A bicycle odometer (whk4 f vsk:5mu/zr0f, ahich measures distance traveled) is attached near the wheel hub and is designed f/u z4kfa5fkmv:0,s r hor 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the ric8fgqdal5j ha ;89 0scv/oq *tm 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 aca */c tsv 5c08a8q9h lof;jdgqceleration change?

Could a nonrigid body be described by a single8*f5h m dq:f4hj5 wkne value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thangu)t - .+zrjfpc*l c6vmb84d t 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 dok** n;zop f,,gars(tr a sit-up with your hands behind your head than when your arms are stretched out in front o(or,;fp*nszagr t,k *f you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockyr*56 vdn fc.9 bgp u4rndh) 6mvjup/5ets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear d u v4c)brf y6d5/jnp 6nu.p*m 5rv9ghsprocket 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 ;o aktf ig r0*ru-7xj4run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics,j xiokufg a4r-r;*t7 0 explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beamb*gb( ydm ;4 .sgkvy;r+3 bnpt?

If the net force on a system is zero, is the net torque also zero? v +tjy0/z)(+vi nj a-m9nqk ll-urin4If the net torque on a system is zero, is the nlzi+j jyuiq /( -9v0natlv+-n)nm4 rket force zero?

Two inclines have the same height but make different angles with the horizontal.j5 ra g5s20mzf The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greata20 5zsmgr5fj er? Explain.

Two solid spheres simultanek3k f 878nj:iwhhmx-h ously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the f3hjx h -m7nik:k88hwother. 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. x).lrtq:-j.j lx z;po They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Whi.;.jl j xopr-q)zl: xtch has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are cons*0c i+hh :p-yw9xl fmnerved. Yet most moving or rotating objects eventually slow 9lnm*chx:0hpfw y+ i-down and stop. Explain.

If there were a great migration of p*/ v;x8l5(uq qt5hq i,acy hnseople toward the Earth’s equator, how would u/ i; nl8q*qqx(5yvchta5, hs this affect the length of the day?

Can the diver of Fig. 8–29 do a somersaul 7wd9hvpv .6bpz/s; 8bl sc64uhb:-wnx)o qmwt without having any initial rotation when she leaves the board?n)uh4s v mxcp6:bbqd 79;w 8 -zo6wlw p.vh/sb

The moment of inertia v aky v. /;nwf5(ja8zjof a rotating solid disk about an axis through its center of mass is/;jvjayvka f .zn 8(5w $\frac{1}{2}WR^2$ (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 ma;d3q3z4n s kncss in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrea;33k 4ndnzsqcse, or stay the same? Explain.

Two spheres look identical and have the same )-z f ke,xk1toijtf5f+av9p 6mass. However, one is hollow and the other is solid. Describe an experiment to determine which ifvfef)- 6k toxz 5,9+jpi1tkas which.

The angular velocity of a wheel rotating on a horizontal axle poin;ouf9am(: bkz ts west. In what direction isfm9 ;zau:(bo k 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 decreasing?

Suppose you are standing on the edge of a large freely rotating turntab5m5nbwth25o+h2qkd phgck(xk 1s ;f* le. What happens if you walk toward the cent km( ns2f2* 1g5;5ph5t o+wcdkhqbhkxer?

A shortstop may leap into the air to catch a ba(bnbc-oj i gj.pb rh45.(d,abll and throw it quickly. As he throws the balrj b4 n(ba(.,gcjibbdp-o .5hl, the upper 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, discuss why a h 7kw1qvie-4rn elicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.win7 - qk14evr

  Express the following angle)b6e-f hkwedo) 16mqk s 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 coincidence. Calculate, s73;5 car b*2bwfiog wt*ta2rusing the information inside twwt a grrs *t722baoc ;*if3b5he Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mom.+w q1 ac+gidon, 380,000 km from Earth. The beam diverges at an aigq.+ dcmwa 1+ngle $\theta$ (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*vi1 b+s-wktozo f 6rl2p*)pj aog9+ v of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the b * glvza 2op+t o19o-wpb* )vifkj+6srlades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball l(.ify1mq/ oy makes 1(1 yf omiql.y/5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diamet+y,) fyuhkpo35j f)i ter travels 8.0 km. How many revolutions do the wheels m j kho,tfyi)yf+5u) p3ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculaf8 :kgn3zyhk 8te its angular velocity in :fyzk8hn8 g3k $rad/s$ $\omega$ =    $rad/sec$
(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-rou30bf.lm g0euxnd makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear s.xf03me gl0 ubpeed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orjg2mu / pbtzd* h98er;ay6vj/bit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp;roc ulp)+aq 1eed 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 centrifuge7kpa9hr,z5i(v 6a wpzpz w*f 2 rotate if a particle 7.0 cm from the axis of rotation z,z p*z 6pw5k2h(iawvfa7r9 pis to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fro/3p-jc yaavbinjuej //:nj99 5 wvhq8m 130 rpm to 280 rpm in 4.0 s. Determic/ janjpuv:n 83j9a 9hve//-bw y5iqj ne
(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 radiuf(ch( ra/s,belb8 uc(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 the4fb 9ozad93 fzbh97k 69rbxqw Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenlyfb967ax hbq4kfw z9zo93r9db . 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 15,000 rpm in :easi 1y2,onm220 s. Through how many revolutions s,2o a emy1i:ndid it turn in this time?    $rev$

  An automobile engine slon1a ; l:g3slrq/+t+ob(kdg z xws down 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 ;nvccgj:5o03maj)ju stresses of flying highspeed jets in a whirling “human centrifuge,” which takesmju5o:; vcaj3cj ng0 ) 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 w8yd ll -bcit0yx72y 9accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a xycltd b-y278y0i9wlpoint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when qydwb(h+xnf3+p7-s r it is running at 850rev/min It turns 1500 revolutions befo+bfrw(hpd s+ x-73qy nre 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 the car reduces its speed un a ap4yq24;cxs5k wq:oiformly from 95km/h to 45km/h Th y:k2s4xqq4 cpaa;ow5e 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 reduces its speed uniformav ql7o*t uz0;/sf0yg ly from 95km/h to 45km/h The tires have a diameter of 00 f7atq0zg;*o u lv/ys.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 2xds*uk uz-k3m:gerz:8k n c-all her weight on each pedal when climbing a hill. The ped e*:uuc rng8k-mxsz-kk 3zd2: als rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on dpqgk7fp)8kc6n en 7d xqu417(yw5 i tthe end of a door 74 cm wide. What is the magnitude of the torque if the force is exertekw 7t74nydnf )q(de1ck8 qi5g ux7p6p d
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–o8p ,x5eqb;qe39. Assume that a friction torque of 0peqox,;8bqe5 .4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mas3z3,bj(j u ovls m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate u v(l3 3jb,jzothe magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tfdh0xa9+w1.dig h1g q ightening to a torque of 38gfag.9dd +ihx0wqh 11 $m \cdot N$ 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 iniynuv +jt y,o v9.q,ma25zj u2ertia of a 10.8-kg sphere of radius 0.648 m when the axim+jn, ,oav ty29juyiz v5 u2q.s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7:v a ,;hn3asn+qejx+k cm in diameter. The rim and tire have a combined mass of 1.25 kg. The m njh: 3ev+snkax+ q;,aass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizontal71/caj *on *g645yxkoqh g cql circle of radius 1./ o4l67 yhojcxqqc5 **kagn g12 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 a-)f :0:7eqz di4c qzay+d zig1mia i;ongular speed (Fig. 8–42). The friction force between her hands and ti ; +maaq1 -::eoi) 0qcz4 zyiigzddf7he clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia 7zrmt bz/m29 hof the array of point objects shown in Fig. 8–43 abo z 79/ztmhr2mbut
(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 total1j cf7cqgfa4 52jwb0z 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 swkq wm22u 6s2dpinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 36 2mw6usk2wq2 d00 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 ot (88.6 f d:r uh-izos irfxkk7 pzoae);6rnr;f 8.50 cm and a mass of 0.580 k o ziz7safrf):r;nrp-h (6t8xe.;8 6droukkig. 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 swingj bhcl:lecwix; 6)0k9 ++p.b2ztjek 0ac ml 5ls a 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-go4zxd tfal,g;,c xkt8,s -7u74gfthk b-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 acceleratiacl4g sdttfxh4tzgf-7,k;,k,x8 bu7on, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rot 15xgc,zo +1tje-oip tating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a fri zpc+5j1o- g,otixet 1ctional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates o7mk36 lruh 2cg,n5bca 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 of the forearm, which hrd.zz+ gg ;f6th ,b7vrotates abozgdf v; 6hbhg+,. z7trut the elbow joint under the action 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 rod, as smu8fvu9rf + 71,)ef*nk 59y af shaucb3,mpedhown in Fig. 8–46.mdmy,n+c79f f bae1 u,p )k9*s5v8uaehr fu3f
(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 o+oyb9hp+ nae3f 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 full turns (x7so uv(u+- xz3 vh5pbrevoluti -vu+ 7z5subpo x(x3vhons) and 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 j l7kq;.wzb) ha moment 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 of 280 68y9r+ b cxscpzw ydw7x a /ogl6k60:a$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass m /w6n0cc;quc5oz lh ,7.3 kg and radius 9.0 cm rolls without slipping down a lanlz 50hmn;quo 6/c,w cce at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of t) pl3sm qi4 d(xe;*ljks(qd (ghe Earth with respect to the Sun as the sum of two tk(e *s 3qsqm(jg4d)x ld( lip;erms,
(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 of 7.50 m. How much net k2 d 0w*wb7x 6rj6q wb)y,ysb*uds4w fwork is required to accelerate it from rest to a rotation r,ky0wb*dw u6rsb2* )7w6 jwd sxbqyf4ate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg startd73d hm. q34f1 hf exf 4db7ltd2cbs3wl2hx4 vs from rest and rolls without slipping domvcl347x 74x .ltd24fbhdq 3b3ffd1 h 2wh sdewn a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on itsb+:ig(rk7ma1u3tg r e tip. It starts to fall and its lower end does not slip. What will be g(k+73e:tmr1ubirga 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 fm9x -zr3-ijia 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular -ir3 fm-x ijz9speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.86jz6kx7ls8 te4m2 kns -kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 15k84ms6 s6z2jt 7lkexn00 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 sij9(;bakbr+zx t/v )hriwc t wds,7i7 0de, on 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 rotation decreascsv9,kh 77x jr)bi tdrt0aiw z+;b(w/es 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 herv s 9i:y:7zwah 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.h9wiz vy as::75 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotayy)a :a: tc+m4xj 4coktion rate from an initial rate of 1.0 rev everymajc4+xa : y)okc4y:t 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rowd81;rzr2xo 07 zumc/o ,xxij 9oppj0 tating around a vertical axis through its center at a frequency of 1.5rev/sxdop9jrz xw/z cim,o07u8;r1 xj po02 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 f 9xo7g36eexqoigure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular mo1. 8eyq hgru-cmentum 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 dh4,kc -k k1mz asswt yk+2.u8cisk of moment of inertia I is dropped onto an identical disk rotating at angular speeds1-kkw +hs2kc4,tm yua.c k z8 $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsj+wh4m,+ob se5 gudt6 at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plat ahl6q23z: ymmr15s;rlh*wvc form of radius 3.0 m and moment of inza v:2r*;q6cm1h3ls ryhw lm5ertia 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 fbf g9p,zute6 cnx217-go-round is rotating freely with an angular velocity of 0.upg29c71 ,bz 6nf efxt8 $rad/s$ 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 wm 74i u6on( p8cmbycn.hite dwarf, losing about half its mass in the process, and winding up with a radius 1.0%4i7nym cp68( o nuc.mb 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 w qd+2vc9+mptw2nx o2 dinds 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 1ykmnxf hs0; 2$ 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,j*xmv-*5. no a:dik lt n7(agd initially at rest, that can rotate freely without frictaol-td7. ix(avgn mn5* *d :jkion. 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 persq1fh0t5xq jx ix.(ze:on stands at the edge of a 6.5-m diameter merry-go-round turntable that is mountxx.1 iz0f ( hqjxqte:5ed on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end7vey1qby.tz( c + njt0 f1rmo3 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 thoyr qz3(1+j n1f7eyt vt.0cb me person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Ea+ mv*/vjwaah*dh 9ajxe 4/su-rth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle / /au*vhja*4m dx+ehswa -j9vorbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/t/ +waxx)cf54ellm8 i$s^2$ 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 cylin fs.5kld+-c: g u7fxjey1)ngnder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular ac:xd.y71+ -lc 5s)nj fgfgekn uceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical di,r p6s/rk yqvm)-rcu9sks, 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 calcular m ur/v6csp-k ),ryq9te the linear 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 speedm7k jo19c swt5zpxy/ 7 of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as gw3e hv(zacyqnv20 63 nreat, 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 uniform sphere at all times,a3cn v3h yn0(qv2ze6 w and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution aut 5zwgp+bi85 0f+1(bu/oyjvys i4 ycz eomobile 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 flywhgzyb(i4b1y0f5 j uc e++ 8wvy io5zps/eel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustra 2)mqfr-bw2 khtes 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 horb7 q b)gsw3+tnizontal 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.ea u .f2qvuay upp)-y2o8p nldk,:u:g8l:n6 vz., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment n8gdnuu ypuz l:yua8:,kf-qpvl p22o6) :v a. 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 vertis5j-.h ykwaz0w)9 ybdp c(f3vcally on the floor, and we want to exert a horizontal ford(0s bk zj f5hp)wvy9.-3ywac ce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with sphyu0 j9bb+b l+-lhex ;eed v=4.2m/s on a flat road is making a turn with a radius The forces acting o+bx h0 l;9julehbb y+-n the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins wh3wz 0nd ;dggyag; te*(1fju3 cv/djc8 irling the rock in a nearly jw*uc jf(yg;/ c ze3g 0n;ad83gdt1vdhorizontal 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 solid cylinder and her arms as thin rods, mai qktvxrrtu n7*(q:) *king reasonable estimates for the dimensions. Then calculate the ratio o(7i)rrnq vq *x*kttu:f 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 consists of tt* r:dwbzt yq0q3(;k jwo cylindrical plates, ofz3 0t:*qb kr(jtw;yqd 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 kp:9e0te 1m ksrolls along the 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 thetpk9 0e:eskm 1 loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butcv v,x; u0.r )ya6jf1mzwzh x4 do not assume $r\ll R$ h =    (R-r)

  The tires of a car mak.p/;j p2c 8l b.bksvr.ttcd p:e 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angu;kp/tb2 plpc8c:b .sd. rt.jvlar 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