and 2lvxm zl +c0u b7wq2u0have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whichwl0bu20xmvl u 2 qcz+7 is which.

The angular velocity of a wheel rotating on a horizontal axle points west- g:6dq+ pd4/hqzrnm ps jmg3 u91uvur,ic(7p. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points eastdqj7+1-z/i nspmpp9 g: hv3u(4crmqgr udu6,, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntabr7;3e wjx48rzga /q0w :fwccsle. What happens if you walk toward the centerecc f:0z4r3 aj8wxg/w qs ;w7r?

A shortstop may leap into theo,mrr10ne0vdk0 utc 2 air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotr1 0cu tv20n mde0ok,rates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conser6yyba3h *vc 2bme*fi3f se/u0vation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operat ff0ume3yb*/2eahbc 3 iv sy*6e to keep the helicopter stable.

  Express the following angles in radians: (2jp*gtl:fiw;*y y xo: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 jn t-rm2jfdf z956zg3 amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and th gzn rdzf ftm2j5-6j93e Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km dd1 ijlw;zn-bb-a x(( from Earth. The beam diverges at an a-;zbaw 1nj-b idd((xlngle $\theta$ (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 rpm4yrk8 ta-q 4s07go-ujwosx 7 a. When the motor is turned offr0ot4 xsu7j87 kwa-soq y-a4g during operation, 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 thew(05* riged 6bpvkt2y e+ tk:u ball makes 15.0 revolutg+*ek0t (r2ti w65yd:uvekb p ions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many re +;wmv4f*7njnsyk i 3vvolutions do the m4* vy3i7 n; vnsk+wfjwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatespb8v:q6zfv3me 0ecqn -7 x2nt at 2500 rpm. Calculate its angular 20 67vf pv8neq xzm:-ebqt3cnvelocity 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. 8–38). ( 5/7liwvn t)qap.uc6xa) What is the linear speed of a child seated 1.2 m from ta x7 .vcu)litn/6p w5qhe 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 Ei 1gu57twc f3rnw0yo(s8,i -icwogm+arth (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 q(zx2/5 xu kt dn,lo6 vkp/:tupoint
(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)4gapgiw3vb su j)r23 axis of rotation is to exp u)2jgbrga) 3vs43wip erience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center:ply2ang 93 m5o*zsmu from 130 rpm to 280 rpm in 4.s5y:u mmgnz ap*3o92l0 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 radiushod8i*l q o0 f0+h6jdz $R_1$ 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 aboard .mt/(bz wb*4q;pf;qaw5xj do c)3jhcthe 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 1dw;* q;. cpm (z)4jjob/x cfq35twbha2-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 fkqie wv))xi8(p9i::e.; iw 9lwvzyzto 15,000 rpm in 220 s. Through how many revoluti:y wxze8wi;) . 9:fzw i(vke9qivp)ilons did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cl:h+y f:,ay3c5g*:mpizk7 nj anrrji12hw)s alculate
(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 qq:/mze r 7j 7btit7oirga. +* lto*5sin a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutionio qm7r*srgqot *a5i:te 7.j/7 btlz+s before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpo5 imrzwmx /3qd*g-7lm to 360 rpm in 6.5 s. How far will a poiid 3r5 goz/-*l xm7wmqnt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is run) l*9rzbnhaei- q8 2h/6r yifmning at 850rev/min It turns 1500 revolutions before 6hz n9/i lfir *-ahm )q28rbeyit 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 th tj)f,p-syc;sl 3+jane car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamet3)-j+;pns cflts jya, er 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 t8nx47:u hvmtnhe car reduces its speed uniformly from 95km47u :nn 8thvxm/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 bi8ju5fmnm+af ( n -m)tdu g4bbmni07 /mzxp /v5ke puts all her weight on each pedal when climbing a hill. The mvn/ma7ut/z g ubb05 f+ xn)85i-mdnmjpf4(m 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 onp+-f (7m8xygn mom :au the end of a door 74 cm wide. What is the magnitude of the torque if the force 7omxyfmp a(-n8m+ug: 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 torquez) b2 7ee lnou fiz-v(i+-y:oc about the axle of the wheel shown in Fig. 8–39. Assume that a)(o yeo u fe:ivzci2-b7+-znl friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massle jbn19qmak i9t8 qi*b(ss rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and qi 9abq k(b98mn t1ij*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 :np*g,koa ds 5of 38 n a,k5pg o*s:d$m \cdot N$ 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 ofn)ohp +qx .89bm)-i wnonbn/y radius 0.648 m when the axis of rotatix+yo h)bn n-9npom .w/b)8nqion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in di7pi ugjt2+v 5)aue.dcameter. The rim and tire have a co).7jud tvau cg+ei52pmbined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram balio ;;8umiw96 wk:fv8e k t4ye)xyu1mul on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2yu86iem;:8) wu 4w1f ;oe yiu9kxvmtk 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 potter’s wheel rotating at consg;1c/6 uoub -m;4rvhyyue 3 i:ppp0lqtant angular speed m p y;rev4: ou-u l/gup3c;qyhb60 pi1(Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment 7tb1 /jqvv: s*l-1o/ji zhfx0ssxht5 of inertia of the array of point objects shown in Fig. 8–43th1s*x bs oz/:t lv5s q1v/hj -7ifxj0 about
(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 conshy 9( h5aevvh2gtrk87ists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, engi/5zvb kpa8lfc/, ge .tneers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady f z b8 l5a/gv,/.pkftceorce 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 radd patu0 i;j;/ef aka7-ius of 8.50 cm and a mass of 0.580 kg.aj k pfti-70/;;ed aua 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, accelerating it ++h2ifg re q1mipa9 x7from 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 ak;*uyd4 ,ai7(tx.zm uvpjc* cnd 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 pr*7jkz(4 , ycvcuad*pmi;xut .oduce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,3cskxja;;*b2 d x gh96s00 rpm is shut off and is eventually brought uniformly to rest by a frictional adj2sxgb s;x6; h k9c*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. 8–45k u,l /:,w2jp*ogccqlmlk pw *,,n5rv 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 iw0 m3(t 7vvpwts thrown solely by the action of the forearm, which rotates about the elbow joint0w(v7 pwvm3tt 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 be considered a long thin rod, as shoi,d;u +b4ctt 2y rmh+-i7*lj 4bajylrwn in Fig. 8–4ij4jlb,-;t*ryrhd u4+c +yl7abmi 2 t6.
(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 consis oyze v nptsf ,je,4kx5a107-cts of 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 accelerates the hammer from rest within four full turns (rev8 /hnh+ye w;qqnp+ s2-gc(dhxolutions) and releases it n2-/ hqep d;h(8+ h+wscqnxgyat 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 inertiagb+k -y db3y h8zikygt9uw/gh6 y0 47r 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 torque oftvs icm:+1,dh;p4 n3gv,fjgm 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 rkn-urn ,pkwdvr+w: +(adius 9.0 cm rolls without slipping down a lank,v+w(p-nu :nrwr d+k e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the /7btdi 9a1 , 0ycsif9stt0esl e+rfa4Earth with respect to the Sun as the sum of two te,srb yasfe0cdttt 1 9 47is0/a9ilfe +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 of 7.50 m3y .9 m, .y tqstpwb6uxdhv3:vhk5 ze4. How much net work is required to accelerate it from rest to ty e3dutbp.vhkv5.y w9 h4m3,:x sz6qa 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 1.8p-o(ch*6cxc q0 kg starts from rest and rolls without slipping down a 30.0cc(x- ohpc6q* $^{\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 m4ldt, 1p1zb cbs1dbgd4 s; 15.rmh skvertically on its tip. It starts to fall and its lower end does not slip. What w5br.;1 4g4, sd11stkdms hzpd 1clbbmill 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 momentuy moouprto/ )/9;sok w* f4 bg8gtr3:km of a 0.210-kg ball rotating on the end of a thin string in a circle of r o9k;oo )to gubr4 /gkfprm *8/3:wstyadius 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 wheeos83ba;wxe b/l of radius 18 cm when rotating at 150 3 x/;8oasebbw0 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 a. puqske,67wb:epe q 1u 1ds0vt a rate of 1.3rev/s If he raises his arms to a horbe.1:6 eqsu uv7wps1 kqd,0peizontal 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 reduce *i0sbpc9d y9gmjtl10 her 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 rotations in 1.5 s when in the tuck position, what is her angular 9pmbsg dy9*0i 0 lj1tcspeed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial ratewc8 09r 7 vny8v4cmupl of 1.0 rev every 2.0 s to a final rat48m n lvury8790pcv cwe 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 axi(t/ .ziuv t zr6)op 67olvg5vxs 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 rl/i5)uz7zx 6ttv6(oo vv g p.of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skwo*u;* t )psp4bwt/mnater 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 Ear v5rlbai 2i2x fb 3v(i6l*tt2zth
(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 inossb t;s )2k+:izt/juo:b ,g)-ngf-4 scxfws ertia I is dropped onto an identical disk rotating at a)/sub:c :;),fsg tz ni f sb+-gjtkoo24 sw-sxngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at -eks bh7a:r i82.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a5.xyri, xx2izxr5yv / rotating merry-go-round platform of radius5ix z,.xry5yx 2 xiv/r 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 merrl:6 bj(m o,oi e,swv+tm+gx5 ry-go-round is rotating freely with an angular velocity of 0.8 ox5si+ bow ,me6ml+,vgr ( tj:$rad/s$ 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 abogkxcnu1rt**1t)*.z5o t4 0uekhng ulut half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rot01 ng.o **kgtrlxuuk*n4ut1 )z5c the ation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excjidn7wq6f )zg( 3r lj+ess 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 tlmbx bgr(6sr96fv*9 $ 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, tha+ yg58h(o4n0 jcvrdb jr6lcj 7t can rotate freely without friction. The moment of ir50jvnr cj + (gb7jho4l6c8dy nertia 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 zer(cv9;f5ouq*kv ; hqc ru85 diameter merry-go-round turntable that is moun;kozqv9erqcv ;r*5f58h(u c uted 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 end of the ropl jn l;a*t3se1e lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What len;*l3net als1j gth of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that thet i5xv ): kkbkf;lvb ;4/ wra1lx/:jyg same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axib w:bx;:5ag/k4xy k;1l/)vj vrkil fts) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates f47avv4y/hwnl,ky m .crom 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 n0po0oc .k +rvthe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motnkoo+ 0 p.0rvcor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0509pk 5 *at)d(b;dh nyfa 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 speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.p ;d ayt5a(b9f*d)nkh    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the4 l/p/f3ux ce3yphx/ fbp qc(/ 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 of our Sun, but with mass 8.0 times as great, were 9*s3 p,mlak;tjzki1 x rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravita 1t; lj*zk9xpk3,iasm tional 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 no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is 3follmx/o()v)dijo*9 6w l hrzed 9 x)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 flyld)l)or )hwi*df(96mvez l9jx/ox 3owheel 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 mc.wq (b3ejl6y(ylu5 $H_2O$ 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 onu 9antma8 n,( sgro vquji0 (2yq2+d(f 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 frictio m( bh l--6 .e5odds+nkdufx6zn) about a hinge attached to a wall, as in Fig. 8–54. Thz-m . 6(f+bdxo u-6kdehd ls5ne 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 is standing vertically on t 0zb)us7gbmdv3 ir0/ 6+th +0n9sv 3ucuixbgwhe floor, and we want to exert a horizontal force F at its axle so that it wil90m v)n0gb+/ 637b sxrcb+i 3zsth0ug u wdivul climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a fla:k8bdiulfk p3/nssf, 7/te , xt road is making a turn with a radius The forces acting on the cys xn8:,ieuf, dl s37t /fkb/kpclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 e9vsk,auw2ah4 v ,t5 ym and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 122 vhs,,5 wv94 atyuake0 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 thinw+tm qcukt45w4) 3sgs rods, making reasonablw scsq w)43kt45t+gum e 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 her body.   

  You are designing a clutch assembly which consists of two cylindrical plat,xko c57 c x7.9t 4ccwflqiz/wes, of masc,754o /l 7 tqciwfxcxkzc9.ws $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 nvdw r(lpby)tl )e3(q, h*q7mr rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point)ylwt*3lq(m,bd qh(r p7ve) n of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do/ gf*s5pinvg 3 not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revo+,u ezauq ew26jn7w -glutions as the car reduces its speed uniformly from 90zaq ewn w 2jug+-7u6,ekm/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s