A bicycle odometer (which msjp rutr)k si(6v g377easures distance traveled) is attached near the wheel hub aigt3pvs 6 (7ujs)k r7rnd is designed for 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 poif(oa5mfpjr ,bj42 fh:+n+x aj17ds lxnt 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 oj dn mb7 h,ff1a2xja+r josxl45p:f+(f either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocity.xfk; zovht+ - $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? be or(5exv- 0bExplain.

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 wif/gvizx ,riqx ,d(9u -th your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to z,x9d ,r(/ivxgi- uqf answer this.

A 21-speed bicycle has seven sprockets at the re1 se,ury*m/t nar wheel and three at the pedal cranks. In which gear is it ha1* rmtny/se,u rder 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 beir le19q/ptmpug*v6wd s7 ea( 0ng able to run fast have slender lower legs with flesh and muscledemsvu/1 tpwpg(7eq l0r * a96 concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long,pf;o 70ue7p sp narrow beam?

If the net force on a system is zero, is the net torque also zero? If the netmt(89 fg0xk ;s sacqk. torque on a system is zero, is the net force zero?f x.( 0;kqkmcasg9st8

Two inclines have the sam*lp9cn ap q,t(wr0i p0 lfzr,2e height but make different angles with the horizontal. T2pl 0 fwlq ci(9 tap,*rr,zp0nhe same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from svc3kg 15rq*g 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 has the grk 15 svc3rg*qgeater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and+ 7+zz2ojthgadcr0cig 654 zn the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rcn +giz4r atjg+z2ohd 65zc 70otational KE?

We claim that momentum axe6 mg; q,;oiu dt5n/ j2q 1hftk,8zzznd angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explajxd; mozz /nz,5;2u1t6if,hgkq8t e qin.

If there were a great migration of people toward the Earth’s equlk;mx9 e(fex mvat293 ator, how would this affect the length oel 3v2a(kx9emf9 ;xmtf the day?

Can the diver of Fig. 8–29 do a sw sa79rysr g99omersault without having any initial rotation when she 9wy r9as7g9rsleaves the board?

The moment of inertia of a rotating solid disk about an axis throunvfq7v3f dn9y*izi3 lb9 9z,vgh its center o7v3*9v9i 93bdfzz qf yin,nv lf mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg ma nd4k4w c0ork6x0 f-txss in each outstretched hand. If you suddenly drop the masses, will your angular velocity f0o4nk4crwx 0k xtd6- increase, decrease, or stay the same? Explain.

Two spheres look identical and have the e;q gndx:c(e h+b8+josame mass. However, one is hollow and the other is solid. Describe an experiment to dg q ed:(b+xnoej+ch;8etermine which is which.

The angular velocity of a wheel rotating on a horizontal axleae 7,f;0b 7(odq zxenn points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular ( zbx q;7efad,enno07speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. fx1 6(c hqgr*yWhat happens if you walk towardg1xh6f (r q*yc the center?

A shortstop may leap into the air tkq 792nf *vecbo catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice v kf b2nec9*7qthat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the ;e -+5u8ftrob ,utcj qlaw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss ontbot u +q;jf-e,5u c8re or more ways the second propeller can operate to keep the helicopter stable.

  Express the following a1f )-ciwyofo*0 )slkoz/wqm 2ngles 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 Ear5y,j*ghv ctz*2a9l)m lg16hpv y2azvth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moonlzl159 2y,a2cpvmvvg t a*y)jh hgz6*, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed a12jbygp e 8i3q2jtiy;o gq 1d8t the Moon, 380,000 km from Earth. The beam diverges at1t8i ;3by8gy i g2oq1pedqj2j an 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. Whewi tn:c,uzx vdh:e 5nu 1l0h-+n the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? le 0t hn:xz,nchi+u-w:v15 du    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to a l1e82 pqs*o 6fjbmq+o+7abja:u 8vienother child. If the ball makes 15.0 revolutions, what is its diamiojesaqlbqa*vj7o 81+:fp62+ b8 u emeter?    m

  A bicycle with tirescyp9; aagd f67 x.a6iw 68 cm in diameter travels 8.0 km. How many revolutions do the wheg . xya;daa79i6fpc w6els make?    $rev$

  (a) A grinding wheel 0.35 m in 9t*5nd 4v1tan ve -rm+z( feevdiameter rotates at 2500 rpm. Calculate its angular velocity1v- f r+n aetvn*e4 m9zdetv(5 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 revo ,9o; vcespeg )g1ew +7x3rvqh)oj1xglution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2ceox,g1 3)s9 +;71gvevxo e)whgrjqp m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun(ti+f9 x. fkmg    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pode-n ai3n).ke-c. gd uint
(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 th/f0 ve6-lt28ovovqlaw +h .1gowlh3te axis of rotation is to experience an acceleration wav30 v qt+/lg to1-e6l.fo28hlh vwoof 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 2) iw0h bse9h(+uxefq130 rpm to 280 rpm ihbiqe0w)x f+su29he(n 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiujgke8gx1m.mnj2qp .5ff aaxz5w7g9 )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, astro:m6om;.2ay vn 4v:v hhggy6vnnauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At themyg y;6:nanvgh:2v4ohvm 6v . 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 i eyt,eip3y7 w/n 220 s. Through how many revolutions did it tur 3w,tepiy7e/yn in this time?    $rev$

  An automobile engine slows down fulf71.f y+gccw(nha k,g 12u qrom 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 stresses of flyc +eo2prgvd2 izs0m0(ing highspeed jets in a whirling “human centrifuge,” which takes 1.0 min mzeg2s02oi+c0( pdrv 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 rpmf el-k a2: ;se skdch/to,9r:b to 360 rpm in 6.5 s. How fa:9-es kca dhlt;2 fo sebk,:r/r will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turo(ttzy ;r, a3vned off when it is running at 850rev/min It turns 1500 revolutions before itz(; 3av,ytt or 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 gnm gbi0 z(4m)zdb6t/ revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m(i zm g6nm0bt4z) g/db.
(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 revolu im a2h8.nl*/;qvzf t7xv5wautions as the car reduces its speed uniformly from 95km/h to 45km/h The ti5ifh nm vw72a*xl;./tvqau8z res 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 cs2qi/h- zi4yvf)9v wqxp v8odv2 l9s7 limbing a hill. The pedals rotate in a circleli2zivqs/v7- )px hdoq49y8 v s2 v9wf 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 the x4ocw4eb622gti(l wn magnitude of w4c4(n6li2bg2 ot xwethe 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 abx v nv +oct;-f-y8hmi- mj1ql.qo)o d4out the axle of the wheel shown in Fig. 8–39. Assume that a friction torq81 vctyl+mohv 4mo-.foj--d ;xiqq) nue of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attac3bwe dpi r6bq hpl39,6hed 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 the magnitude and direction of the net tored,3pbb6wi 9ql3r 6hpque on this system.

  The bolts on the cylinder owu 12 3hicwnux36i o/head of an engine require tightening to a torque of 38hxw1c3ouuo 3 /i6nw2 i $m \cdot N$ 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 ofzgu 5il.btdh 3r0m :0jzz)i3e inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its )tuh3.z:ib3z izjd m5 e0r0 lgcenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamete/eown3g/n vho( - b -;wfq8bgdr. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (wh//hb-;8nof (w o-gvwdg qnbe3y?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizzyucomk/juu85q:8/ x ontal circle of radius 1.2 m. Calculax ou ukyj/qc58/:mz 8ute
(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 .y jy1j i6c9-xcm 0czvrotating at constant angular speed (Fig. 8–42). The friction force betwxyv-cm90cjj6 1c . yizeen 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 oe6ceyj-rc:x rs*aj ;1 f the array of point objects shown in Fig. 8–43 abojac j s y;cxr:r-*61eeut
(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 totalhdwzsf+ iwc+a 21n 6i 3t(k7h+bra*de 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 cye-n)awy.c9y) kmp4n plindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fayn)k mc p9wn-)4ep.yig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mal tpwn0 llh864ss of 0.580lpn w8 46l0htl 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 bd+ 8ppg 4o0c 9 lw18fbuuu5qawat, 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 tangenti; 49q0us+t ih1g l8rnw3ixf nkally on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required tx r+4;t0u9i18inslwqhn g f3ko produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor r;fkwp yxx ol;e8 ,u6l*otating at 10,300 rpm is shut off and is eventually brought uniformly to rest by ,wpl 6fklx8 oe; ;*xuya 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 accelerates a 3pgl2 jsvp5ijl1b :06 4qfi6i r.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 fi,feqp2tco )p57ov) norearm, which rotates about t5),qo vctf ni e7o)2pphe 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 r5y5 d+geddrr01 g3w/ ,zop xvqod, as shown in Fig. 8–4ddepw,qdg 5yx0+rgz 5o/ 3v1r6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two m/aji mk-(rj l77yc8n passes, $m_1$ 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 ik 6qts2x oi,v nu4l(hp7xy l1z;5oc5 full turns (revolutions) and releases it at a spehi( viucp1z5xy ko;ltqoxs ,4 67nl 25ed 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 moment of -m v2e4/ abkuko,d5cpinertia 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 d j1kqj p)i 1-ll5c/pneo7n9i( m;mmjtevelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping ;bpvi; yb;8atm(o f .udown a lane at 3io 8;tby(uf; v.mab;p.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetictp g(y4l xtjs57 udf)* energy of the Earth with respect to the Sun as the sum of two teu)gdl*4 5x pf7( sjyttrms,
(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 ymz5cn4o2f .0nsytr dd-p9 :bkg and a radius of 7.50 m. How much net work is required pf9 2s bozrd-n cy5d0my4:nt .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 mass 1.80 kg starts fr;:jo ndwu,l0p om rest and rolls without slipping down a 3;dw :nlo0 p,ju0.0 $^{\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 ve:q-;wi rp,.t-mj:) hhynrg 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: Ue.qt rvrjmpi h, w y):-h;n-:gse conservation of energy.]    $m/s$

  What is the angular momentum of a 0.2hy9 cohz 3v2ka;;wr-n 10-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at ayz3r nvowhkh2ac-9;;n angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular morh9k , idfnt/6mentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm/6rftkn9h id, when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of ;/ll8*omftv a1.3rev/s If h* ;fmlav t8l/oe raises his arms to 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 o 2v at1n41zbx1u-2x nqgx0ri yne shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotat 4ux-qvztny11x12gnb 2 a0ixr ions 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 sp 07vzww3st )vgin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final r7vv wgzt) w3s0ate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a8iho(w okgwr*zp1y; 3 frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel afteoyhzwwpi; 813(* k gror the clay sticks to it?    $rev/s$

  (a) What is the angular kokl 36 j45jofr2z/l lmomentum of a figure 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 momentum of the E7 oicjf5kh7+ darth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an ip4pev y-51o medentical p1o-45 yvmeepdisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 12 (y0 jd ost+,1uzmjef-uu5gcz,gx a 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 pgn7b b0zekusj0;c,.lq jk ; qy 8r5qz1latform of radius 3.0 m and moment of ky; un8. 7kc;qj0z gl15bs qq b,ezr0jinertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular veloq+f rrzk).i kmj9:6k+e(fplocity of 0.8zqfor+ +.k:fkp 9(krmjl)i e 6 $rad/s$ 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 collapse4i6dmz w,7x uqrv2 sw;s into a white dwarf, losing about half its mass in the process, dmix v7r,; 64uwqs2 wzand winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds bc+l1wo un.ckcp *5 f0in 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 u + kjmnkp68 ;)x*. v7sgd/evtpru pe7$ 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-kv 1 ysd4)(o zr*irta rest, that can rotate freely without friction. The moment of inertia) sot r -4k(1r*yaivzd 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 merrhdnm5xg s1 31ry-go-round turntable that is mounted on frictionless bearings and has a mome3 nr5shgxd m11nt 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 twe)1 sk dhc:d h0 df7d 5ybsw095s+nhphe 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 length9dd7hpb5csdeh ks:h 0ns)0w5fw +1 y d of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such )7z)eo3 cq dhythat the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about itsey )zocdh7q) 3 own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 3kn xj 2xxj .ghml)p-,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 cylinder of radius 0.20 m acqcl2l ew9,orp 4vb546 ooj3u6 yrs4dnxuires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcuo,ry6bu4d 9 e2pvlw4c r536nolxs j4olate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050lf yzir )k) -c1e3rc g;(kmj,p kg and diameter 0.075 m, joinem )-)rzy gcp; ,j(1refklick3d 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.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed o,fq-ber/a;( cu-);xsr ej e fxf 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, butx*3op szx6x7x bl-fdf 8 o+kg+ with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere-fkzbxlxopx7+*36 + 8fg s dox 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 ;92ru0kyh ll1 ayx (ulit to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrickr 1u9l0 yh;luayl2x( al 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 illustrates anbp)v+/k.2 f) euqu fnu $H_2O$ 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 h 3*lydwxhrd:;e /v-ig-tn a *vorizontal 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., wefj-v c+q enpe;x;6 3kv ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, 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+3;ep-v qn kxfc;jve6 rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It isy6dl1-sy: t.v+ycdq7 7ewmcn standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step againlwc q: +-ymved71y.c7ydtn 6sst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wi+fgqzc.w iqm2 9lyv ;-th speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist anl 2q-fy ;+zwgc9.qv imd 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 boa:5dza lbljg2l -71negins whirling the rock in a nearly horizontal circle above hia: al z57dl2ogbn1 j-ls 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 soli rx)af0;qvihobu.g 6;pr 5pm, d cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly agai moxr0rqbpp ;;ag.5 fv, u)6hinst her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,9su. .) e2dc:mhhfgu cvhr4g-9qe t4 j of mau9d.qs4:9 egurj vm gh .4 hf)-hct2ecss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopedsrxj9z( +qno(i2h sw ; 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 poiw h;so(+nq9(jxriz2 snt of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume gcdp zq8(c(;z $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed ue)r 0 e8u-eeow7qo6ixniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the ang6)exe0 uo8reiw7oq e-ular 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