A bicycle odometer (which measures distance traveled) is attached ne nh2+uo--t64qvxoh 2c aixu:zj7b+e xar the wheel hub and is designed for 27-inch wheels. What happens if you use i :uq2u6xx-zc7jooenh+ bt4 x+v2hia-t on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a fxrdm3ar/+g : th(axxx6 9.vxpoint 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 component of linear .6txm(hdav + xxxrg/fa3rx :9 acceleration change?

Could a nonrigid body be describeiung5 qh,*l1lw9 ksm xl0j l d-/u;kz,d by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater to .pulh)/tj( ts:h0mbv rque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind ,qr 0 .jav:gzqyour head than when your arms are stretched out in front of yorq0ja,: g z.vqu? A diagram may help you to answer this.

A 21-speed bicycle has severi0vk(bq;7vd n sprockets at the rear wheel and three at the pedal cranks. In which gear kdv7;vi(b r q0is 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 l4-s2bwn - lk/bwc/6isqkr +mf ower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational -mfq4ksk/6rwb-ins+c bwl /2dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkersp7 mij:/ak 62xzllj-o (Fig. 8–35) carry a long, narrow beam?

If the net force on a syst w5:+ig)7mcx 3 yhrynt4ta;ghem is zero, is the net torque also zero? If the net torque o xtwm ih htc)3r+gyy7; 5an4:gn a system is zero, is the net force zero?

Two inclines have the same c0 wyue)17q2 bjiyx+s height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball)ecjw+0 1 2by7xy squi at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling -hw 1c3kp*sgi6 xh 0nj(from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which h wi0g3chnks xp1hj6*-as the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius an/igjej. 6dp hakke4 ,8d the same mass. They start from rest at the top of an incline. Whichpk46. khija e/8g,dje 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 an)umu:z)ux6a 4mi bl + y+dhuc*d angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explb+4+*l )6mu umd)uay: hxcuiz ain.

If there were a great migration of people toward the Earth’s equatorejisl f3t g0 *g8dcg.9, how would this affect the length of the9. 0tldj8fc3 egg*igs day?

Can the diver of Fig. 8–29:0 g 5peeb8chhl-m1su do a somersault without having any initial rotation when she leaves the bo-bhl: 50gue1s8ec mhpard?

The moment of inertia of,m hv10 p,.up .feqp q6bev:mv a rotating solid disk about an axis through its center of bv. h ,e,: q.v mvue10fm6ppqpmass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in ea af/p+- w10glgy7gbl rch outstretched hand. If you suddenly drop+lgfpgaw lr-/70 ybg1 the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same massaua ty5;4y/vd .7u6u vdt dl+w. However, one is hollow and the other is solid. Describe an exp+avlau ; uy d7.v46d5 tw/tuyderiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west.1-n)f( o rnzn xq7 ,:m ofdfwcp)(ne3d 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 or 3f)p7nx) :d(or(nn1nw- qfc o fmd ez,decreasing?

Suppose you are standing on q6 t4x.pwramhg*,enz8s p:/ lthe edge of a large freely rotating turntable. What happenwps, apte4zrl:*q m/ g.68nhxs if you walk toward the center?

A shortstop may leap intoxex p(nr h,(+ gx.icn) the air to catch a ball and throw it quickly. As he throws the ball, the up(e) nnxrc.+gpix ,x(h per 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 angu :xqqt7kap-(cc h .zl+lar momentum, discuss why a helicopter must have more than one rotor (or propeller).:x7 (z +kphtqcl-a.qc Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angleh. emadf;a/bb j+ a:a8s 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 cotle 8,b 4ficv )*zj2cf pan7/zincidence. Calculate, using the information inside the Front Cover, afb4, / ce8znzlfc*)vptj72i the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00lg .5szd 3d0y(hz a9xw0 km from Earth. The beam diverges at an g5 .(xyddlwha3zs09 z 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 motor ispu0q qtfn/5 tzk:(vq. turned off during opervp/fu :tz5n(0.q k qqtation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mab)k*2qisg:7)einy8vdu* qo6 8tjkr)l) ofu ykes 15.0 revolutionsiyf* ko8*7ou r bqd2:)t i))jqsuk6yvgl)e8n , what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ho i(0/0us au6w qhh9obvw many revolutions do the wi/6b9avsqho(0w0u uh heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2 i xuwvvpl100nif;9o-500 rpm. Calculate its angular veu vi0ix0w n-lfvp19o;locity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one b;hk6g2.eemfcomplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center? e;bf6k2 heg .m   $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a)mr 6gwdln/ c4og 8f*q; 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 po13w(r7cf sw jx 0 adsel1iij1g7 bhl2,int
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axisa49j r; uk25xelkph -a of rotation is to expp9ej5a-k4 k l2;aurh xerience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm tuagbhia83mj ib+ i30o vc*p8+1myt 7o o 280 rpm in 4.0 s. Determine 7htbmum* 8i1v0 + i38opag oi3bc j+ay
(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 radiuvaybz lvx tg+1f 3ws6ce.21e :s $R_1$ 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 pn w bd;4fx8 eogn/ g26rpy1-mer-d.llut themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thourr2g6;d8 mb p nx-.lgyd of/41nw e-elght 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 unqff;xzl;q69q l( *xdb iformly from rest to 15,000 rpm in 220 s. Through how many revolutions did itfxf 9qlzdx;*b;qql(6 turn in this time?    $rev$

  An automobile engine slows d:pum:ne ;oc6l own from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses2t *2 fum7poqt95paug of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 mippa22t9gm5u7uf*t oqn 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 rpm in pod /dw.b3wx (6.5 s. How far will a point on the edge of the wheel have traveled id(p od b.wx3/wn this time?    m

  A cooling fan is turned off when it is ru3y3ycw. eq235zqltiy nning at 850rev/min It turns 1500 revolutio. yqwezlt53qy2c3 3iyns 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 revolutions as pvj.d8x a90njr f+xwl t (fdr-*o tn80the car reduces its speed uniformly from 95km/h to x 08-nlf 9xrj t p8fdv0rntj+ d*woa.(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

  The tires of a car make 65 revolutions as the car reduccl0f - an,wpy1es its speed uniformly from a10lfnw cp ,y-95km/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 a h v-bobvb(.wgz()0von each pedal when climbing a hill. The pedals rotate in a circle of .o vwbh0)gvb(abz(v- radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55pxx+y; e doz7ztg:(,p N on the end of a door 74 cm wide. What is the magnitudz e7(zpx:o+yd, ;p xtge of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig+b 7cj7g82 p1tmg uzo*+lz:j q b:nwum. 8–39. Assume that a friction toz wmu+cpuj77:bl j*:+ ztq1m 2ng 8gborque 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 masslesyco4or;m;6c )pgn, ais rod which piv;; rm6o 4 cic,pa)gonyots as shown 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:57k v(cuk kzva +te5t an engine require tightening to a torque of 38a v u7k+5(ktvtkc:ze5 $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the yj((1;ba7 wj+uwgm ph axis of rotation is through its1;w(7mh g (jujpb+ay w center.    $kg \cdot m^2$

  Calculate the moment of inertia of a big2)5q,ac jaigphga-gi6so ( 7j u) 5hmcycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub caa2g u,ihcg5po -i6) 7g(m ajj5hsg)aqn be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatfjgn6iqaa0 4gs9 yp/(g.,*hoz/ j gd ced in a horizontal circle of radius 1.2 m. Calculateo *yf.jga ggsp(,qd9na 6g4iczj/0/h
(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 a bly1 ;nwmo9mr px*8l4t2j7e,ijl4xo t constant angular speed lon 4i29y;t*xo7 wp1ejjlbx4,rm8l m(Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment oflpg s 88wcfs0i+oi:)j bq6p8 e inertia of the array of point objects shown in Fig. 8–43 +qw)jl s8 :ofcies i8pgb0p68about
(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 h ajkzd9r,g:5.asg,t whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindri1;wvfpubw:a7os2pr a:/)xaq r g zf5 )cal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a1b/p:o usrwfxawzv)g p5;r a) :fq2a7 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 g/ pf:rchgogt+ 0l 4)eof 8.50 cm and a mass of 0.580 t pe+c):g0gh4l/orfg 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, accelerating it from rest top*4y/bm 8jbz9 .t lceq 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 tangwn* y:o)nd;pq entially on a small hand-driven merry-go-round and is able to accelerate it from rest to a :pny wnd*q);ofrequency 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, 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 is n.5bna3//lvry wvf * weventually brought uniformly.arf/vb w*vnwly3 5/n 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. )qeyv wqb sp( md8oeua-cvw ,8302cedbpf, .98–45 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 *mzy, .gdg/ izof the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated *gzdm y.z/g, iuniformly 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 0dw/atpktd.5thin rod, as shown in Fig. 8–46d.tp /a t5kw0d.
(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 hbd 7y3u4et ).,:h2sv csy bjxconsists 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 frdzxxvoi7u65 rt,l ,;som rest within four full turns (revolutions) and releases it at adx x7lz ;u,6 rts5,ovi 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 momen-k/dld*plw -dt 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 torque evcg1l/ txrg e0ln(+.fbg w*8-mp yol5 b:ix.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/f j0dj9vi gs, radius 9.0 cm rolls without slipping down a lane at 3./g0 i 9jvfs,dj3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of 8n3bev 8ibu1e zon0c: : igan9the Earth with respect to the Sun as the sum of two terms,:i 91c bue 8nn :3eg0obn8aivz
(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 ouro u0slt7vc- d wm24 w+1-iv5ssoqk7f 7.50 m. How much net work is re 5w-omv7sv4lq1ow0kudu cs -7s t +i2rquired to accelerate 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 massgzn4 furo: d8d9f-/ wlf5c /a4fgkc) br x-q3e 1.80 kg starts from rest and rolls without slipping down d3uxw c- kff nf zc)g5/a-gr:o 94/qb4elf8rda 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertthy5-hqwirhsv-w ;e,lfe 7 .. ically on its tip. It starts to fall and its lower end does not slip. Wt h57w-,ee wl -iqrhyhv.; s.fhat 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.ym4tlmu95wjii .:o h-210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed olwu-mj4oi9.yi 5:m t hf 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel gu7/b:v a(ul xof radius 18 cm when rotating at 1gv/a:u (ux 7bl500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of /3jp +9zou yrd 6)qxuz1.3rev/s If he raises his arms touuy) qz d9jx3p+/zor6 a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertik:f hq4t 4z+bea by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what i4+b efzq t:k4hs her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rzlb+ ,m tdh n;k7r(;czotation rate from an initial rate of 1.0 rev every 2.0 s to t;mzk7 +bhzr ,;l(cdna 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 cent s z991dfwts;fer 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 shaf 1stfsz;wd 99ped 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 skater spinni9hoi(sm )7adz1z) yp tng 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 momentugxc zph aab5,/x*ki;t 6dl7h-m 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 oa42, 2eptpzy ,b(r 6iei;nwuof inertia I is dropped onto an identical disk rotating at angu 26a zro(we;2ib, y4epp,itnular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 p8l ;kq(v 6nuc tpb3hv:lm r0-$rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-roundz9( tkf(3 g5 cebu8 ooxks:la1 platform of radius 3.0 m and 5t1 kxa3(k 9s cbe8ogz(f o:lumoment 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 ro5 ra 5 xkkfmlp:llwd9l:-qqk3 oygx874nbh4 ,tating freely with an angular velocity of 034fqy-rk 8xh9n d g7o xl:a w ,qklkp55lb:ml4.8 $rad/s$ 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 col 7n7qpv0 hlrk;lapses into a white 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 0q p;r7nkv7h lcarries 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 l/ hvu.qx,vo4mt(beiy6 c*f r.1 2j guwinds in 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 ,2k :jg kslm0q7v.mkbp7 mb t.$ 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 f inhpglqe(myh)p+77k, 6 r:v without friction. The moment of inertiq76:rgl7p) ihveny(,f kp+m h a of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merrp282/ol py.(3xlb 7)dk8mgvgpzn bxz lmw b*1y-go-round turntable that is mounted on frictionless bearings and h3bv*m.)p /nxpbp8mb w lz(zd2g 27 x8ykolg l1as 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 lyinfvl 5: ta7zy/l+7jrksg on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? Hovjl5:7+ k7afz/yts rlw far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. dyt3ac psqicl)t.2767)r yetc)jlj+ Determine the ratttpyil)t .)7cl7j32+a )j dcqrycs 6eio 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 particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates f;3hupx brcxp m425cg3rom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinderhrz .97o 4ksyaua 4oc, 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 deli. 4r y 7,caak9o4houzsvered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0wj+xz bo9c 3nm: v07tyi8oll2.050 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-vo79tz bin0x8wl:yc o 23j ml+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 the2ib n7ju+* cfa 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(7ji1quaasb ht ww(3vg- /k(p 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 would its rotation speed be? Assume that the star is a una(a ph -tqwu3gwij7v sk1/b((iform 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 energy storegj( * /wng (-kgw,ziqza*ur4x d 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 g4 a/z g-wxwqr*jgk,(i *uz(n, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an ixy1n)g;) ge k$H_2O$ 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)bmyyg mm+ge35* 9nfd;ph8 b7 l is 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., we i;z :xejvjg(k3epo x:;gnore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the 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:jz;e:k3ej :xxv;o gp( See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertica ym whzh xy0tki04:*m*lly on the floor, and we 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 4ymwy h0ih:0z* mxt*k height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a tiui 9o i+8x;v crhzhy7e tq r5872*iaturn with a radius The fiyzi h7teh qv*x 5 r8a9i8 2u7+co;ritorces 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 slim bbx: 4i2hc/.u se1ozac 2)bbng of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requie2 :hz) sbbxb4bu cm 1oa.2c/ired to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyaii.+a.6 gf tvlinder 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 arm+ag.ifvt . ia6s, and with arms held tightly against her body.   

  You are designing a clutch assembly e -v7nw2kr awxmtk-)3 which consists of two cylindrical plates, of3nt2 )w -vmae -xrkk7w mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 l r+g4mj 8bwz0iooped rough track of Fig. 8–58. What is the minim8 wb4ir 0j+zmgum value of the vertical height h that the marble must 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 dxl, op5k w/piqt)c(m 7o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as2.fa 6z5,oy +x0nt g6ur akbsw 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 tire? ayxswao b25uzkf 6 ,.6g+ ntr0$\approx$    $rad/s^2$(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