A bicycle odometer (which measureie4 w0o-,.akt1 zgq8 hh:aro is distance traveled) is attached near the wheel hub and is designed for 27-in hrh .ikz twioe84 g:a-o,1a0qch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotatga-c-gih8 fag87(rbvcltg h,03 ka7ves at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s a,lgr0g7v-7g a 3i b88 gf kahvc-h(tcangular velocity increases 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.2zoepq9vtky9h t8sz6f 3zs8 value of the angular velocity $\omega$ Explain.

Can a small force ever z4zbdp*57s 7ctvsuh,j ( dd9 1k9vhp oexert a greater torque than a larger force? Explain.

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

Why is it more difficult to dow (( *tky*nwia t*(ach a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to a*w ihktat*y(*a wn((c nswer this.

A 21-speed bicycle has seven sprockets tm *aob(nw-y/ at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front s*boy /awm(nt-procket? Why?

Mammals that depend on being able to run fast have slender lower legs with flesh+by x9t :ckjx; and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is ytbj9+ ;kxc:x advantageous.

Why do tightrope walkers (Fig. 8–35) carry a i-az6dyeu6. r6ccl-m long, narrow beam?

If the net force on a system is zero, is tlug3h:h c1n0e 9jkij/ he net torque also zero? If the net tojin:kl19 jh0 cgue3h /rque on a system is zero, is the net force zero?

Two inclines have the same height but make different ) zpvvlw wh65) 7x.o i9i.sqwwangles with the horizontal. The same steel z)ww9hvw .x7iw .qo l)s i6vp5ball 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 rest) down an incline. q5d0yt nht ;3(yb9s:e4 5prtdjgeix 8One sphere has (8t:h53yt 0dg;nr4be9qe tp sdjy xi5 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 cyliny +kc,fxi.w1x5 s jnn2der have the same radius and the same mass. They start from r.nf 1c5wyj n,2 sixk+xest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most bp 5dymqwnn,e3a 2m 8bv3t0; +w6jxap moving or rotating objects eventuallymba nqmp; aj 28+dn,t0vwebwp6 3x 35y slow down and stop. Explain.

If there were a great migration of peo/k7z; qo: ktd mfh*l:fple toward the Earth’s equator, how would this affect the lql:;h 7kkof: fd /tzm*ength of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation mfacafsm*z.y sl +nr8nt5 vu777 cf *6when she leaves the board?77+* m8fcmauf5c 7narf6sslzv*y.t n

The moment of inertia of a rotating soj0vqkma:(z7+ pp nuv, lid disk about an axis through its center of mass + ka0z7pvu:(j,nmp vq 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 mass0rgw,fxz4o 3k in each outstretched hand. If you suddenly drop thxgr0zw ,o4f k3e masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same maa:7uw 9thnt1j/v q( erss. However, one is hollow and the other is solid. Describe an experiment to determine/:97u1(rhte twqvn ja which is which.

The angular velocity o7bh qbid 183un.o8c kff a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, de8f8bid3ocqun 7.b h1 kscribe 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 f0 0vmwzd;db-edge of a large freely rotating turntable. What happens if you walk toward 0 bz-fmwd ;0dvthe center?

A shortstop may leap into the air to catch a ball anda 06yl ;qmt7yl throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite directy6lq 7aml0;y tion (Fig. 8–36). Explain.

On the basis of the law of conservation of an xrs,v 9dn9a9fl7ja)f gular momentum, discuss why a helicopter must have more than on v9a9 a)fsd9,7j lrfxne rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30zzbe;wp,+wn-2y: heo $^{\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 amazi fl g4jm,fbe)ven4+wz-0 nqa5ng coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the gn5eqe,0vjw n l)a4b4f-m +zf Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam ep9z*-2 zmv 5d yv:)mj9jz6oqhzd+rvdiverges atj:9*z6d pod+zzzm vr-vj952y q e) hmv 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. Wwxxojr+ 0 f5e++nf .nmhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades mef+nnjf0r5wo + xx.+ slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mhu w/u16egp/kakes 15.0 revolutions, what ihg1/k /upu6wes its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions dv9v5s ,qlon duq.13cto the wheels .vvq31dco s9 nt5lqu,make?    $rev$

  (a) A grinding wheel 0.35 m in diame hrxoad9s/20dcyu+, (82zvb 5 k komneter rotates at 2500 rpm. Calculate its angular vel( oy9h2b/u2cdo5k vzm,nke8xr+da s 0ocity 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 makj5vuie 4 p,: qo*s j2moh*errtu1m+;mes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the liuu*vo;pjrhsm4e jmm5,rq21i +t *o:enear speed 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 o) fks4kby/a3+bc8n war-ly(6 ljxxy 1f 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 a point -4tr(zgd w +b-r)j*u*s kxwzj
(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 from7j e *kd.:d)fgm0p gpm the axis of rotation is to experience an acceleratp gdm: k7mpe)f.0jg* dion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to)e9wnlxn 6 77wy)ccj 5ps299qrav hco 280 rpm iv97ycr p nj7n569w2xs) a l)qh 9ewccon 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 radius a+6fs kum5fy .l el6r9$R_1$ 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 themselv5ufcc z,+xk8wr; l5vves 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 rvzwv58f c5ckxu, l;r+evolution 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 yo* puvv*o.u +,b4lgayd 2nh)15,000 rpm in 220 s. Through how many revolutions did it turn in this d.o, vl4)+hbpuu ya*2g*o nv ytime?    $rev$

  An automobile engine slows down fromb +v e3;deuhdf ,cv*a0 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 flying highspe 7nc)lqw2:rfrfzn y,3ed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 qf3:r lr2 7ycfnnz ,w)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 uniforme-)cba(z3nnp y ai 5kl0wihl.)u. )t fly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled itzan3 0)y)- k)c ilnif h.w( b.ula5epn this time?    m

  A cooling fan is turned off when it is running at 850rev/min ypoen 3lp2*k -It turns 1500 revolutions before pk-y*o pe2nl3it 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 ts- h:if-lx x3phe car reduces its speed uniformly from -hx -3 :lsxipf95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65m fpqmb0dc :7(ktcwn;f y b0:; revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m. (k0p :yq; mnf; :fb 0cm7ctdbw
(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 q:78 jx evjh+aputs all her weight on each pedal when climbing a hill. The pedals avqhj 7j:x e+8rotate 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 5f;:svgvk irz.q9eg3 on the end of a door 74 cm wide. What is the magnitude of erv9.:i;gvqfkz 3s5 g the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axa r32)k(51livg qo jxple of the wheel shown in Fig. 8–39. Assume that a friction togvrp3qio)(klx 5a2 1jrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a j jf3acqh3 76hpgju2k8 mc*f1massless rod which pivots as shown in Fig. 8–47cmjj 1gu8* k3 62fpfqh3jahc0. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head omion+/ojuhw9x5(n 4 dm0top *f an engine require tightening to a torque of 38 9ti *w4+n nmhd5opuo j0(/omx$m \cdot N$ 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-)qz7 p v+fyg i8bv7-takg sphere of radius 0.648 m when the axis of rotation is thro 7py +at)i7v8q gv-bzfugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclmice2 h :e lg,jmydx75jv85/vk qz/c/e wheel 66.7 cm in diameter. The rim and tire /kv5v8j72 ,cxicm dj q5/e: mgylzh /ehave a combined mass of 1.25 kg. The mass 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 horizcbu2j nebkl j0;2b+* aontal circle of radius 1.2 m. b0e2*b + an;uklj2bcjCalculate
(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 potte qfs/d)1jkl q+r’s wheel rotating at constant angular speed (Fig dq)k1q fj/ls+. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the arraycde c s /mnvtor peuhzck p4+852 2l0kd1m;*c- of point objects shown in Fig. 8–43 abo cc0ku k d+h e2v4emco;dt8* 1csr5ppnm-/2zlut
(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 atomsw-iq 6gj p-v fg:g1n,v 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 cyliq (*,iwk ;nnvmndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is tov,m*kn(i w q;n reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder wit mavies04y 6;ih a radius of 8.50 cm and a mass of 0.e v0yima6 ;i4s580 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, acxj 2(o0w jath:celerating 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-e8k1mwt/ 6vo,4nyqhigo-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 amyk o,nq 1wt/68h evi4 mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating a 8spy7n ts22hxt 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torquep2n2s78xt syh 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. 8–4s1fy;o8 x+oqj5 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball i rna+b weigi6 n4n)q 0s0vrp0,s thrown solely by the action of the forearm, which rotaten0 rb 0gsnp6,i0vwnaqr4e+)is about 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 bladexm 8p2.(ftmcr can be considered a long thin rod, as shown in Fig. 8–46mxfr.2cp8m t( .
(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 qcz7u t9x-x9qj 0 - jv6xh.rh-t4 -hw2ld uadbtwo 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 wit6y lcx0+)isjmhin four full turns (revolutions) ans6m cxi0)y +jld 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 a moqo *, gztj4mz,ment 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 dtp6,j 7h:u bsj4ami7 yevelops 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 ro38k1 pyqg /owulls without slipping down a lane at 1/ poqyk83ugw3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with rf.nxy s id: ,(xskf:-despect to the Sun as the sum of two d.-xnf fi(kx, s::sydterms,
(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 yb.hlleq,,lc m/b5z1 a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotatiobll,ec 1yz5 ,h/b.ml qn rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mas8g 5sbp 1h7o /jcs2jz8nd qi4640z 8ybgykiv ns 1.80 kg starts from rest and rolls without sl jhz8 ps y25yos4n61bg8cbvqi0/8g 4idn jkz7ipping down 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 its tip. It starts to fall and frs-d:.p b ,l*,qyi kbxg3 ;pdits lower end doesfyd3g*;pkxbl ,rqs:.,p bid- not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end f8 snav8r8di - g)ra3hof a thin strin a8-id ngrh)8sr3fav 8g in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylio06t pmwe8st.j9xs. indrical grinding wheel of radius 18 cm when rotatito.8ip 9 mes6jw .s0txng 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 fbr7*t job 1+kj:1cgea rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatiokg 1 :cb*7 jrf+tje1obn decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mof6 qp-0iudq: 8iofd2vment of inertia by a fac 2q6f0q fvip :udo8-idtor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can incr *nci/lq-fkc/ease her spin rotation rate from an initial rate of 1.0 rev every 2.0 sf*k-cqicn / l/ 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 rotating around 1exsh14y3xn s2rbr.j q5sj e2 a vertical axis 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 then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm451snb e1q2 2rjjhxs .yesx3r , 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 momentup0js)6jz5fukc-ps e, x8 y5vr m 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 on6q1pss jf1qqh: 0 g1the 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 diskj;avsv4(/ mnw of moment of inertia I is dropped onto an identical disv mv/;w4naj(s k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ctvhb rm/tjm1fd00) pg 3nne( ,6vxw)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 rogb.(;wud0xiio4 p h h5tating merry-go-round platform of radius 3.0 m and moment of inerbwu po5;i dgihh 4.(x0tia 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 velot5q1/ a9srqcqd17 tl4kwy4t bzt(ac;city of 0.a5 s c(l11ra44 qc ;7bty/dtzwqt tk9q8 $rad/s$ 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 twprg2lrbic 1)97r1 h a white dwarf, losing about half its mass in the process, and windincbh 27l trir1p)wr1 g9g 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 in excess o5 wd6y6zah2ljz k jjgvei)l7.6ar u -1f 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 w 2f-ly7w ifa-1rj :-h e1xgyu$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely witho80y1v8wlpoy2j:u og mut friction. The moment1 8vy0uylo8j :p2gwmo of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round x)emx/(q e hg94za4)p7 zebvj turntable that is mounted on frictionleszb z/xhmqe)eg vax7 4e(9jp4) s 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 end of t ad; a9casmtiq )o9;f4he 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 ropeads miaa;)994co;q f t unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces thzn:le4b(o:jsz t0* .s elr 6n *lttr*he Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treatrs b*z0n 6*tle*s.l(h znttj 4 :lero: the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerate4hy5w af 6srt,s from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone i2b0hhdpec0hx.fs vkh4 5 /yg*n the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of hcxsphg4.h hvb*0 kf2 /ey0d5from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050m/nx:2sob )kt)lso/ t gay:4ra n+i1l kg and diameter 0.075 m, jolxyoait:rs) /):nto+1 n/gls a b42m kined 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 ssj2:cfv- 7tzs8 hp3xcpeed 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 great, wertd jhgeu5-p8,p7-hbrl/ft h -e 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-qua5bfrpu-ph j-/8hd, g7lt- h etrters 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 as)-; u8wx lcxxgh .74dwcojr5 utomobile 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 cylindg) hw.rul-jx xc8x7 w d45;socrical 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 a,nwrfl/o* w(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 horizo(f:3st w toqe*7th)edntal 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 (l90- -3 g. umsmqsik-b7 xqvcii.e., 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 of release, determine (a) the angular acceleration of the rod, 3 b q-mis-v0.lxcq7um g- i9ksand (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 verrm 9*ivtu wg5s:ll5w+tically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against w:g+lm5t 5r*swiv 9wluhich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with t0 ,slenj nr712j sy7oa radius tj,2ne1nolrsys7j 70 The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and beg3;f/ pos sxg4iins whirling the rock in a nearly horizontal c/gp4 fxisso;3ircle 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 thidkf;t0f0gm/ 7m +cx oqn rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretchtfkqfm/0 omxg0d +; 7ced arms, and with arms held tightly against her body.   

  You are designing a clutch assemblhhfwz6y04 h k+ zde/f9y which consists of two cylindrical plates, of mass h/d0 6yz+wfh9zf4khe $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 cidtd;muz p2y xw:)-8i iny22track 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 the loop without leaving the trackmczdiy-;t puiyn82w ) 22 dx:i? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not /rs+ p-ber efr;73w7,idukc cassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car re1/icb-qcdl hn s (9+fc ,r05ka,s gy sjq-f0laduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) W1fg(-aasyrls0b9lj+,,q5q nckh id- /c c0 sfhat was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s