A bicycle odometer (whiy6wif 9-fw7.g(n ssuj ch measures distance traveled) is attached near the wheel inw.w7 jy9f u-g(s6 sfhub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at coa fou,g7d*6(cgewzf-nstant angular velocity. Does a point 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 acceleratz7docf*uf, w -a(eg g6ion? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describem+l ( jx:hc-wqo om*b0d by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a grearot 0 ;b1hrnv(ter 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 do a sit-up with your hands behind your head tj8xt:6krtfg a 4pc8o 4han when your arms are stretchetajfx8ok4:grc84t p 6d out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets of6jc0l+o gp. at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small.oc6 fj0p l+og 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 being able to run fast have slender lowery ;nxc*d*8)9rz- ljbbobl u : i7miv.g legs with flesh and musclxu.gb*om9b :nc*-d8b7il ;vz) r yij le 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, narrod(wt:1qrw ay/ w beam?

If the net force on a system is zero, is the xfx4ck37t or:r+tx.s.i6ok +x .uoca net torque also zero? If the net torque on a system is zero, is the net force zero3xfcxs.k.r+:io x4a x r.ttk 7ooc+u6 ?

Two inclines have the same height but mab)1nbcd 8ce2hq)lgjnv: x ;k)ke different angles with the horizontal. The n)bhc;k)c bxgn)v2j1:dqe l8same 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 (h2at+r em ny+w)g+ (gqfrom 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 greater total kinetic en+wa++(ym)g2rtnqeh g ergy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They7wps;sq:k gwbag70.e:65 fr 9hlm gx b 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 mp:q69arw;. ef7s:w s7 0blgkhb5gxg total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most2wtp5uvefla j88t )3p gy/8e cv13iw j moving or rotating objects eventually slow down and stop. Explain.a233ep 8) pc/g1ej8ut5ywvfv jiwt8l

If there were a great migration of people toward the Earth’s equator, how would zg33 eph+gi g8l2twf,this affeli3,2 wz+ht 8g 3gpgfect the length of the day?

Can the diver of Fig. 8–29 do a somersau-hdn x019 h9(p0dnvu/ e)mn:jcq 2il4et picslt without having any initial rotation when sh i1cde-c4 nn0 spumji)l2t0v9p(:9hhnqe/ xde leaves the board?

The moment of inertia of a rotating solid jsv+l xz6qm:/kd;2lsm x+by 1 disk about an axis through its center of mass/smll;mx 2 jz 1s yk+vbx:d+6q 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 mass in each -qqs /z59ru-s p 7ljxboutstretched hand. If you suddenly drop the masses, will your angular vesxls/j-5zb 7puq9 q-rlocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollordh 69t*uk1 z4gb jz*p08jjlew and the other is solid. Describe an experimentbhzk*8tur jj0l* e6 9j1p4gdz to determine which is which.

The angular velocity of a wheel rotating34ppq2:c8xu;tjyj d j 0qa:mf 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 y: jc48 qu:p j;2dta3jfqmx 0pthe tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. 78/bkwxmuhp 8ra 3db0What happens if u7k8xbra0d 83ph b/wmyou walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he th zd.rtrh4j2zael6v7 5rows the ball, the upper pte42jdz .hl6z5 7rv raart of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of ach. ops8bh,l8e-f .wsl1tfu5k* y; kqngular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter st .fqpychw*l,1oubsf st588h-kk.;e l able.

  Express the followingvud2tral j9.e 80(+i2weqq fe angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. C qm8bpiebo 5yzy)a: c2n2c0b) alculate, using the information inside the Front Coverybna boqbecmcp28 2)y05 )iz:, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km ea1i tqw8xz m1s9s. b/from Earth. The beam diverges at an anzb /s8a 1xqswite 9m.1gle $\theta$ (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. When thz q fd71dpt9a3e motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as thedd731 fp tqa9z blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If4rru1( vpqoirssfs5hk6( 4y 7 the ball makes 15.0 revolutions, what is its diameter?f(16ss7uo k i5s h(4v4rrq ypr    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hv cloo9z-xx- f+ hq90kow many revolutions do the wheels make?xf- oqovh09z+ xklc9-    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rt4o:frpcdd9sa9+ yue j5 jnda(583w y pm. Calculate its angular velf cs95 dey9yp3o 8t:dr(wa4+5j jadu nocity 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-h5y8ba5*j f en vgc7op8iet9 complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.gnp9- 5i 5aec *7ojtey8vhf 8b2 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 itslhiv5nm:j.:q k2m. 3:tb gorj orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ac*lc gavb(6 ,r 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 e. w -65ehllm3s lvu3urotate if a particle 7.0 cm from the axis of rotation is to ex lsuhu.l el363 wm-ve5perience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from. g4puu6 5x.w 8w h+nulkpv.lyrcj/d1 130 rpm to 280 rpm in 4.0v/8+5.6 x j upgwl1yhprn dcluw .k.u4 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 radiuhoc8kevd;3h s* :yf7.nkji63: fu pcf s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselvewkpg yca3;5+ka2;lh0x8y,sti p3jw ss into a slow rotation to distribute the Sun’s energy evenly. At t g 3; yj5ip+, p83c;2lwksk0whaatsxyhe 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 unifoma kgmon l(t52c 03b1zrmly from rest to 15,000 rpm in 220 s. Through how mag3o51anl (kmt2m 0zb cny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. o2 w4nbmyyz5(nw.w.-exd 9n9 eiv nz 1Calculate
(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 highxu.) norl 72aaspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finan7aoul.)xra 2l 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 acceleratr ,j6ri8qo0x xes uniformly from 240 rpm to 360 rpm in 6.5 s. How fario,xrxj 6qr08 will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mj wrmwlip 1,14in It turns 1500 revolutwprm,14lj w1iions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed933a,3 toy .t dbdjqiozp.9 ry-jljt3 uniformly from 95km/h to 45km/h The tires have a diameter of 0.t3 tzp9qtdy3.l39ao3.jo brj id- ,yj80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed+uzcf ;( ofaafj1q9( 0)bg cth uniformly from 95(cocfq 1(uht9gzj a;)af 0 f+bkm/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

  A 55-kg person riding a bike putsuww .i j 9vk) h(eb* qhj,kl97,2nrepn all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm ) k7l.,kq9*bw ipnhh9,r j(2venejuw .
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force o2ylwm l wqsw,::wbe .5f3ovg5 f 55 N on the end of a door 74 cm wide. What is the magnitude yq2lwv.gww: l : ,smoe 53bf5wof 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 axle of u k:vbo;e2*qx /h7nxvthe wheel shown in Fig. 8–39. Assume that a friction torque of 0.;nkveu 7qox :x /2*hvb4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass a(qojnu9gmx6n9 3b 5 fm, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal positiou6 oq 5xgnj9(9b3mfa nn and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteni.i ;si+n/m o fzu4:l tr,+hgv+zf3mdw ng to a torque of 38 +;.3gvm iu izs+h: /wmno+4 tzr,ldff $m \cdot N$ 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 ulj9u )u ksp7pcz241z0.648 m when the axis of rotation is through its2cu zkpus1 7 j9)l4uzp center.    $kg \cdot m^2$

  Calculate the moment of inertia of s:ehge.b-c.mol-gw tqx8w;; a bicycle wheel 66.7 cm in diameter. The rim and.;xhw- o cgs le:wtgbm;-q. e8 tire 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 horiz7)gbyf6li .pdontal circle of 7gfd p.l6yib) radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angbc:xo v9fx46 up8(m avular speed (Fig. 8–42).f8mcx4p avxbo (:v9u6 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 t/(;ms -yryo duhe array of point objects shown in Fig. 8–43 abom yu- yr(do;/sut
(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 total me+tqmk-8pu k*m j /f e47l0treass 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 correct rv-l ywyd3;s ua71, ezmg :z 8w/kgek.qrq m,t4ate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass o.1 4ywys3/ mqa,mwv8kdktruzgz - e,g7l: qe ;f 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 aj2 ta +an,5ksc mass of 0.580 kg. Ca+cj a,ts2n5 kalculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3zm( rrs9 u.ukv yu)bs,ao5i38 $rev/s$ 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 smallidywn .x b.42khfj11jbr 7 )bhj3 xxk+ hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15k4h1 br jkxi+2by1n)jf j.dx.bh7 xw3 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpmlssnb)v5w4,lvr /w o 3 is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2osvv5 3r,/)wwn 4s lbl $m \cdot N$ 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 ox7d on8i g5aowl7s*.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 o aklsl(mj3f*y11c6 dball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, 6y(l1l1msf*ako cd3jFig. 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 bmnx )0,-cntlplade can be considered a long thin rod, as shown in Fig. 8–46n)lx mcn0,t- p.
(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 con(y8.s 8oyf5rl(r p9rkdv mc4 psists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerp/u.x(1zl7k vnu xn .p(/azhyates the hammer from rest within four full turns (revolutions) and re v npx. xl1hzun/y.(pkuza/7 (leases 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 moment of inertia ob ;e z1t b5e,6-x0-zbkialc dqf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile enginef whzj 4z/3rsmxr+nnqg:(w 46 develops 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 downd+pbn;pn0, 6+ryy3j9qbajd l f/cc ,o a lane q9j nr+3lnccb0pjy ,d o+ , yfp6/b;adat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eartp3z/z4idu e 2p6uje:g: uaiy2 di-l3kh with respect to the Sun as the sum of twgd 2 ule3yijep 6iz z34k-u/p:d a:u2io 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 1640 kg and a radius of 7.50 m. How muchtmu6gt/;g 8lz1re 6s xs9g:tl net work is required to accelerate it from rest l :em1xr6gtt ;s8zsg6tl/g u9to 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 * fk(bl2ygl.5,v xpafz2h.rvjba6 6g1.80 kg starts from rest and rolls without sfv,kgah2lvbz r2. 6a 6x*fybjg(5.l plipping 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 z/lfnq.oz 3w-p:i k m-to fall and its lower k -qli .-pnzo:z3/wmf end does 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 of a v8n)l2 gvlm 6pthin string in a circle of 2lm )8vpgl 6nvradius 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 cylindricaljq3iv nm fit:99s ,4ym grinding wheel of radiuyn: 94ms,fvqmjt9i 3is 18 cm 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 r,rmi2q v2g 2ug -vallb r1p-,cotating at a rate of 1.3re-rug2-vmrq,l2v1 pa b,2 gcliv/s If he 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 one shown in Fig. 8–29) can reduce her moment of inaty,)yguy. v ,z5 jkabf //;sbertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations,5g fjuytsb/y,ka /zab vy.;) 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 rate2+wj0rdn(m am of 1.0 rev every 2.0 s to a fina2mn m(rja0w d+l rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a s248w) bvgk-itswi/s 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 clayss84)t bw2is -vwk/ig, 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 angulay*om*gt0 dwi: r momentum 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 E3w 7)da)d/73tynsz lm7jzr ularth
(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 dr;nt w8jtw279y;hxmy yopped onto an identical disk rotating ttmjywy27xnh w;8 9y; at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ttj5e.5 k -ejm2xvv v82.4 $rev/s$ 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 rkqg6m :n pv2ha/3u4 qsotating merry-go-round platform of radiu:6 kq43p2haq /nsvmgus 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 freefre7cp roa ,r- gh7+gp*n x3l1ly with an angular velocity of 0.l7, rnrhr7 gcp+*oefa-xp 1g38 $rad/s$ 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 collac+ 8/bto29vyl p(i.v+ vfxjwjpses into a white dwarf, losing about half its mass in the process, and wincl+wj2xpo iv/tvfv (8b.j9+yding 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 wind6gnp5csf+v l 2y1 il1cs 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 8p qoib:you139 u ghxf0lr**i$ 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 q:lp+6d .,wwz5f ezfd can rotate freely without friction. The moment of inertia of the person plus the platform is.6:l+z 5 qdw ,dfezpwf $I_P$ 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 tu/ 6 07qui 8rvyrdce +wnhqug5.rntable that.+v0rud7y c / wh68eq5 ruiqgn is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rollswfdt 7 5ou.y7o on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center offtw o.y7ou5 7d mass move?

  The Moon orbits the Earth such that the sa.rixc-2bd1rd5 qu9.qrwx stgsi*k96cz kt +6 me side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angq659. d* skd9.zrxk c 6 tcru2 +g-rqbsxiiwt1ular 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 h w5p-qw(- 6:o5b jgb dc 54tpqkgehf1m/$s^2$ 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 sha.9 kc,bb1 xnvgpe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered bbg vnkb.1,x c9y the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg iws,avv f*g f hg1*p7+and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is relea,*pfa *gv7+hs1 figvw sed from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular;w 3zfn axw94(je(pzx speed 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 1 bfu8ecnwm+ mgg3 bw 0-.g (vz8m(txxmass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational colzbfm.w+8u-( mxn g 0e1bgmc8vw3t x( glapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile99 hund/s3b( rsgnny ( is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of b(/ 3 9( 9nghnussrnyd1400 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 illustrates a* n hwu,tp3 vx 7ty31rsc-vtd-n $H_2O$ 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 horizontal surface5s9qj e aaz3 7.xnxz)2tcd *zh 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 p5hrbj4hj4 :m8euz ,g tww.k/(i.e., we ignore friction) about a hinge attached to a wutr4 zgb j/ .p4whh:w5j8,kmeall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiuh3alw/vxzw( +s R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step againsva+/hzx(w3w lt which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn w0b ;0xqh( in2neurri4ith a radius The forces acting on the cyclist and cycle are the nor(4n0 uib;hi 0nqr 2erxmal 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 kq,6c+pe9oxdz8: ljxb5 5 etfand begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm xdb,9lkpxqzcet5o+fj65 e:8 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 .rg*ra v pimwj::w 6y)thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and witp:*a6ryww:jr .i)mvg h arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cyl)h gf3x,uu)4 a 7mfcidindrical plates, of mass uf) f7x h4c3um,i)agd $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alyg+ 4; ,vd* eevoqmxr3ong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leavq*;v,3geyx4 evmr+ od ing the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do npuya gs88x3/ol ejc :+ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car re;q4qa7zuc o :aowf1h-duces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the ang uha;q:-cw4za1 qfoo 7ular 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