A bicycle odometer (whilc :6p s8n*cbojhk,mr 7 s15hvch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use8rhc5 c,k1:lbjm7o 6*vhs spn it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant .qur)da)+ nu9 2(l0uu vc6r; vh nzoux1uqj8x angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’x rculx+u9qzv.vr;u6(0 u d2u no)qju )1 8ahns angular 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 value of the angular veloc wz29lwczlk ea+q*1yu16 q /y v4r8sq5k tbft/ity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force qeky6o*a,r9j ? 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 l(q( ntv9jf9.fuh e,soh n 2/nbehind your head than when your arms are stretched out in front of you? A diagram may help you to annu.9 /h2 le,t nf9jhns( oqfv(swer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal :d5zq.q6oe uo cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harzd6:.o eqo qu5der to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to raziyh0 t49/ z4xypoz6un fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig.yazx64 hp4z/o i tzy90 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, narrow b8c m7 3joxm izf.1rsr7eam?

If the net force on a system is zero, is the net torque at7c th;m1( ol;siv i:8.x yexnlso zero? If the net torque on a system is zero, i hm(ceot1.87txs ;x;y n:liivs the net force zero?

Two inclines have the same height but make different angles witaqnv .d+ )b zzhxa 24zeo*(fd7h the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at tze*a7nd +24 a)zov (. qfzhbxdhe bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline.z4jftus8pv 1) 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 spfu)v ps1zt8j4 eed 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 * 2mkwyqtr0iq yba4 f6;yn7+zfrom rest at the top of an incline. Which reaches the bottom first? Which has the grebfiwt 7ma6 4+y 2rzyn;yqkq0*ater 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 7ih3 yzar-g)u9hubz 8conserved. Yet most moving or rotating objects eventually slow down and stopb9-y gau 78z3huihr)z. Explain.

If there were a great migration(srvj0d0d; +t(1oaqcmz l(cj7g a6u r of people toward the Earth’s equator, how would this affect the length orm1 ;dr sd(+6q0(uacagjtlzj 0o 7vc(f the day?

Can the diver of Fig. 8–29 do 7oa49)vlvc n5pd7 fek a somersault without having any initial rotation wh)nv5pcv49f a7k7 ld oeen she leaves the board?

The moment of inertia of a rotatiovo88rpj bs:(8c -/eud-zv k ong solid disk about an axis through its center of mass -o/s88 vz b -uk8(c:o repjvdois $\frac{1}{2}WR^2$ (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 -+ /ff+ hn; dx(aeuitja rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity; jne+fhx /(a dft+-ui increase, decrease, or stay the same? Explain.

Two spheres look idef 2ocbmqn /z pe:3 5fzi3g/vmw1 ef,i0ntical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine w3f/32mw/cf 1qp:z b eg,fio 50vize mnhich is which.

The angular velocity of a wheel rotatin/)8 dmyhyxa:l b,y/tdt5vqi3g 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, describe the tangential linear acceleration of this point at the top of the wheel. Is tm/y /5 hlbxdtdy,iv3:a) ty8 qhe angular speed increasing or decreasing?

Suppose you are standing on the eds 2lksk(rg klu8jl23x 3,xcqph n58-qcmn+c8ge of a large freely rotating turntable. What happens if 8kq3km 8lkqssju2 ,5- 2xgccc(3+ln xlhpnr8you walk toward the center?

A shortstop may leap into the air to catch ah p9 uj6nm9mqv8pq3l8)xn5 m t ball and 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 opposlp q6x q 9h38pmmnvt9n5u)jm8ite direction (Fig. 8–36). Explain.

On the basis of the law of consjykz ;3q9 ub2iervation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more waq;j zk b2u3i9yys the second propeller can operate to keep the helicopter stable.

  Express the following angles in r ejul agi3)*b;adians: (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 Eamn -j7yl3uc ec 4a,v9b l16zkx7b;mx dy0r-t arth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and thezrk-lj4ny7mc;l 9vm6 7dyx 3b a0 teb,1c-x ua Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km frocl p nt;t/li0h:h19ev *rc/h,t k -fwkm Earth. The beam diverges at an ang ,cwhtrkiklte -p1hc9f;t/ hnl:*v 0/le $\theta$ (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 6500te w /92k+zzn7b 4gm mhb.lppqq,q z507/ykqs rpm. When the motor is turned off during operation, the blades 0q7h/zt q.bq2skmn+pgmk/qe7 49zp l z ywb,5 slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball make9f 7bltxu8xeo-ef, 19ne,lp w s 15.0 uf eewo , 1l8x9-lnexbf ,97ptrevolutions, what is its diameter?    m

  A bicycle with tires 68 cm (t)k0n zi jwjf4:3wh zin diameter travels 8.0 km. How many revolutions do the 0 zjw:3)tkfhnw(4j z iwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatph+y ,bpu9 m2se its angular mpbu,9 h yps+2velocity 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 revolution in 4.0 s (Fig. 8–38). w(z ,c4lvb, lg2* eiku(a) What is thlz*u,kgi(2l,v ecb w4e linear 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 of the Earth :cb4yzc, ;yuu t-jaz.yek/lat76 s3mj ap x .*(a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed offhpj)*+nom(b rhw )z y5a ue7j:q36as 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 if a 9u 2erz)1pdl/4 vbmdv particle 7.0 cm from the axis of rotation is to expddple/ m)u4v19b z rv2erience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpmiwui)k(o d/0o to 280 rpm in 4)k/o(0 dowi iu.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 radiug3 4 ph)jnczo0s $R_1$ 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 spac3snmgd/0lw7tgv tk: pk- /2h3jm2xal ecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolut:dt3/l0ahkx s2t nlgk3g-pvm/72jwm ion 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 from27 qo+tynrry;s zx+ y. rest to 15,000 rpm in 220 s. Through how many revolu7syozr y2xr+n.+y ;qttions did it turn in this time?    $rev$

  An automobile engine slows down from 45( d6 gdopbip+500 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 highspeed jets in a whirjoj7+9 hwj3 h+/ 2yjtv cv)znwling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its +j7c2h3yvjvo twnhw jz9)j/ + final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to / :3d+ crzxylhe9do o/360 rpm in 6.5 s. How far will a pohoc: 3x9r+ lddyo//zeint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off (g4wcy*gn+c4 r 8j bncwhen it is running at 850rev/min It turns 1500 revolutions before it comes to a sgw* c84njry+gc4(nbc top.
(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 revol :u5buvr10 b8qu cz s*oa2xz l6uu;b+eutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 rs812q6;vl5z*zuux +b uob:u u0acbem.
(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 uniformly fru*nvyk.wgb 2 b42 gpf*om 95km/h to 45km/h The tires have a diameterfnbpgbgvw24k *y*2u . 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 bi fj.n8joj;m i+/ izww/ke puts all her weight on each pedal when climbing a hill. The pedals ifn+w //ozwj.im8jj; rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the 2xhp*r 47:jc b xacmyot lp8ilo8(v m(/bxw+/ magnitude of the torque if ly c*b+/4p r( 8 jh7 m /pl(vwoo2xtmx8xc:biathe 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. Assume3 lo1hm1.vom e thah1 31oev. mmolt a friction torque 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 endpdoqqaz1.i.n fmb1-bhu ra *; rz; d*8s of a massless rod which pivots as shown inb*hp.r81 iaq d* qnur-dazfzo;;.b 1 m Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an:0t2 tqmr+vv6u pk q)p7 3kdxz4j+ ltb engine require tightening to a torque of 38 p4k3)q2vtkj dm+0+p :tq t z xu76vbrl$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when bk 1ba+o4 v8msthe axis of rotation is throu 41m osk+8vabbgh its center.    $kg \cdot m^2$

  Calculate the moment ofi g5- (lz-duljn/ ixk9 inertia of a bicycle wheel 66.7 cm in diameter. The rim and d/ -l9 kuj -x(lni5zgitire have 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 horu6yli r*83qtr;euv t1uh 3m8 jizontal circle of radius 1.2 m. Calculatmturey8v1j*6ut ;lu q3rh38 ie
(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 rotatingb6 f8ua njt8 syuxo-j1 9wh*u0 at constant angular speed (Fo-6* u8xuty1ua08b9jj f s hwnig. 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 inusyzim.d) z+ av3c,8s0h9i jyertia of the array of point objects shown in Fig. 8–43 azaj3u9 .+y 0mys8h ds i)zcvi,bout
(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 o54 iwtds7uzugn ;8 e,cf two oxygen atoms 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 spinning at the correumfz 5t(2gqn;5 nsxb.ct 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 f.;n g(s2b z55n qutmxis 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 cylindery3 zdm- u7laktyy* 2hjy-7f:o*6ny az with a radius of 8.50 cm and a mass of 0.2aj6 -t*m7:h d 3yf*ul7yz -oyakyzyn580 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,zh .lj*zc/ 8of 9r.gl gkzd9i bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hay( bci90vrg:znzyo9 eud(3 o*nd-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) si(coonzgb 9 :* 0r9iy3vzude (yt opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is 7 9ivi*:mhxe eshut off and is eventually brought uniformly to rest b:em7 *iiev xh9y 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 accelerates a 3.6y/t(g uzmw)teq 9;rz 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 sole9;(sy(whdlx( /5 qyynmxr ;s bly by the action of the forearm, which rotates about the elbowlxnh5 qbdy9 yx((;ssyr;mw (/ 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;; i( tpyeuz:csiw;8u rod, as shown in Fig. 8–46u ;;p(tcsz;i:8uwiey .
(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 masses, 9/8zfdow 5-4bqov9hx5 lyjcpgfj: u- $m_1$ 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 fulba26 f+ cf lu4rf,ttg*l turns (revolutions) and releases it atb uff 6a24cfr,lgt+ t* 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 ha: pm6i6mg)lyu4 ufax(s a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develop 4 eusy6*lui9us 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 down a laigec)og8;mw h0 /fbplyho5 q+--z6 kis *x; vane at 3.3h+px -c* fzm6;ybe/i 5vqgk gos8;olh)i0aw- $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of4po7 h,x,y0akxz k5 *5aswjbt twp,j y*ok k4xtw5xazs5ha 0b7,o terms,
(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 16lu ln/s4hn,w) t:wass59x.5ts;v e yu40 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylin,t.tw )5nuv5y l ausses4 9;h/sw:nxlder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg star9z9)bqk2sy*e b7 a9guqme q,pts from rest and rolls without slipping down 9* aqee97 ,gup)z 2ybsbqk 9qma 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 f:s ndud0.oyb ek cy)*k7/g tg,all and its lower end does not slip. What will be the speed of the upper end of the pole*y/:oudc . etsbg )dgk7n,ky0 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 t /4)j5y(u zwd 4ebp l7f0mqldthe end of a thin string in a circle of radius 1.10 m at an angule(u05 y7p)tdlf /mdlw 44jbzqar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifocxqw4e8-xf q )rm cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpxqqfwexc8-)4 m?    $kg \cdot m^2$
(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 rat4euty , aad e-s3b. pc4ha3xv7e of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rou3t xh ap4yd-v a.acb,47se 3etation 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. 8–29x,togd.r ;hx9 v u85acjhr: 2k) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck posi9 : ,ur2a8dh c5ojkxr txgh;v.tion. 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 increase her spin rotation rate from an initial rate nhg)k:lsko4ij)k to:( . )tx3n/e mzz of 1.0 rev every 2.0 s to a final rate of 3 tmz)gs n tlik : .j(kx h:/)4no)3ezko$rev/s$ 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 cvs)ln+8*nny v 89ci3 ,pwq28q tat kx.nnat p0enter 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 npq .n+ 8a lntyk wsv)xq8n2c3p*i,nt08av t93.1-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 momentum7dskiu-5*d8f )qxh. ub9v zltm/sb/ b 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 momentu vd;x4fdu7q /p+d orfsu74:ibm of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of d(v*9u: 4mq ltk4/pbl,gji xlmoment of inertia I is dropped onto an identical disk /ilu4 9bmjv,k(xpgq* 4lt dl: 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 29z aud*d9 8pgve-fm z7.4 $rev/s$ 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 pp4l .xu3 8jmastands at the center of a rotating merry-go-round platforpjpal.483xm u m of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freelihj g4wf.9iy2y with an angular velocity of 0.i.gf49wih 2y j8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half i q34cw(o5 pr6 vorvo.lts mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass cwvo4.536vqo rcl o(r parries 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 wirfr1z5esb s k z-q+ ed5r3t/t-nds 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 j o:js0dc )j-b$ 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 i-6coq7agw j *.w ks6:h4 tammwithout fwa.i s:6cka746 *mw gmhotqj-riction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round oe )hg3z2glj(m i7gq2turntable that is mounted on frictionless bearingeq2igjz(gom73hl g) 2 s 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 roper vh:re1a17vr rolls 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 rol:1vv7 rer r1ahls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alw h,; 2w;9lanige6zv vgu-y/ zu*hl o)jays faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In g vvywgnuz*el 26)lh, iu ;/h9-;joazthe latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of f0w/ 5vr g60wm8ba.a stamb+h1 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 mr: 3q ll6mp6ho acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered pl r6oml6h3q:by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each oc3hau4 tekc4 u .,enzedk;eeosq9453f mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrkke equu3;4eans,o 4e54 z 3tc.chd9eical 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 of th bwp sz0v d97w+ qq3gj-x8ttr,e 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,f wy.07u1 fq2j(ndlubx4ca0 u were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radif20lj1(ad4 xu0 . buufnc7qy wus 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 automobile is for it to use energy3uv)2 6bz uety stored in a heavy rotating flywheel. Suppose such a car has a total mass ov)tub6 yez3u2 f 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 illustrateshv:+ehl2 +r wy 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 rolline id03qe;ibm:2uucmeg4 s78tg on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., yxzfkbl y;:a)5.6bxw m6aw t3we 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, det ylayfb wxx3:6b z6kwmt;a )5.ermine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It i .1p7 frlgv2:/9rekfe ( bzgf0g a;smns standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step againstg.9 sfl;1nmk0f ar/r7fe2(pvb e :zg g which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wite *tc2sbbmlf11nso ;( h speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cyc(nom1ft2 b ;bselc s*1le are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins wd96 2),zu+jpspif w4yfsad- whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requiredyf-di)ud+pwfwz46 s,2 9jsp a to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solidi eq wu5/oe7n1 cylinder and her arms as thin rods, making ne7 1/o5eq iuwreasonable estimates for the dimensions. Then calculate the ratio 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 consistk- aty*;9)o opjkc 7nhn 6il8fs of two cylindrical plates, of mass notoacyjn; 8i)f*9k7- h 6lp k$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a1- 3x 9xwgmea7m i-reflong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reac- 1rmemgxiwf x7-3a e9h 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 afvr3s.yx 23)k -a-fitva7l5fef;dug7tni b )ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduc .*vriyzn )x1kes its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a)y*k z i1r.)vxn What 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