A bicycle odometer (which measures distance traveled) is attachedi,v;u j9s zxe/ 0vpt3q near the wheel hub and is designed for 27-inch wheels. What happens if you use jx0 i/,ze3v;tvupq9sit on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point ; t; g2plmcr mc(qvm),on the rim have radial and/or tangential acceleration? If the disk’s angular velocity incrc;l2m ,gmc t)(qp mrv;eases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angularyxjfpk b,:1f5 velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger fonbdm64 o(/0 ekzi hcln4) *kehrce? 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 yoh7u3xy) 2e4dnf.:hwizn- lv sur head than when your arms are stretched out in front of you? A diagr3vihxzfe.w- )n7 sludh 2yn:4am may help you to answer this.

A 21-speed bicycle has seven sprockets at thvdm lm0r(6fjb l6fas ux3: )dh*1a8lse rear wheel and three at the pedal cranks. In which gear is it harder to pedf8rl3dhsu :vxf*sm1 d()lmj6aal6 b0al, 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 beingc d/kloopg 4u6-0zl5 8agg:i r able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dist8ok5auz0dgc- ll 4 6r/iogg:pribution of mass is advantageous.

Why do tightrope walkers (Fig.h4 jfqh)7p3kw9q3gpa 8–35) carry a long, narrow beam?

If the net force on a system is zero, is t24y.7daqs0gq17 zp 2elu,: ljqtneb yhe net torque also zero? If the net torque on a system is zero, islpd14t yugnzyq 0eas22 j .7lbq7eq,: the net force zero?

Two inclines have the same height but make different angles withubxgm+6gm2ej; m ex+ /onc k07 the horizontal. The same steel ball is rolled down each incline. On which incline will the spee jmbne602k+mgo c/7 guxm+ex;d of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fr: t, hv 3qn+rrp3enr2gom rest) down an incline. One sphere 3hpv+rrreq n :23,t gnhas 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 fta7em. t).(zbkz binc7m. t i*rom rest at the top of an incline. Which reaches .te 7.kz7m i *( czt.)mnbaitbthe bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving orkf7 g7s8lwp5r 1so*fc rotating objects eventually slow dow7osws 5fpg8 1k*lrfc7 n and stop. Explain.

If there were a great migration of people towa+bgn 8wb yi3goi, c6jw .5q/ptrd the Earth’s equator, how would this3i qiy 5 ,g.8cbw/ontpgj+w 6b affect the length of the day?

Can the diver of Fig. 8–29 dekj5m*dju z 5o. 7nst)o a somersault without having any initial rotation when she d*omj57zn5s k)jtu.e leaves the board?

The moment of inertia of a rotating solid disk about an axis through its g3j-.g a v gj4(csgjw3a r0kf9center of ma9g3j.0cggs 3( jagvkjrfw-4a ss 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 s+;wrtmj s hk qnd: p05c.,t1pune 7zn+tool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explai: wrk ,1nnmc++z.p j;dh 5unt0 qes7tpn.

Two spheres look identical and have the same mass. However, ;nkw ,hecb8v6 one is hollow and the other is solid. Describe an experiment to determine wh8 c vbhen;,wk6ich is which.

The angular velocity of a wheel rotating on a horizontal axle poinmuar )lk-jb6 u4lb.l /ts west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceler-.bl) aurlj/b 6 mkul4ation 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 turntable. W9pkni3jk 6q7jlb-5+dp g2 txbhat happens if you walk toward the center?npb7j t 93xkj kdqipbg6-+ l52

A shortstop may leap into the air to catch a ball and throw it j3f.q7 5ahe.x,wcim lquickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rmx3cfhjl.7iwe 5 q .,aotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law o/4auwsm3i 3 vkf conservation of angular momentum, 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 stamui4a wv/k3s3 ble.

  Express the following angles in radians: (a) 3z8d hti h:t q,ndi/;x2eo8z1 w2mno*v0 $^{\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 ofh. r8 g(co (ay2m;jovl an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radio;2 .lo(yr(hvmgcja 8 ans) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mobjg3o a kj/5r43i2vb ff:m-woon, 380,000 km from Earth. The beam diverges at an angle 2 rk4bjw f fob3im3v :g-j/5oa$\theta$ (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 r0kj)7sqbj8emh - 6c otate of 6500 rpm. When the motor is turned off during operation, tk6 co70bjj 8semq h-)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 bau/qajm)l55 o n:i5f uvll makes 15.0 reafm5iuoj: u)/vql55n volutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8..7kr lo:+dv kn/t4 )bcu2bkno0 km. How many revolutions do tc uobbolktr/4nk) 7 d.knv+2: he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate vkeexe 4+(n.4p gj j5dits angular ve enjek+5v4(e d 4pj.gxlocity 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 complete 8u)nt+lrf kh xkuq3/,z 8wcx( revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 /8c ukf(zx3nq ur, l+wxtk8)hm from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocitf)t bx) -dte4ty of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of qdaw4* d:fgt: a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate iffd0w 28i hl1xr phb;lcq .4/ agu5hre5 a particle 7.0 cm from the axis of rotation is to experience an acceleratil ; 5g5brhrc.042aq 8hw1eihlpfdu/xon of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about kk*q. e1gxn ,vm 0fj5jl 4ovo/its center from 130 rpm to 280 rpm in 4.0 s. Determnjg kl/o5mkv1jxfo .04,v*qe ine
(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 radiuguhd 4qc7n,p ,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 put sf:e f7h-6 qbwthemselves 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 -f7ef6whqsb : 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 tbt:g*99dznili gk, /lo 15,000 rpm in 220 s. Through how many revolutions did it turn in this ti/9bl:kg* dni9 z,glit me?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculalghuant( ah ;- ggu66k r31q4zte
(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 fl.smjhyr*o+ nqa q 276f9j-jsq ying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 2o 97y-js h.*m 6qr2qa jnfqj+s0 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 un ;jo1xp,(6 ls+y/leiff5 e elhiformly from 240 rpm to 360 rpm in 6.5 s. How far will a point pe e /1i,l flxs(56yhe;+ljo fon the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when7j/):zhkl y6 d-ga xfs it is running at 850rev/min It turns 1500 revolutions before it co zxkjg)h -yalds6 /:7fmes 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 revolutio:f0q 5wn( gklyns as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.fg kq0wl yn:5(
(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 uni;g6zk 3 ;2satq/v zeuyc 7u.poformly from 95km/h to 45km/h T7zpy oz ug;tq6va.;e 3/us2ck he tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a 8j3rix o 5r13g9utd t.qcg3ok hill. The pedals rotate in a circlegr8o 3tri3 31.u9xqoc5k gjtd 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 the end of a door 74 cm wide. What is tol*,8i (2ckr bv 9jo atl93lsbhe magnitude of the tor9la2rl3 8tjo( bos9,c *bvlki que 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. 8–39. Assubx:j5 hml8;pome that a ;l 8p5ox bjmh:friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of m:e6ifj9 9di j djj)g/mass 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 t/idjf jj 9:6d9m)ijgehe magnitude and direction of the net torque on this system.

  The bolts on the cylindepjxof6xrg6 qe,-rx+2z2q 9 p2ift0d hr head of an engine require tightening to a torque of 38d9zjtpigx xer+2602 q62x prqfohf-, $m \cdot N$ 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 mp1, z6e:ak y+tox9,vd b-bsp 86 ez)pu3uzath when the axis pe pze8va)9 xo3u16+t: ztk u6,dhs -by,zabp of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whews7.np)cai/93d j4tbh y5s wael 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass 4yn7jwd cs/w59.tba3hpi )s aof the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is r ozjj*4q .ch(motated in a horizontal circle of radius 1.2 m. Calculate hj o4qj( zc.*m
(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)gz vhi6g(k+k at constant angular speed (Fig. 8–42). The friction fo g(giz+)kkh6v 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 of inertia of the array of point9bcgz:x)c a-kuyve8 4 objects shown in Fig. 8–43 abouyccazb g4 k)xve u-:89t
(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 ato) k,jx;lfi)yc7ipf s: ms 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 cylindrical satellite w qc heo85wt*/kv(. lpspinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass *okh.w(el pct5/wv 8qof 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.5 6j )-pi6a x nhb9gcabyx9.487b 6bwpcnsayr5 n4 9 byx6abhgp-p9.jc)ns5wcr 6xaba 6iyb8780 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 to 3: 9sr(gi ct(mr $rev/s$ 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-go-round and isjb1j31 2kw hqrsj/0go 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, /kgs job1 wq321 jrjh0neglecting 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 eventually brought c/ 325 ngq oag;6mccv kqtj/t) iwkp81unifor5tm t6/ij;o2g1p nk) kwgc cc8 /q3vqamly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerate-2 qevhu) qfu.a /:wbzs 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 solel6hk(x cqr km xjw(gk8..,(trty by the action of the forearm, which rotates about qt6xjwr r(hxt.mk. k(8k,g c(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, p,h4+ m(cxwz sas shown in Fig. 8–46pcmxz, s h4+w(.
(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 consisy8 hyb4 lvub.te o*w-2ts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns fy/ zr)dv e7;zg yq:k9lb/z(h (revolutions) and releases it at a speed of/)d zvz(bf/kh eqrgl z:7y y9; 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of + 0wjjvb4w/gjinertia 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 /nv3:d3 0u9pc guac ymof 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 l e4+(jo1 ww;jqxw+/ u r3ldesmane at 3.(xd3j r+1+ms ejwqoew/wu;l43 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the 6e0s w)ch(r kxSun as the sum of two tscwx 0rk6()he 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 workigb3tm mx 867f is required to accelerate it from rest to a rotatx7bgf 68mi3 mtion rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 ugv8 7k2b d: wp6.fj9bm:wflv cm and mass 1.80 kg starts from rest and rolls without slipping down a 30.9wmpjd6b78.ufblg v wvk: 2:f 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 ver9i-dkq.2 up kpt(xk:u tically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use cpx9t du:k(2kqipk-.u onservation of energy.]    $m/s$

  What is the angular mom.+r).ttrnlu.dd r u r2entum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an anguu rlrdt .r +udrt..)n2lar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentumk +ej--wq8x;fbmvlzkim i8 41 of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rota mf 8;kqkz4i1mj v--xei wlb8+ting at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ra6nqs;w34fon/ t g ;;nj0zvvdate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreas;;n wtfv3 6v szj4n/; dnqg0aoes 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 rs1*otc z (+93mcw.(yckkj mwzeduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2 .z t3y(cw (wco mz9j1+kmk*cs.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation xf6n)/;obk l0gpcw9cd l78ma d (li 0frate from an initial rate of 1.0 rev every 2.0 s dglwc6 (oi)07mp8ll f k/cf;dn9 axb 0to 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 axis43h.0g urvdn o1)qvms through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter thenvu q0rn.od3)m h vs14g 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 skater spinning at/cfu 2paevk ad /ilj+j cr5)ct,*kl/ 1 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 Earth6jkwj5qu/h /fu juir:4 f. /qv
(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 cvdz214fkk d: disk of moment of inertia I is dropped onto an identical disk rotating f1dd 2 :4vkkczat angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at:bo)az b/bs; k 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-bmhj) j u1jb2 stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 9jhjjum- b1)2b20 $kg \cdot m^2$ 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 freely with an angular veloaxasbhy 35v;1cx)f ubt :0 q/fn78o zecity of 0.8 u7axn/e;f1vqfb asbch80zt)5: o3xy $rad/s$ 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 eventuazpq95v+.fps h+35weg 6dd d 9te2sqzc lly collapses 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 carries away no angular momentum, what would the Sun’s new rotatiodd p9p g55 v+z3 eze +c.9qds2f6wqthsn 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 windskb64ff)lug.9dfug,,y t a0 cz 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 )6exad -5gybb tjbsx *;nr;*y$ 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 free 5iv17dphvy 4wly without friction. The momenw5v47hyip d1v t 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 the6c*gtjqd6o;s y+o+ q g edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of iner6q oc+sg+d;ot * yj6qgtia 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 ryz *h.,fe.m ubolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from th.h*fy, . ezmube spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Detq y6uv;3fdj* ap1a fjai:rf ;0ermine the ratio of the j j1a f: u;f3rfvaia*0;y6qdpMoon’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 from rest ata1 8knjv sj1f8 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 th.m*)ywz t2u 8vegm17ul 6mahme shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by thlvw1 7.z egmhy*u 6 2umm8)mate motor.    $m \cdot N$

  (a) A yo-yo is made 8ru .5h+ jhjghof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest. h8guj5. h+rjh    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular spe/eybeiwr9- jk0.k3 d: )zimb ked 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 8.0o3/gn z4nys(d times as great, were rotating at a speed of 1.0 revolution every 12yz/g4 o3sd(nn 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, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution an da4)pypy 5i;utomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such ada5yypin;p ) 4 car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “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 illustratzhi 86rrg haa:3va2u5es 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 ozjm tcr)sh432n 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 ig 5)fx9:d z+chqrd*t nznore 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, a fdtq:z5)r+*z9x dnhcs shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we w6byu54yy5e vp -fpmmn0 t ne*;ant to exert a horizontal force F at its axlp pt5e n 4*6n ;fvm0-yybme5uye 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 speed v=4.2m/s on a flat roq9wa 8 0rdg5*kqfp k7sad is making a turn with a radius The forces acting on the cyclist and cyd kfqkw8g q7rs9 ap05*cle 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 lengub7jsuzkz7.) /8l zltth 1.5 m and begins whirling the rock in a nearly horizontal circle abuu zts)7 7z jll/8kz.bove 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 cylind u/)gawut (n8oi8tu-a er and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratiu-unt8u w( ao8/a )gtio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,zb.acmj 3zci 1.r/g 6v of masv/6cm.za 1.gr bj ciz3s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 F )h6t-/ydajpe buxo2 qb+ 0dfw1 7sv+kig. 8–58. What is the minimum value of the vertical hev0-+w e1bfdj b toh/kaq+6xp)d7 s2yuight 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 do not assumeon 2 +r*svk( 8qce3jsn $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces aylh. (hybz,6*f 2cls(ff*c t its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was thzhtlyfshb*6(c(2*aly c ,f . fe 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