A bicycle odometer (which measures distance traveled) is attached near the cm0 eqge4p/:c0 e4r vfph1uu7wheelemvehr fpp0q1g c:cu47 e04/u hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doexcp t;3jcg):c o8k1x ts a point on the rim have radial and/or tt c1tgc;oxk 8 x:3)pcjangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a singleeem8gzqe/la(kn .:1u value of the angular velocity $\omega$ Explain.

Can a small force ever exe0it1 r xfm-rtz it21v+sb.)wert a 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 dr:.b hwahpr,n988 gtto do a sit-up with your hands behind your head than when your art phw8.rr8b:9h a, ngdms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel an0 y).jhuzq j5ctf8b3b+)h0+ rt kpfb od three at the pedal cranks. In whicu0).hkrfjpj+qo hbtztby+c 83f0b) 5h 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? Why?

Mammals that depend on being able to run fast have slender lower legs with f -c r*s3dogl2rlesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamirrcg23dls o-* cs, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carryf ,bt )d;d+gxngmmc3.as 01waou u+zkaw(7f2 a long, narrow beam?

If the net force on a system is zero, is mh 80 rpwigtxg6 cz7 n(zxn777the net torque also zero? If the net torqx7c n 8mh7r ingg7p6(x7wzz t0ue on a system is zero, is the net force zero?

Two inclines have the same height but make different angl2f4 1 lcudrlkk;vzmu: jsr68 7es with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed flm8r4 1dsukcv :rzl jk62u;7 of the ball at the bottom be greater? Explain.

Two solid spheres simultaou*+dri - f9kkneously start rolling (from rest) down an incline. One sphere has twice the radi*rku9dik +f- ous 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 same radius and the same mass. They start fromti z. vxx*8ih4.3o xss rest at the top8xsvt soxz4i3h.i*x . 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 angular momentum are conserved. o0 2gfh .ig*aoYet most moving or rotating 2.ofog0gh i a*objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equaty7, *i f+y9u fbfwpxk1or, how would this affect the length o1wfb+fyfui,*7x k9y pf the day?

Can the diver of Fig. 8–2y xouuzj7:.. x hf64cw9 do a somersault without having any initial rotation whey z6j. fw7:uhuo 4x.xcn she leaves the board?

The moment of inertia of a rotating solij; quj5 3jpkg te1, (;ksugj6td disk about an axis through its center of mass is;p jg6k us1juk jt,qtg5 e(3j; $\frac{1}{2}WR^2$ (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 rotat1,zhhb0 *yh e3f7qyt ying stool holding a 2-kg mass in each outstretched hand. If you suddenly drbthfy0 hq *z7ehy1,y3op the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look ident;wcgp t6qiargkbqp3v4))3 t5 ical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to detqiapg)wpv tb 5c3k)t3 g6;rq4ermine which is which.

The angular velocity of a wheel rotatiw89 axig r;wcxkht1,n7m t5k,ng on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tacwt 9ta kgx8 x5 h;7wik,nr1m,ngential 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 turntable. Wv u..hc.t x6aqhat happens if you walk toward 6vux.t qca.h.the center?

A shortstop may leap into the air to catch a ball; +e80fs95mtucb qa: bzyra3 vjf+sxr 0:qa .e and throw it quickly. As he throws the ball, the upper part of his body rotatv qq5m0+yzjsaf:af rc; x8e+ 3:s9rba bt.0uees. 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 af2,zr,qom h+):1annemykbv : ccaz) 5ngular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss ozz1 qe ao,k5a v)bm )mnc:h2 r:fcny+,ne or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i it iq8zq(d3iox r*q40xo.i(in 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 co12(u2 ogsc04 xl:q8zbtaa qroincidence. Calculate, using the information inside the Front Cover, the angular diameters (142z r 0s qux2qc(g abolt:8oain 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. w5v8/8wm rr*maxgo .lw.a: qaarp-+c The beam diverges at an anglr *a+a-r mlp/g:v5wcxawmo.qw88.a re $\theta$ (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 rpm7zm zkee(iow 1+x 3v5a. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? 7v5x+(za wo1 k3izeme    $rad/s^2$

  A child rolls a ball on a level floor 3.5d1fmu0dtow ) 0b;s1 gz,x9fniprc y,;ba5mx; m to another child. If the ball makes 15.0 revolutions, what is its dia mf ;50uz ,1)x;gctr m;dy foxbbpn0ai1,dsw9meter?    m

  A bicycle with tires 68 cm in dib+8su zlgtchc6m5le +c+9 yy *k rs51hameter travels 8.0 km. How many revolutions do the +5+ cylb9hhu zlt+r1y*5 sc ckem8s6gwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250dk/ 9p6yck xth2 ec53n0 rpm. Calculate its angular velocity i k6c encd3p2h9y/xt 5kn $rad/s$ $\omega$ =    $rad/sec$
(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 x,u6-gb rr3 ugmakes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the lin6 ,bru-rgxu g3ear 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 vs hutz,q1vq ;-elocity 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 speedi9wo ke x qq++g9yp7 mlmw0rh2w,2 y+v of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrildz.e, wbzwoj(c zu , *sl2:8ifuge rotate if a particle 7.0 cm from the axis of rotation is tojui *sd le2.z8:wwoz,l b c(z, experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 l1hw f*4n,lolqp on r/bh.;5rrpm to 280 rpm in 4.0 s. Deterhw1;bp/ro hlq r.nl,5nl4of *mine
(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 radiua5qz i*h4m9uc o m(-2d)t htgds $R_1$ 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 the Apos 2y8gy7tx, millo 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.s,yg2my 7 itx8 The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,00)ztmxhy8 jqc ig9z+- 7l b/fr39hc2ug0 rpm in 220 s. Through how many revolutions did it turn in thig+cr7 3-ilf/jb92xuq9c m8 yh) hzgtzs time?    $rev$

  An automobile engine slows down from 450 a5yowu1/ ki rkc.fweyb(a7;.0 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 highspeed jeyf6mk r0(uuhb a:0 k,jts in a whirling “human centrifuge,” whiy0,(h uuamr:6bkjfk0ch takes 1.0 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 uniformly from 240 rpm to 360 q(hc o35brr/prpm in 6.5 s. How far will a point on the edge of the wheel have trave/hbro5p( 3qc rled in this time?    m

  A cooling fan is turned off when it is runny,1s9zf/t/ xmwv 7vkg ing at 850rev/min It turns 1500 revolutionsw/, 7yk9z1vx /gtfv ms before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revhxrr qlx+y i: r8c-x5;-c1kbpolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires h-18p5qrxcrh:blrc+y - i; k xxave 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 uniformly bva2j igfsyxz:ty,an-6*g 6 ,from 95km/h to 45km/h The tires have a d*gvzjyafx :,bg6at- n yis2,6iameter 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 bike3z66yp l 1yooc.xz7/r/hqpf w puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radzplyhc.6o/p/7f3z1yr w xo6 q ius 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. Wh(n-u; bz gb 9*feshp: dkckp5rzqv91)at is the magnitudecp(: b-q;h z* b ufgdsn99r1)v kk5zpe 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 of0cm(4vyw) h 7zsh f4qc the wheel shown in Fig. 8–39. Assume that a friction torque of 0.44 s4hych fm0)7vqz w(c $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, azf;(t z.z,gyr re attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of;z yt.(zrfzg , the net torque on this system.

  The bolts on the cylinder head of 0)ps gz1v0opban engine require tightening to a torque of 380opzs1 gv)bp0 $m \cdot N$ 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 10xxbgrh uj5t+)z br3 6yf;h8b(q,6vwm .8-kg sphere of radius 0.648 m when the axis of rotazb(v hg6ry85+tm6qbx; fw r u,x)jh3btion is through its center.    $kg \cdot m^2$

  Calculate the moment of inert1 sn -1geodzk5ia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass o skoeg-1zn1d 5f 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 oa7;6:xc ;c9l r*htghxkj -dbhorizontal circle of radius 1.2 m. ckcxbxlr ht9 g;od;6:7j a*h-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 angular oy4 a;hb5ia rbsl4fdo(- b+(w speed (Fig. 8–42). The friction force between a+54o -b s 4w( irybh;l(fabdoher 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 of9 zhhu36tv- mx inertia of the array of point objects shown in Fig. 8–43 abouthxuv-z m9 t36h
(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 tox4+ m31tb1yktc6bh-ybhc j:,ow v1dqtal 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, enr( -7be5uozhwgineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 36 z 5orueh-w(7b00 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 fe t,c9*c6+doc* rpc hcylinder with a radius of 8.50 cm and a mass of 0.580c+9dc* h tfceo*,p6cr 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, ah:m)r 5vah-x m ).lhudccelerating 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 ox* t7seud(.k fn a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume thed*xuf .ek7 s(t 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 10,30:j lopvm+e3f 00 rpm is shut off and is eventually brought uniformlyve 3+0mpf:l jo 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. 8– uo6ihq9.bh;o, rolas9 0lcc*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 btvrk ro(c 8i1b;q+ qmo.) )c+dr 1wxfr5o,vhof the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleratfxw vcb qt.)m8ov) rh 1i,qc15rrrod ;+b+o(ked 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 conszy3k12va j he;idered a long thin rod, as shown in Fig. 8–46. aj;k1hyzve2 3
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,/ bdsoj4ms(mw,j.gg , $m_1$ 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 rlrw t+l j )jxr02rjj(2est within four full turns (revolutions) and releases it at a jj2 (xtr2)wlj+0rjlr 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 oiy1;9bh1 l xt+hbe ;nuf 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 torque of q(m07ahtwnt4 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0x/ )pn:rq ve*w cm rolls without slipping down a lane at 3. :wqx*/v) pren3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with rwi )4i.ckhz i))a 4g-shlaq9u-u0 qhfespect to the Sun as the sum of two ) 0l- )h)-h ua44iwqqkcs z.uhgii9af 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 ar59t*rs vklw ; mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerar;rt vkwl9*5ste 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 rollgj5b6 4s/nx:kv s,qqss without slipping down a 30.g,qs6n4ss/: kx5qvb j0 $^{\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. If, l)h p ujsh zafj7po-6agfpg(*n ;5(t starts to fall and its lower end does not slip. What will be the speed of tua -)z6 (hlh jpg5a ff(g7,pjo*psfn;he upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of ops6-tp:0f6kv18dxl w*ihbja 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m w1k6dp:sfv-th8bj lx6*i po0at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of aokpts.2 ; i/oo:*wdqz 2.8-kg uniform cylindrical grinding wheel of ra:.o2 po/wq o*sk itdz;dius 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 a platform that is rotatin6*u b;s7p,vltkdf- y gg at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0 lg,-y;ks udf67pv*tb.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 her moment58qvi14 r.cx unvlj3q of inertia by a fa513vvil .crx 4ujq8nqctor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rabsyyfq-:roz8hr :lb (c0hn 5z/ f(z- wte from an initial rate of 1.0 rev every 28s w-bf qrc r:y5hb 0(n/(lzzh:foyz-.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical a/y;xz+j otp r90npc05mp8hvhxis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and djn/rt8+yczxphh0m 9p vo; 5p0iameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular m51bizh bq9mx i( g;i*pdo/9r womentum 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 angularh4xbc -s+i .ew momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto a+7y2lsg::jzii vm) 6s4eh;t xjws50mvv t h 8in identical disk rotating at angularhwv ytzvjl s2+i4 h:vms8j76m st i;ix0)5e:g speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsw -,wlvw, nqn- at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stand . l;sz/ aa9qj+ylj0ovs at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inerti vsj l0+q9jzoa/ .la;ya 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 rot; 9/ zq+b7pdhkfffce +ating freely with an angular velocity of 0.8; 9fekzfcp/ f+7h+qdb $rad/s$ 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 whqw,ief2(d +rp0mnv i/uqt 6q+ite dwarf, losing about 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 angular0 ivpq/tdr2w f(e i,u+q 6+nmq 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 i-qivigj,p3p/y l f/1ts; cco6n 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 1n en7nmwn*9 -a;)y2g 3kssutomab6p$ 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 platforss6)bhv3 i dph- ma- -upqsqh,*:gm0rm, initially at rest, that can rotate freely without friction. The moment of inertia 3pah h-u6v:)s hmisdmq,b-*0g -qspr 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 inik5bj:3-7m ii2licv 2yi 5 0j*gcye-m diameter merry-go-round turntable that is mounted on frictionless *e2cl3 m-v 50kiiygji i 7inbc 2i5y:jbearings 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 k4w2t(k htc( nthe ground with the end of the rope 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. Thek4tn(w2k( thc 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 Ear5n :0j p*a pz-f -1ojqunvve,z6egi m:hqob00th such 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 th ,p m0-: 5z0z*aq-v1q6o inge vbjnpu0e fojh:e latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratwao+g w;v/5urf xeb1 2e 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 j)pco0zu6h/k 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 hpj/k0c6ouz) delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two sol 6xx:jou:ur.qid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reach :6ujuo.:xq xres 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 anguta. kpovd 1vn .si+fd e+q3cuz:k1z- 7lar 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 rota6 d8in ipew,gb(knp/ 4ting at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a ngp,e( 8n6iikdb4 / wpneutron 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-pollug .lu90-ibk) jikh)hx tion 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 diam iuhh9lj.kk) b)xig-0eter 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 illustraoe- kee46d ,u;kg2gbn5 qs m0m(+pp bstes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface ou dqlkw,9qwc,+s e zu) yx-+(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 leng)1ka 9,(mqy9ez0qc; q xhkmod th L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horikeqk ;m (h a,dy1 )0zxqcom9q9zontally 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. Ib8wcc7ym ; r 6n2zvk3vt is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a nmy k3v8c v7r wb;2zc6step 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 flat roads)3s nqp oj)2h is making a turn with a radius The forces acting on the cyclist and cycle are t hs3)n 2j)spqohe 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 apn5;2pq;vmdueq8px *vy5 es 0 nd begins whirling the rock in a nearly horizontal circle above his head, acce dpe;50eyx qv; 28 nq*umpsv5plerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skaterpb7 (f xmiu.(vzfqa++’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with armsa(i7(u f xqp.bzf+ +vm held tightly against her body.   

  You are designing a clutch assembly ,+w5f ( k4 bc-l;mvmkqmmf 3zar +a7yawhich consists of two cylindrical plates, of mv( mfk+5mblr wyz7kcq;43m,+af a -amass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopc v,5i ,o9dgjded 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 without leaving thidvo5j c, ,dg9e track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumdbn d,wglhp 8h8b70 b- fow.y)e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revoluti .h+c czs7qjvfj4m/z8/z d 9mqons 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 /8.9qjsz4zmh/dcm7 zvqfjc +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