A bicycle odometer (which measures distance traveled) is attached near nopxzzy,xpw+ nv5:1z 6h9j -zthe wheel hub and is designed for 27-inch wheels. What happens if you use -nw zz ov95 nh:x61zx,y +zpjpit on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does )j q3l jt3fby(a point 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? For which cases would the magnitude of either )fjblq(j3 t3y component of linear acceleration change?

Could a nonrigid body be described by a single value of tj3t twz ) (n4qmz6s8odhe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largyo-b t ;4cs;u u;7th4d3hd.9onc grtder 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 yob+zb, mjuj (;xur hands behind your head than when youxzbbu ;,j(mj +r arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has sel5mg,yhb3 eap.c hz 7+ven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large reh pg eylz37.a, 5cbh+mar 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 fasau*woq6timxo-) y;g4i ja2 d) 0xqgsc)y.+sat have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distrixu 4wxgy s)) );-ao6cy02. aioajsgm+td qiq*bution of mass is advantageous.

Why do tightrope walkers-*; 7sqsfxm m-ecr5- :gm7ymoyvwmu 0 (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is nw2(h61 r(j/ ofcjnnhn+ )lp fthe net torque also zero? If the net torque on a system is zero, (6jo j(ncf /nw21+pflnhhn)r is the net force zero?

Two inclines have the same height but make different olrv+,/znok3angles with the horizontal. The same steel ball is rolled down each incline. On whi+o/zv3,kolnr ch incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fro(mqs0e*a txv;lw) 4i+s oiq3im rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the botto ;t+qwi(is3alsio *x m q)v04em 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 th:zvm3(:6qg m reht*ege same mass. They start from rest at the top of an incline. Which reaches the b:z rgmmt3(q vgeeh:* 6ottom 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. Yet most moving or romhjn()kcpd 7 b r8o,c9)9ejhetating o kpjnj7ecbe,o dc9(r)h )hm 98bjects eventually slow down and stop. Explain.

If there were a great mi+ y.*kb veot(zgration of people toward the Earth’s equator, how would this affect the lengy(+tvzo k*. beth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial zh/yjrd7 ushym(+a ;/rotation when she leavey/; as 7ryzu/+ (djhhms the board?

The moment of inertia of a s4zfzv x-y:/u rotating solid disk about an axis through its center of mass is z 4s-z:/xyufv $\frac{1}{2}WR^2$ (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 rot z x28a5nt*3 ygusqhm0a- g2akating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, willkq02- s2z axg8aat5muny*g3h your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Hxhpskbl p)s5)wc3c 7g,x /lu8mh py/ 4owever, one is hollow and the other is solid. Describe an experiment to determine /psm4u bkyplhg/8 ,h )s7cwxl5xc)3pwhich is which.

The angular velocity of a wheel rotating on a horizontal axp. -fl5r d4xe h,kyjf;+(xxjqle 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 increasing- qfp(5ed.kx ;lh4f,rxj+xj y or decreasing?

Suppose you are standingk(vaqmpqrd /)b(b :c6 on the edge of a large freely rotating turntable. What happens if you wq/( q:bpmd abv )(6krcalk toward the center?

A shortstop may leap into the air to catki f/7qt80 ozkch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look it7k8q/0 ofzkquickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatc;43/t,. vacbjft iv sion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can oab4si/t,c jvfcv 3; .tperate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 *2-wgwmlq /d y$^{\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 coinciden q )(s5v8kxxoj1owjz :ce. Calculate, using the information inside the Front Cover, thekwo( zo 85): jxsvqj1x angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed )ior ; aul+ti 2/mka)qat the Moon, 380,000 km from Earth. The beam diverges at an ang)aurl ot/mk);+ai i 2qle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is tf .pw,6pe 2rszurned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow p,e.6z s rpw2fdown?    $rad/s^2$

  A child rolls a ball o ouqe:ejl(s;d 55 lq(ln a level floor 3.5 m to another child. If the ball makes 15.0 revoluti:sujoe ll(qd;lqe55 ( ons, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travelb 9/1gdeiz8itk., xo g;s 3ysdgg,;pc s 8.0 km. How many revolutions do the wheel, 1 dzg id;ok ;. pgsbtxcgieg8y9s,/3s make?    $rev$

  (a) A grinding wheel 0.35 m in d )(obd:f:hgx2 sbunw*iameter rotates at 2500 rpm. Calculate its angular velocity in 2ow) :bb*(u xgn :hfds$rad/s$ $\omega$ =    $rad/sec$
(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 mak9 4z ryaph*llbp(bc g 2o0 4e03ezy,jdtot-.hes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed ofdh l.peebp z4yy0( 3rcgh*9l t0zj2o4a-,otb a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around thp.x oatrm019rv) i8g re Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear s g g2vgv9b3b6gpeed 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 centrifuge rota.et(b/pubqu(6cfda ):num + xte if a particle 7.0 cm from the axis of rotation in +bfu6ub(t/:q umcd x pe)a(.s to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm topf*wy gmenss j. d*r(b9 4z*f; 280 rpm in 4.0 mgez4 s* j*(sp; nf9w.b y*rdfs. 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 radiustd)(3e2webnqv -nb .u $R_1$ 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 Apollo spacecraft .s hj/u/e 008tlt yd(yba8bdqput themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, th(u 8abdd be/qty 0tys80lh.j/ey 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 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 yyn ;as(gi 8lll+.pm6 from rest to 15,000 rpm in 220 s. Through how many revolutions dynlgi(;.amy+6 llps8id it turn in this time?    $rev$

  An automobile engine slows down from 4500pvm-hbapxs88+ m 0 p8ksx9z:x 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 forc7t xqqk) 4sz 3zm:8n*qb0dvq;fn /jr the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1cn8xr43*qvq7q f: j/k ;dzstnqmzb)0.0 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 uniforml/b (c unt:8ncpy from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel h tb cpcu:/n8n(ave traveled in this time?    m

  A cooling fan is turned off when it is running at 850k hetklw*vj68 on9 ,r,rev/min It turns 1500 revolutions before it comes to a stl 9,e r*vnkj,ok6wh8top.
(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 unifc8 7,qy .rjui/unh 2f(e t-aveormly from 95km/h to 45km/h The tires have a q rn 8hcufutv2y-,e(7./i jaediameter 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 revolutions4lr+-w wz3a j;3q ccvy as the car reduces its speed uniformly from 95q3cwc3;-+ar j 4wlyvzkm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal wheodu mh,yz 4px8kj(fb5jw: g79t 8ba:en climbing a hill. The pedals rotag9mu8jode:wb(,zya4pt b 75 kj:h8x fte 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 qj3e 2+ow 83q(s przbjecq3/ nof 55 N on the end of a door 74 cm wide. What is the magn +cr3q(p e z/w38n 3qje2qsojbitude 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 when 0xs*m8 lh0z wkc9wht;e5q9 yel shown in Fig. 8–39. Assume that a friwyqwt5n z0; 0k*melc9h x9h8s ction 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 massless rod2s;0ee3j-llc :2ktk(qy6 f lgsw0 vr p which pivots as ptlvw0(k0;lg efcl ss2q6:ye- k2 3rjshown in Fig. 8–40. Initially the 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 anf ipk1x80blu7 0v t*gq engine require tightening to a torque of 38 up0b8qvglk t*if70 1x$m \cdot N$ 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-kaft0l8.ndsra4 kn 0(s g sphere of radius 0.648 m when the axis of rotation is 0a4 t. sfnk 8anrsl(0dthrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 62m 34cob3p vpj(rqv(h(d mws99uk ;yx6.7 cm in diameter. The rim and u9 ;c pvr9hwbvqdjp( 4sm((y mo2xk33tire have a combined mass 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 ik :s(n4hs 7lj8 + eaqlziqqi:,;u yi7ys rotated in a horizontal circle of rad7e +h:,i;ns4(iy7aj s qiqyz ul 8kql:ius 1.2 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 constant at-);g lfa peo:0lp +uungular speed (Fig. 8–42)uoup t;l) :gep0f+a-l . 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 inerttc/4ad+an+w c+ed6 hi ia of the array of point objects shown in Fig. 8–43 ab+nhtc+dci 6w4 /ea ad+out
(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 ,pa-iyp a:/0sboe ec6 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, eng(9 0+jv tln3kwzf6 v;da)s;cjsha( ugineers fire four tangential rocks jkznt(9 w+;vcshf)ad;6agv 0 j3ul (ets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a massmvj :*k0so*rmohzttb (g b2)z70el2v of 0.580 kg. Calcugz*z0tb :mbtk20oj)m rv (*ov72seh llate
(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 5qx+e20wsqw0ite;ok bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry za4b g8 ,zpve;)ungljr.f1( mjpai 0/-go-round and 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 l01m( gbperj.i vuf jg z;/8,a)p4azntwo children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and in,p5jj9/kv n ro8 bthriy:o..s eventually brought uniformly to rest by a frictional torque of 1 ty,n8p 5ir jno. j:hv.kr9bo/.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 l cm+)1viw3nnpi) l1x 89zcop 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 throd1h2+bdnt b4t1 (al yrwn solely by the action of the forearm, which rotates about the elbow joint under the actiohrdd t1ayln12 + (4tbbn 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 rod6xleizq9 *2uu7st bb5 1p ley., as shown in Fig. 8–4ey7ep*sx.l b692l z qu51b tui6.
(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 consistx36grv l wxo60s 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 fulltj fn(4 .s;wihqzt9d5 turns (revolutions) anjw59htq.f4; dn ist( zd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momentt u+q j4 rpn7+me-+kay of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a tb z sb93;mbx193z9dp(vub c xyorque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping d n6 t;lfq.k6nkown a laqkln;6nt fk .6ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect cxw s dr0ekbrk*u 98,)to the Sun as the sum of t 0x) k,* rrb9dwukesc8wo 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 radius + fifl: a3*moyof 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 a3 :foml*ai+fy solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withoboekk dq/ 451n(mq(lout slipping down a kb(e5lqo4k/nqd 1o ( m30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tmk)o 3 sntid4,q wpt -h68os*rip. It starts to fall and its lower4dt h ,-8nks3)w rsmq*iopt o6 end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of hri8 n/ idp**keg8w*oa 0.210-kg ball rotating on the end of a thin string in a circlokh8i/wndpi*g*r*e 8e of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of ukq*2ik7arp e l .-xs2a 2.8-kg uniform cylindrical grinding wheel of radius kui.kle2qs- r2 a7px *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 rots3r f p-el;4tzgey2f.ating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of el.;g4f3 ezsy rf2tp-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 reoa7;. n4-fl8rx j rpgpduce 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 r 8 al4.jpxop- r;n7fgposition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial n mi)c56 u5ini5c jc7j q19iimqf*uv +rate of 1.0 rev every 2.0 s to a nij+ ucn5qi uqcj1mim7 fi55*v6)9icfinal 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 af*mz 5v 6 lber1lwmeel 6:2r: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 flatw e vf6 5 lel2:6rblzae:mr*1m 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 momentum of a figure skater spinning at 3z(h i*w:73qsctqtoi e 4c 6du4.5 $rev/s$ 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 thehzmb3.34ler (vypw qmup:f *v ; bk78m 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 dishle.6pb1, qol (3ayf sk of moment of inertia I is dropped onto an identical disk ro,s1ybel36o( l.pfa h qtating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4w99myy:*hsssl *0 i jh $rev/s$ 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 kgh3cfel+5/xdf k* ydo 4 stands at the center of a rotating merry-go-round platform of radius ocd3 dhlx4f*+5e/ kfy3.0 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 5: u+y livme 1jhfccts, u6cgj34,,dvrotating freely with an angular velocity of 0.v5 v1 u sd6jycfleu :,+t,,mc c34jigh8 $rad/s$ 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, lo3+22rvget dphrf 4 5qjsing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming therph dv 4g3jftq re522+ 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 excess of 120o0pii::cuh 9 +dh(2xisu gul 7 $km/h$ 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 ux /s 88- u2bq5h/vkuhnjb-gd$ 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 freely withoo7i 1 (q:9a4ltbww p,dh:dmb-;bt fvgut friction. The moment of inertia of the persbw m,( 9oq1dtpb;f:v-d7it:g4 ab h lwon 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 persiuv; xbtqh,z (5 )fbe5on stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionle(x5) teu;i,zh 5bbqvfss 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 rope lying on thee1 a+nk-g3 ri rwkaqv3d88t 2q58dobl 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 length of rope unwinds from the spool? How far does the spool’s kird3w8kaa-8nt3b5dqe o1l+v g r2 q8center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earnjk2na td e-pk:-t*pu ip c )ank)7 /ctj;3m+oth. 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 partaa3jk/-p nu+p :)* ktmij)7k cpteoncdt-n;2icle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 mo.0e .+ugowph /$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder,r 1caq 3c wd m26k 06wcvtxn6jf3ilu2 of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant ang a63jicu0 t2qkcr 1xlw d 2cfv6w6,n3mular 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 0.0dj)-lc :0mof-s oh(eo50 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 ohms jf (lde-0o: )o-cend 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 ar4,ez4e gt2kangular 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 mass2xmqgb e,6 k7o 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what woue,o7gb mq26x kld 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 avph:8a2.yl pc lbg;,g- wc ti/ low-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 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinuv :w28pa;p hi/,ytb c.g-l clgp.”
(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 ld:ctrvw;1 5i 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 horizonta tjekc3p mf6,3l 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 igno 7h 8r(blse vokf8*w.u/5dlh fre friction) about a hinge ful*vesh8krwbf7l . 5h/8o(d attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of 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 onyj4 on,2ys ye4 the floor, and we want to exert a horizontal force F at its axle so that it will c 4,y2oe4yys njlimb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/3f/t-r jlr(u*co :q bqs on a flat road is making a turn with a radius The forces acting on the cyclist and cyb(/ rq:c* tflqj3ro- ucle 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 beginfr0i85-i4 f g4ir. jgst njls0s whirling the rock in a nearly horizontal circle above his head, accelerating ij -04nl g rjr5s8ft4i 0iigsf.t from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and herie y t,hxocg 48x02-jv arms as thin rods, making reasonable estimates for the dimensions. Then calculate the r0g t,o-eycx 8 4ix2vjhatio 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 consisty exsi2h xh.jcp1 2:.ttb 5pb77, hhqes of two cylindrical plates, of masyp.bpxxc : .he,heh1hbi7tq5t 2js 27s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 thees8 /melppb84 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 of the loop without leaving plm4e8pse8b/ the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but 0us0ydx .21uv 1hp mglcn/pd 8do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its/p+yjt:ybq 3h g/::* k kwbzip speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of ea *zbk/3y:/i :bhq + pk:wpyjtgch 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