A bicycle odometer (which measures distance traveled) ij8vovekkb 7c +;dr 4x3s attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicyd4co8eb3xv j;+7kvrk cle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a payu.r s ac- 0mra9z/q6oint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? Fsz-ay6 r.cr0umq/9 a aor which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angulartxm bh5n29 y+sl- wtn1 velocity $\omega$ Explain.

Can a small force ever exert a greater rqyhd *-)*vl/i9s5xh wp6 g catorque 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 a1s mva zsf)p(s5 yyd57 sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thpsfd ) s5y71ma(zsy5vis.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedalraj+0lbs -.kbqqv:i 4ky ;bq* cranks. In which gear is it harder to pedal, a-+q. :asvylq 4*i bkrbjb kq;0 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? Why?

Mammals that depend on being able to run fast have slender lower legs win6g(0t + xuzw s,+hfufth flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageoust +0sw+6 xhfgz,u(nuf.

Why do tightrope walkers dp(gmex8 ev5,)iow2taf:x+z(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero,ey7o0 mnbqp*h+h y, -v is the net torque also zero? If the net torque on a system is zero, is the net force zeron +ey7y ohhpmq ,*v-0b?

Two inclines have the same height bu b 3vffl,e d,05q chtsy,l88ogt make different angles with the horizontal. The same s 8bhedgto3l,0c5ff ,y,vql s8 teel 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 stam86 vqyfk97 u ck z/c.guiow2:rt rolling (from 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 7/ .o :cqu2f6vc9uk izygkw8 mhas the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They std5 :f4zylwd k-art from rest at the lyzfdk: 4 5-dwtop of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and a,2x0apm;wl5(v cdtdxpf,,- (jkjsf lngular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop.pljj sft ,fak;xl(5 (cpd,w-2m, dxv0 Explain.

If there were a great migodvi+bvbr( :mvq:9f g*ox/ o ;ration of people toward the Earth’s equator, how would this afod;o(xgb r m+vv9: *vqf:b/o ifect the length of the day?

Can the diver of Figi(:eqff2tvojn1m+i 5 . 8–29 do a somersault without having any initial rotation when she leaves the boari+ f 1eo:mitqj2n v5f(d?

The moment of inertia of a rotating solid disk about an axis throu(o/jq gzz sq 5at:/m2 /zz/bhxs 0k+pngh its center oosat50q//z /z nm+hzgs: /j(xzq2p kb f mass 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 amk: ho4,1c st*c9gns;q hq(s p rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly d9 qm :scct4g(ossk;*p ,1hhnqrop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical a-.zfkkeyqyz 14 h 7e7mnd have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is wm yhf7yz7zk .q k-ee41hich.

The angular velocity of a wheel rotating on a horizontallljj).,vy lf::fh . 2sit. sif axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increa :s,)tyi2j hsf jfvlilf:l...sing or decreasing?

Suppose you are standing on thx k :jgjk9ebxyl/j086e edge of a large freely rotating turntable. What happens if yoj:b jel kg8yj69/ xx0ku walk toward the center?

A shortstop may leap into the air to catch a ball and thr 5l6kaqht v3m:ms xh(;ow it quickly. As he throws the ball, the upp :a5lqhh;xt (6v3ksmmer part of his body rotates. 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 conservation of angular momentum, discusss qk6. za(o.g usg8;zwlf0,gx why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to k g;luoafs068gzw,g s xz(..qkeep the helicopter stable.

  Express the following angles i 0vki kz2q1b 1*e))itd;cmuvfn radians: (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 coincidence. Calculate, using yhaz5 su4,cj-the information inside -zh4j,5y ucas the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ear z hck.owiq*u/ ppx871th. The beam diverges at an angluwz1 ik87/ x*hpoqp c.e $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at s:6 k5fub1s9ajqn2fo a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the an f9k ofq5a2bu 6n1s:sjgular 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 theb0 ccqtg4a8- h ball makes 15.0g0qhc c-8t 4ba revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rel(ulaw 3zwznj/ qc4u;p* dxj7be (n+-volutions do the wheels makj + 3; (n-cqwbux e(zaunz/ljpw4d* l7e?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm.5nqq9m4a6qj9 5ijh et Calculate its angular velocity n eh6q9q qij94tjm55 ain $rad/s$ $\omega$ =    $rad/sec$
(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 2oplw+ejd4)l,,b gjxp d9 h)cone complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear ll p hgx9w4 ddp+),bojc )je,2speed 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 xi*xq ,aw mt.)velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a g hn0j1ec1h w*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 centrifuge rotates ;+kmpw/ h,n0 tmbkd; if a particle 7.0 cm from the axis of rob p 0knds;km,h+t/w;mtation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itshqu d gtd:(i91 center from 130 rpm to 280 rpm in 4.0 s.9qi h g(1dd:tu 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 i e)ib :i974,bgfpceb2x qup+ $R_1$ 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, astik wd)c.gl.1y)rs gu/ronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder withwi1k ./) durlg) .sgcy 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 uniformt36)k3hiknvxv o6b v -ly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? k6v6v )kon ti3v-xhb3    $rev$

  An automobile engine slows dow3x:5 q; ceqzc j/e)aurn from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying higq7-wcrwq27i4 ce zz6v0gz *zbhspeed jets in a whirling “human centrifuge,” which takes 1.ivrz7 q4 e2w q6bz w7zz0c-*gc0 min to turn 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 uniforml1 fzbm82zvq hq-j* 4ajy from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this timeqzmaqj-8 jz*4vb 2fh1 ?    m

  A cooling fan is turned off when it is running at 85 6op+ygu)jdz-yl1(+ xtt, dwk0rev/min It turns 1500 revolutions before it comes to a sxy (1d ztyu-wo,lj6pt++dg) ktop.
(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 uniformly0 lg+*mbo6s -azne (-ia7y1raro*msw from 95km/h to 45km/h The tz1- y l*areriso 7gaos+(*n-6b mm0 awires 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 uniform+otw c+6bzt 0e ukc(1qly from 95kuoctwk1 60 cbt(z++q em/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike p v/q0 )7s na3mnl ;rin(zv4bx,pf0apz uts all her weight on each pedal when climbing a hill. The 0b04 / 7 s(lz fmvn3)appnaxi;znrv,qpedals 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 the end of a door 74 cm wide. yhd f72o. 1fzwr7qw ofb(b0/i What is the magnituh1f i/fobyr .bw07z7w(of d2qde of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume lca,:k vxf+n5that nfkvxa: 5+c l,a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to16town z 6vomtyo)l:1 the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the6t6 o)t1y:zmon v o1wl rod is held 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 of an engine require tigh2nps5k,rxp eli3;nb x ke/f s6cr; o(0tening to a torque of 38ppr /;s3rn 2 iso5xeb ;knf(,el0kxc6 $m \cdot N$ 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 jn (lko b(jty .1u7ah.sq,9eiradius 0.648 m when the axis of rotation is through its coineq9.ts1j kl(jy7au( b ,h.enter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. ( c6wb8a,m8iej8xy ok 8c*w ssThe rim and tire have a combined m o 6isj8wa,kyc8e*xcs mb8w8(ass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizo)a0 i j*ts s dp)3pvqd0-m7cgjntal circle of radius 1.-dqj ))*vjms70ti 30c agpps d2 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 5ffy r80 l/iwca potter’s wheel rotating at constant angular speed (Fig. 8–f5y w8li f0c/r42). 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 of inertia of the ak*p hk 6/ rshi.9qgj)urray of point objects shown in Fig. 8–43 abouj/hhr*q .)ukki sp69gt
(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 consi4.: gvo dez. l8y5obststs 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 cor69j)v8kh m.2ar meelx rect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of halkr 6) mx8mj2eev9.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 unifoqi9pzs 34fk uz/ 61tddrm cylinder with a radius of 8.50 cm and a mass of 0.58pu f6kzdd t4z/1iq9 s30 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceleratg+ 2:0 scqrmuring 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-roufdjmdtj -y79:3kec 1v-qt p 3xnd and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 st-y79xd j13ef:mvktp 3dqc - j. 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 ofxrocn 5ny;g 3v(.pkd ,f and is eventually brought uniformly to rest by a frictional torque of 1.,y; opc(.gkxrn5d3 vn 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-kg ball(8 ezazb t6aw4 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 fore g5mm r l0:,t,aoca+nh-xmu7 carm, which rotates about the elbow joint under the action of the triceps muscle, Fig+rnmutghl : 0moc ca-am,75x, . 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 n31aby wdljd;j3 qbed+6/ m (ma long thin rod, as shown in Fig. 8–4633n1m ed+ j;l/ (qmjd6ywadbb.
(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 twod( yphl g89 b;esx-tihuos 5)7u r70ed 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 -p ghil1s23hf ud9 ny/x; v:qphammer from rest within four full turns (revolutions) and releases it afy- s3qhniu:p;g2/1 d l pv9hxt 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 mome; tviy*pwla ixre-,w-d4 +bkpu:dnuy0 w 093bnt 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 developsjttfwda 2s7u;(*+s oww9 7 mkq a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 khd , bxh m7m5a82-qwdyj *v)tdg and radius 9.0 cm rolls without slipping down a lanemh vm w dabq-h28dxd*5,tjy )7 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect 6qyi+ w+b9)5fve pbmwto the Sun as the sum of twm)+qwy i pv+wbe6 9f5bo terms,
(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 radv(* skiz6w7 rfius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylii s rwv*(76kzfnder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fdg3. fn1rds i*rom rest and rolls without slg.sdd 1* n3rfiipping 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 vertical 7/zbt j9; :)xzfgr:6bzh q5pdsmnn1jly on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just; znt1bd69msgjn 7bp5jxq ) rfzh /:z: before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.*8(kbhmht vhb g-zen)ql3/-o 210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular sp*qb)betol(hhhn8zvgm - k-3/ eed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angulaucv18q )97qgcn g:l 5(5aodp,zurvh qr momentum of a 2.8-kg uniform cylindrical grinding wheel 1zdruc q,lv57qugq()a : gchpn95o v8 of radius 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 side, on a5*prscc6c c8z*i sg i2 platform that is rotating at a rate of g6zprci 8*cc2*s c5si1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shod-ebl 5bq ek z)s8m-x7wn in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 wh )ebe-lz7 dsm- qkxb85en changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can i1d zraa4vi-b.ncrease her spin rotation rate from an initial rate of 1.0 rev ev41ibav-. drzaery 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 ih*)fct,s y2 earound 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),i s2etf*c hy center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skaterfq+ ohl )79bb3f, ogqd 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 Eartha7fo v;luz10 xy b7c)q
(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 di+lr8g; * x5*nnat:e7hep yujy sk of moment of inertia I is dropped onto an identical disk 85nayeet*u:x*g j+l;7pyn hr rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2fed h b6l*n qtahq9.amiq85,7.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating mer ; ,m :xp:nqzioh358, gfv3i9up hqwwhry-go-round platform of rqwq3h,xuo3pig m: h9;wfpv :n 8z i5h,adius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an a lk,w,gxc,j/angular velocity of 0.,x,, g/clakw j8 $rad/s$ 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 vm-7us2 jbau0633qi u e6hzyr a white dwarf, losing about half its mass in the process, and winding up with a bysre-ij 6 q7u2u z0hum363va 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 winds in ei s qqtu-h9kf9a7* gh0xcess 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 j uj ,v8v 4:c+m,njqyv0h mf(y$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freelyi+70vvp sm gjy6(7 aea without friction. aeg (sjvvpa60i7+ym 7The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-k ) ):xlqzqp/w5 a.vppwi0 .jfm diameter merry-go-round turntable that is moap ).kpf zxiw5qp0)v w/:.lj qunted 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 28+d.vv2, yhbu wxmwrls eu6*with the end of the rope lying on the top edge of the spool. A person grabs the8 us+y* x6l v2m,ehdu.bv r2ww 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 suchinr:fs:ow5 .gf +k +(9xvdehy that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a pfdi 9vnxeh5(o:yk s:wf+r.g + article orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from y6y 1/5 tz kr*ejc-bjmabr:s,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 s u,ceczo 8ni. ,;yjn6chape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest overjzn6y u ,oiec.8;ccn, 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 solid cylindrical disks, each of mass 0.050 kg and d 40/di w r4wo99orbvuqiameter 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-lonvorw4rqbw/i94o9ud 0 g 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 anv jbniu:. 2cv/2 yi3lwgular 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, bute5j)z ( lpo+ xilqbo18 with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo g +ilbz1(x 5jl8 eqp)ooravitational 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 lqcxnx9+t)* lfu6 l99p,4t8lv y uztemow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50l8z96ttx4x,pl v )uyn9ul9qm cft +e* 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 +6da0z rwwr j3$H_2O$ 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 rollivvhn6,r x36. p1zj60 kycqjhn3y;y vjng 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 fre arkp .n9+-yrh0cj-zvq62rrxely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the 9r j2h+x- k6. aryqvn0-pcz rrangular 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 veruzttsd qha 3ci-o9n m:--84aoq)i)pytically on the floor, and we want to exert a horizontal force F at its ty9i -tpa8o4 ma -:snqdoq-u)3h )i czaxle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/ac ,e.qae9p3 zs on a flat road is making a turn with a radius The forces acting on qpe, a.a9ecz3 the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins 8s:* 0u i)ewgss6lgcswhirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where doece s:*s0su w6ig)s8 gls the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin, ge-(h7fft*jf5 pigb rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched armffg b* tgpf-, hi7ej(5s, and with arms held tightly against her body.   

  You are designing a clutch assembly which consistscv h;5pg zg2m(td94iy of two cylindrical plates, of mavc25 yihz9t;g4gp (dmss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rou 6kr5 tzu0qzb-gh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reachb0z-q 5tr kuz6 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 ashie9an* 7(juq9n3y oosume $r\ll R$ h =    (R-r)

  The tires of a car make w9k p(rc na,f,85 revolutions 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 each tif ra,knp( wc9,re? $\approx$    $rad/s^2$(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