A bicycle odometer (which measures glwyt;e5)n8 qzq-2w+ q k-pysdistance traveled) is attached near the wheel hub and is designed for 27-inc )qnygwqw 5sp2z+ k- t;-q8leyh wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloch5f n7 hmgo(1tity. Does 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 eithe715tm(hhogfn r component of linear acceleration change?

Could a nonrigid body be described by a singlep; fnu: :sa1uma1;bmk value of the angular velocity $\omega$ Explain.

Can a small force ever (qym0bj ,gz9a exert 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 difficultn4d*)e 7ucpul*t qzy2a8vu5h to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you7582uu at * n4eq*vhup)yzcdl to answer this.

A 21-speed bicycle has seveclf9-q/:kt qf6gyy1j n sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket?-k1fgl fqq6j9yyc :t/ 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 slendd2hd7 u0wb )ider lower legs with flesh and muscle concentrated high, close to tb207 ddiw hdu)he body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bea(hlw;iry 4z/t 8u8m kjm?

If the net force on a system is zero,hz*x 3ahm** sx2s6 z (rrkgx4u is the net torque also zero? If the net torque on a system is zero, is the net forcea*(h sxk2r6gm4*xu*z s xrh3z zero?

Two inclines have the same height but make different angle7 g l*jxct.c(ds with the horizontal. The same steel ball is rolled downcd(clg7j xt *. each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Oneb;2e5r6rlbdkt/x5a)t4y o tun; 1:xh,zzp jc sphere hne5zht6kd4; t/r2c;r1x :t u)bbolp ,y zaj 5xas twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start oh vo29w;99v d ok 7ojwsx:x5dfrom re59x97s;wkhv2 o :dw9vxoojd o st at the top 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 consn1 k/pbs x*(rid cw((jerved. Yet most moving or rotatin/ jxbpwd*r isk(cn1( (g objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hgy/qfk zdtt4:.rowb4i z/ s, 2 p(8akwow would this tagt (z 4zob.,:py2fksq/wr 4d/k8 wiaffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any /c;2+z.c xx*dlh i 9rcfbhk2oinitial rotation when sh2x .cbz h;xdc+*rck2hoi /9 fle leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cen se -xi)e0hqs :pi,/nxter :s-ep)i/0qhnxe,x s i of 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 sittiyapy vpic6 1-;ng on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity pyp ca -v1i;6yincrease, decrease, or stay the same? Explain.

Two spheres look identical ppof0sim4 x) 2and have the same mass. However, one is hollow and the other is solid. o i04f)pps 2xmDescribe an experiment to determine which is which.

The angular velocity of a wheel rotating o-z ; je1nyvo3p9 j9leon 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 y- 9v93pe nlo ;1ejzjoeast, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edcly0 yv3mj(d ;s(,p ppge of a large freely rotating turntable. What happens if you walk toward the ldpv 0ympcp(;,j( s3 ycenter?

A shortstop may leap into thznp n vka+7p+*8m:aow e air to catch a ball and throw it quickly. As he throws the ball, the uovnkmaz:7p a+w +n*p 8pper 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, 09uxa2 4bb1y vqkf;rsdk-ua .discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.x1ka s2 ;yk.uafd u bvr0-4q9b

  Express the followiny5e2alrmw.dwi*4byjmx , z + -g angles 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 Eart* q9sxvsb1-phl/( fqxh because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular-qlx q b9xsfhs /p(v1* 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 ke7c 3wj0n/jr b( hc ,jj7qe)dym from Earth. The beam diverges at anqc7j he bc e0/7dj( jrwny,j3) angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the moto .)ubjmroh ol2u5lbig 3y fkkwy4. 4k;. 1ap;jr is turned off during operation, the blades slow to rlub1yykp hbu5.i.m.ka4)f4 r3lkg ;j;w2oojest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes eh/k v)rl4ez815.0 revolut)rvh4k 8e/e lzions, what is its diameter?    m

  A bicycle with tires kbf ew8y ;c *9(ugh :h3hdgwv:68 cm in diameter travels 8.0 km. How many revolutions do the w *w;3edc:w:g(hybf8 9hghuk vheels make?    $rev$

  (a) A grinding wheel 0.35 m 9ekg n5aom87l pve7j tyl*)j,;f p: xoin diameter rotates at 2500 rpm. Calculate its angular velocity ie)5tklm;:* xjlfv ngope7,p y8 9joa7n $rad/s$ $\omega$ =    $rad/sec$
(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 onehz /*t.p 4: e*z,8 kxgunkxgom complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from th m*8zg.hxp :outx*ke nk,gz/4e 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 Eaxie 4znq 35pr,9)b gqsrth (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 point3oz9djn nfu s0y2o3s:
(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 centric kc g l.e8t2ji(wc.pj(h*pui9sk/5 mzr 5t k+fuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 8e*p /l+u gtrtcjk(k iph(5 .9iw5kmz jc c2.s100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheelunj9 wye , p5p+ *00in9xnnzdb accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. 5uein xn dnb j9+9zny ,p00pw*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 t-bu7cfsu2;y $R_1$ 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 M96yxsz, hw6cboon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute zx,wc6ys bh 69the 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 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 15,00hn jcjl3957bg 0 rpm in 220 s. Through how many revolutions did it turn in953cljj gh7bn this time?    $rev$

  An automobile engine slows down from 4500 vl1 -2vylk0yh-jh c u-xwmm)(o 11tfszd:q :c 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 highspe kp 5x+v58zgpyed jets in a whirling “human centpzg5 8v5ky p+xrifuge,” which 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 fro7,y ;mjolae i7k/fb bn7:9c xom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thby7b/x7 :icao,ekj7l o;fn9mis time?    m

  A cooling fan is turned ofe ckh / xpx/:l(a6 mrms7/swv/f when it is running at 850rev/min It turns 1500 revolutions before it comes wcps6m:amh7/ xs lvr/ //k(xeto a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniforml ct 8e,2 lytk5f8dj)kly from 95km/h to 45km/h The tires have a diameter tclt8,k ylkfj8e 52d) 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*)*vnyvd: ) eas 2pf xbl)zvp*t5a4x y speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.*xf l epzsxbvn)y * y:a5)*t)d4pv2a v
(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 when c7d zw1 +glbxs3limbing a hill. The pedals rotate in a circle of radius 17d+wb 7s3 lxzg1 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forcep; di4bpdhn. 8 of 55 N on the end of a door 74 cm wide. What is the magnitude of the th4i.dn b8p; dporque 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 s9atdq lc j- nkph*) pa5t8z(8thown in Fig. 8–39. Assume that a friction torque of 0.4 *pk)jl-zn achdpa58tt(t 8q9$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass mn5zj yg yv/ ,ju*pa 8u- trqb/n6xxt:+, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is htr x+yxq-,/g:/*6t annp vbuj u8zj5yeld 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 tighteningmvy +:f0yoix4 to a torque of 38mf x:y yi04v+o $m \cdot N$ 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; 5m f3+gxtay tle4l5j.8-kg sphere of radius 0.648 m when the axis of rotation is thrx4jtf+lm elya ;g5t5 3ough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 c:0/gu ct6hympknjeb:1p eh* w)/( okwm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass yp1g(p)n jhmc ew:k: 6b/*kwehou0/t of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball4en*o1t. s48 iuh.eg qncvn( t on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculacqn.nt*14ei8gth o4 su.e (vn te
(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 spln,*vh81 *brlgs7bzl eed (Fig. 8–42). The friction fo18l hgll **rbvzb7n s,rce 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 of4p ,gkq39myo j inertia of the array of point objects shown in Fig. 8–43 aboug3k4y p9j om,qt
(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 whos e7hpo i :e5;v(a3j g+nmtwz;me 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 cylindriw/mj2qp 5uyd.cal satellite spinning at the correct rate, engineers fire four tangential rocketwumq 5dp 2j/.ys 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 y)*vvp/ dqyik 3qlg (,a mass of 0.580 kg. Calcui(/y)qp3gldk*vv q, ylate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest l1v*y:7j.k37z68sw fkevr wgkf eg6g 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 tangeng o1fnds:45bgf;ce/;p yl(fwtially on 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 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, nef15g 4dogyl e(/s:pc nwf;fb ;glecting 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 ii s* e1q:6qmyfs eventually brought uniformly to rest by a frictional :myq*s i fqe16torque 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. 8–45 accelerates a 3.3l9-cf81e ut ja e zcjyb33g7w6-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 acttqpr*ub-8 5 kn7l*r axun7.il ion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fr 7x. l5-u*qrku7 bnpa8ln*triom 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 consideref4xq00uca ju0d jw9e / z:sq3vd a long thin rod, as shown in Fig. 8–460d/ u9a xf:wqevj0qu04jsc 3z.
(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 m*(oo5 wy o.nbjasses, $m_1$ 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 withinclt2r4s mt 1z 4dr4n(b four full turns (revolutions) and releases it at a speed of 2(mbct44nrs4 zl 1 rtd28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inert ;+psfdwzm5,a8k kg mm) an2o1ia 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 tor: x i1 /,a,thshf, voanx7+xwfque 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 down a r3 +m-c-xs/ lpalg sy- v9c/lelanelvymcx-9+prss/ eg a- 3lcl /- at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic en lw0peq;ta02 vergy of the Earth with respect to the Sun as the sum of two terms,q 0w2p 0l;eatv
(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 ofqsvmf)/,0v ud 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 ,qv /umsfvd)0 solid cylinder.    J

  A sphere of radius 20.0 cm and masst *uzv*3vjq:bro+ 2pv 1.80 kg starts from rest and rolls without slipping down a 30* *vpz3b2u+:jrv otvq.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 verticalu2x)x1 wfc) lh0hz/bjtwk,6 l ly on its tip. It starts to fall and its lower/1wtkxl6h 2,) c)f 0xbjh lwzu 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 a 0.210-kg ball rotatin.ai4g kxpk s7nv:5arp0)6p yvg on the end of a thin string in a circle of radius 1.10 m at an angular sp7) 0sa5yvn:v gaxrkppi k. p46eed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whee1st j/7zwz*, xwvppi*l of radius 18 cm when rotating at 1500 rpm? z7tz vw*xp,i s/j1wp*   $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, oljap, wa u6yihe24hw) 3gu )3yn a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of jha6p)awy)34uy ew,ghlu i23 rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her m:4 d.gjflj4yx oment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rot.44:gl jdj xfyations 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 increa0-tbwc j 0gjvqz.ovf8lhr 1k83b 97vn5 +nlbq se her spin rotation rate from an initial rate of 1.0 rev every 2.0 s toq039frg8tl7 5 vj8.1bwjo bqv0b+n-nzlkv hc 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 axis through its center at a frepx-xk 5t2l1r ui wc1az3 3,pcfqx5tilxcr f1z3-k2wau c3p, p1 uency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure s ujh ;uhovo/ 5 jw3x-)n*0khnmkater 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 56qbi3llyn ;bw dazv.u(t ++u 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 ogm o)f5aw ko+nua+ 9g*nto an identical disk rotating at ann5uwa)ao *g+f km9go +gular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 50khytan )zf5*9m vi lat 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 k. pra-ed55qrstands at the center of a rotating merry-go-round platform of radius 3.0 m and momentqr epda-r55k . 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 fre tn(6s/lcqvwc4 p++a tely with an angular velocity of 0+ caq(+4wtcs vl6 nt/p.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half ij+ao6 t( ptg0.fvj+e kg32gsu ts masj6 3gjeo(2 +ptt.g+f avs u0kgs in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new 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 i3 qfewlqbr )::m ae43in 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 f1dd t(:ylkha4 jh5p; $ 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 restdx gzt//yd0.m 1evd:+b cl4hm, that can rotate freely without frictionhd: lvb t0+1c .gey// mxdzm4d. The 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 u95o co0 :kai 2)x33gomwglxb6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings)x axwmi3ou og59: 3blg0k 2oc 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 withvs c ksl2f,1xu/v(/ex 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. The spool rolls behind the person without sl/fx xvkl/ ec2,vs(1u sipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the sameaki))ntn /up,3 aawx2*wj xh. side always faces the Earth. Determine the ratio of the Moon’s spin angular momexk.)x*2 ,wihwunanaj/a )t3 p ntum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m.nzuj2tmsrnxt o6 jm6u/((l* 1 bo;mg /$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquirh7ijck:t0b xvu 9p+ x,es a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque d,9:v jx7k tb +ch0puixelivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so/ xiwjhq662kc+ klml1lid 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 linear6i612mllk q h+kj/xwc speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of 19-kguurfci+ ;peq q: 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, wao 6hb5/k c, s4fahj .k55q(fiay o5myere rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron starih.6sc o5aj,f5q( yyobkh55/ma4 kfa of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ene2 wr,;ae i3fwnrgy 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 fwen2fari w,;3lywheel “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 illustra -uv9 p7edqyw.1wcn8w tes 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) it w9 n17mkjff )n6kx *j+jjq+u0nl iy)s rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., w fkhmq:*t2snjhk1 i l8a3r;7x e 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, detera2 87rk*kf1lnimjqt h 3:;s xhmine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and wdirn0e60yfy sr 3ea4,+wn6aye want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h,wnriyar0 3 6 4e6ysyfn,0aed+ where h

  A bicyclist traveling with speed v=4.2m/s on a flat road isg1fhg*plq/p0ut 0.x bj njtk7 fju 75+ making a turn with a radi7+p 0p5f/n *ht bqjxgg0ul7.utjkj f1us The forces acting on 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 bj-is 52nk cza(egins whirling the rock in a nearly horizokaz jic5(s 2-nntal 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 does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a sowy,5wvia 9/lawh9a h8lid 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 arms held tightly agaiaihyhww9 8 v9a/a5lw,nst her body.   

  You are designing a clutch assembly which conq3/ixik1* dheafm) b;sists of two cylindrical plates, of masih;1mx qfk)*3ide / bas $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rough track of Fig. 2 ls8l fvm sd 95;y*kp4qps8ggzoy8k8(ln i+h 8–58. What is the minimum value of the vertical height h that the marble musg;8kld82pyvhfy9ksm 85qgo li(zp s+s4 nl*8t drop if it is to reach 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 assume 2wgh eqsaw/gc)6 i;y9s:wq 1s(tr,xh$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the carqew 7pf7pgfb+r0j+ . : ao-ikrvjb9u, reduces its speed uniformly from 90km/h to 60km/h Tv +bjp paef :qwifr.-rg9u77k0 ,boj+he tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s