A bicycle odometer (which:) 5.,yrl fhqi9nlib kq d3)aj measures distance traveled) is attached near the wheel hub and is anhq)j5,9 klr:3qifdb i) l.ydesigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular 5rg ,rod hvb7 kaj8 ubo510(nnvelocity. 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 ao j 10vabrod7nb85unk h(, g5rnd/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 aegrt jw a2 ,w(;h(eh1 mki(le(ngular velocity $\omega$ Explain.

Can a small force ever igyx5v7 xikm:3nv9 me52njj. exert 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 do a sit-up with your hands behi84ym yszs4 +d :/ahtiind your head than wh8z/ 4+ys:ma4yisti hden your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has se(k* fdpr/co+m 7e7x) t:zqn lcven sprockets 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 smalco/+):cmrl( pt xd*nfk7q z7el front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with 8ioiud d25 n4 coyj3tw4cpl0559 addb flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageoutpc3iddn5d4lw54ab2 8iujy do9 0o 5 cs.

Why do tightrope walkclzlj,ssb/l ;k*bz 65ers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the netmt1 s;+oh8gpx tzn 5q. torque also zero? If the net torque on a system is zero, is the net znt;t .5x8gs+q mph o1force zero?

Two inclines have the same height but make different angles with the hch gaj0* dob3vnuuuye).5p-4m)c,s norizontal. The same steelemn-ga )h0u b*jo5cucs)3vpd n4 ,yu . 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 rollioclzkx9 224w+ nxc 7qhng (from 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 energy4xxochzk w9qc 2 l+n27 at the bottom?

A sphere and a cylinder have the same radius and the samek qqna5 :qf opy8 h74i/54qcyc mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the5/q4:8cf7qio5ky y hpn acq 4q bottom? Which has the greater rotational KE?

We claim that momentumep g( b fz*sm3r2kfqj+ p(06yq and angular momentum are conserved. Yet most moving or rotating objects evp6 2qzfq( f3 bk0m*pe(grs yj+entually slow down and stop. Explain.

If there were a great migration of people toward the Earjs1wgro w4m 7 pel6 /o(hj1ap6th’s equator, how would this affect the length of the day4(or asjm1pg/7eo1jw6w phl6?

Can the diver of Fig. 8–29 do a somersault without having x drc6 j(8d0k elv7wsaz8c+;a any initial rotation when she leaves the boarda c;8jrv wzcs0d7e6adx l(8+k?

The moment of inertia of a rotating solid disk about an axis through itoy-3zplq/e- r g*v6 yds center of mass iyzg6y*l--po3 v /edq rs $\frac{1}{2}WR^2$ (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 23 .bwju,abs m3-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Exp 3busb.3mwj, alain.

Two spheres look idenfx.rsp*i* r:hxgorl;/r 5 b5w9tef1vur .yp 3tical and have the same mass. However, one is hollow and the other is solid. Desceri r13bx*. xtr5r5 f*vgfsuly;9 :o. wrp/hpribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horiz2u 67ejtwg/ taontal 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 the angular speed/tw2 egu76a tj increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntab3v ag,/yotja8d0i 2v jh5hi (jle. What happens if you wal2 jj 5 oi08vtdy,ahg(va ih3j/k toward the center?

A shortstop may leap into the air to catch a ball and7: 0cs3xt inri3nzf ,w throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will nfinzr ,itxn3c37w0s:otice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of consevgo7) tp9va,brvation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secon,v97avbo gt) pd propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 7lflx y4rg)+8fon 3ggnbu+1z$^{\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. Calculx;a ;*b ycwd i9nhu08tiqlg56ate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen ci;iyg9w6q0 hl bt*;5 x8naudon Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directef4igi x 4q+y(yd at the Moon, 380,000 km from Earth. The beam diverges at an yix+i44 gq(fy 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 ratev776qqpj m (i :az4oee of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular ov7:p( q4q eezami6 j7acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level f1yb3ktbz c5n5 loor 3.5 m to another child. If the ball makes 15.0 revolutions, whackb5yntb 31 z5t is its diameter?    m

  A bicycle with tires 68 cm in diameterkju- ac1vhsf;q7o c31 travels 8.0 km. How many revolutions do the fscqu7 ja1 o3kh-1 ;vcwheels make?    $rev$

  (a) A grinding wheel 0.35*walojws*oc 2 gd.)t5 xio*e; m in diameter rotates at 2500 rpm. Calculate its angulard i2le wc *ogax; .s*o*wj)5to velocity 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.6sxdk5gqb0*kiz4,p o 8–38). (a) What is the linear speed of a child seated 1.2 m from the cente6i*x5,o kpgz qkd 04sbr?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around pku m43+yjwt*the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed .k ;()lbrdq li ttek 1/wj:2zuof 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 particle 7.0 cm f-hh6kaf cs u+k7a k*y.qkday1ix*-6 prom the axis of rotation is to experieyaxk6ca y- u*h.h*d- kq p176k asi+fknce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates tj8)glgq4 2u ta-u3cv-, ig sauniformly about its center from 130 rpm to 280 rpm in 4.0 s. utg c8j)qga2l3sag4viut, - -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 dun4lkm;2 y2w;,cy s *r 8mgmw$R_1$ 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 n to hh,m(ok;)spacecraft put themselves into a slow rotation to distribute the Sun’k,tmhoh)n (;os energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The 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 uni9*eg6z,dbd l* ;dafklformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it, *kfzl6bd; dld*gea9 turn in this time?    $rev$

  An automobile engine slows down from 4500 f8m:k7d42 ;qpux2 v1wxj mx xqrpm 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 flx-: wq/o,un r+jh0me1liaj*iying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions -,+rni0qow:e mjh ij/ a*1lx ubefore reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpe2h2qeb +9q m +t,sah4kti0pzm to 360 rpm in 6.5 s. How far will a point on the edgemhpi be4+9,02 htqzsk q te2a+ of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running mfh7:*np m*k 5 e/rbfpat 850rev/min It turns 1500 revolutions before it comes to a stnprm5f*eph* bmk /7f: op.
(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 spee by*kx9txct7 4d uniformly from 95km/h txtk by 9*47ctxo 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 65 revolutions as the car up2;pwwo(sef;999v wh bard3reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.oswph ;ar9e;39pf(vb9wdw 2 u 80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing 7e)*yy4 fp sln zem4gz;g) 8uxa hill. The pedals ) ms;nz p)4el*g 4yfz8 yugxe7rotate 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 53z)j cg,k/n 5. ouwfw wf2,ohh5 N on the end of a door 74 cm wide. What is the magnitude of the torque i32/, on.,h) j wzh5w wcfokguff 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 ; g731nmtlksl-/uoqcaxle of the wheel shown in Fig. 8–39. Assume that a frictio sm7l n;c1 gl3k-qt/uon 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 ends o4 -g 2 1(.zrcfupzz6h2hfex8yj9yvmmf a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitudep.9h 2v -(c2fg zhryyzze4xu fm18j6m and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a tor* fze7xwd0t9sque of 38 xwdz0ef97*ts$m \cdot N$ 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 ine,c j6jkmwna) ;rtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its ,w jan kcj;)6mcenter.    $kg \cdot m^2$

  Calculate the moment of inertia ofdwk/zk rp ;72xr-.-qzpfmh q2 a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The massd- kh/mx w;rzp k p.-272frqzq 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 af()(2 o nvvhcaulz72 u horizontal circle of radius(nca o7h2u )2lv v(fzu 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 bowl0 4w6dmwwnm63 lo+km/7wc f gy on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N t7 n6lw/cwm 6 wy34gmmf+dkow0otal.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point o;akv8y:gh*s objects shown in Fig. 8–43 abou:savo ;g 8yk*ht
(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 conrwv/- 9k 9sdn;qj- l izej+t11ca0vzgsists of 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 pb1vw(t-02bdu0 kp jhcorrect 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 required0b2k0 vbuw -t (1jphpd 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 cylind. v2 f6hoj 8 d)u-oudx+r8h0ons5sffder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate)ou8 -dd. ffjx os+5o2u hhs0nd r86vf
(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 reoqs2,74ftq o zst 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-go-round and is ablejy 7e0*k8 m; lsehn.1npqypx* to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite eam k8hx;nyq*nj71*elp s0e.yp ch 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 shut off and is eventually brouh)upj(ms; 7v vrv3/ bwght uniformly to rest by a f;/vw r)p3 shb um7(vvjrictional 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 3.6-kg vh4p*m)c+j 4)ibnkq z;g/u h dball 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 5ugud aoj022qp1/ vhvsolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleratp15vdhgova2j2/ 0 u uqed 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;fg 1ekewxl(:,e em8b se /t.q thin rod, as shown in Fig. 8–46.xwtb:sge8 /ele f ,;(q ee.k1m
(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 4 -3em zlvkcmj*nd u2h;,c(z nmasses, $m_1$ 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 wphs;bgb58t +p ithin four full turns (revolutions) and releases bpg; b85+hstpit 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 mbau-q19 kpd7 soment 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 develops a torque of 280 d0m 0okgw q,x1/ hh (aza4ihu4$m \cdot N$ 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 1+1ddrz(jscr mw2u .z lan 1+zjd2mrwz1 s.uc(dr e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth ts.sf yt024 qcj1-c yd8ubgz1h(w 4sw with respect to the Sun as the sum of tw fs4wgc -c dhy bs1(wu02t.y4z1q 8tsjo 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 ohrz 05ipa-1c df 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of-chai pd15zr0 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wit1.uxmq6c2 xp d/ )dmaxhout slip xdd/21 x6.q)ummcxa pping 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(5(kr,ukh/sc3xd1 j( hpfq mj its lower end does not slip. What will be thj(ck3dh mu/pf(,j5s rk 1xq(he 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 mocbl w*kbua hb/yy )+e23(er5jmentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular spe+ycke*/)by 5rjw2b ahb( l3ueed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform d;m5buy( ezd.b/ feu) cylindrical grinding wheel of radius 18 cm when rotating at 1500 d5;/bube f() mde.yuzrpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, handfk) x/gwy* l-zs at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises hfl)g-/xkw*z yis 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) cg+ ;pv+catl*nan reduce her moment of inertia by a factor ol+gnctv;* a p+f 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 increase her spin rotation rate from an initial ratbo - /h16q hq0c(1 efphjq9qbge of 1.0 rev evj p0h qfeho11h-9/ qbq6qc(bgery 2.0 s 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 nai;1tg-0c8ovgw) q b bu31 -a spkh3j2 z9fwtrotating around 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 the1;- pusbat9q3at8cbnfg2z iw-ghk0o ) 1jw v 3n throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning a2k fb ;j3q7jqdt 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 mo-89 ;axyv1dxc ft/svfs8 mv6kmentum 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 moment of inertia I is drtv3yrfl mkm h g5 *99dt,itw:2opped onto an identical disk rotat *9hymfm3tk9tw , 5drtgl:v i2ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 kox2pz6r m ;.g$rev/s$ 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 otxbi p/5k*l oz5(yp3y f a rotating merry-go-round platform of radius 3.0 m(5z i/xpt 5lobky3p y* 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 freely with an anhp, n ss0l ;33)oltjhpgular velocity of 0.8jpslos 0 3hhn,l)p3;t $rad/s$ 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 itkbgh6h7/ 0 1uzh bj9ecs mass in the process, and winding up with a radius 1.0% oz1uh j k6e b9h7gb0h/cf 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 e n+7rivj/ h6kzo8u;c wxcess 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 fcc y.aptig3*6hjtdqc6nin0zl4h -p * v,4 u.$ 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,u14 u wx2gcf 2y2xdnw) that can rotate freely without friction. The moment of inertia of the person plus the platforx2 wg2xf21udu4 wy)cnm 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 tux 1aqx2dn6 3rmrntable that is mountedxman2d6rqx 1 3 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 rolls on the ground with the end of the rope lying on t 9gepdk*h6di1, )zfluhe 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 sp6z,1f)he*uddi kg9 lp ool’s center of mass move?

  The Moon orbits the Earth such that2no2b s mgw41:o d3d cbh*5rde the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat thee*b ghd b2c3m1 d2o4s5w:nr do Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fr vx*xutem/qf8;)l ) goom 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 in the shape of a uniform cylinder of radiudngeyjb,o++r jg) .og4th( /gs 0.20 m acquires a rotational rate of)hgr/ggj(.nye+4 , bt+oo gjd from 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 cylindragc1 tp2: d.pzical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of itst2:zpg pcd.a1 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 the rear 6sa+ smkn-xv c -bo fdc.w:l;)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 Sny.y5 cwcym5 8un, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular moy ncy5.y5c w8mmentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low )heh7t;fj wwp:9 dsp4-pollution automobile 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 cylindrical flywheel of diameter 1.50 m and mass 240 k;wwp)j4d7thfsp:9 eh g, 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 illustratezok 2,i/ a-ahklwwp0aca4c621q h o, es 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 horiul64fifng rp(m)/.j. ay qbs+ zontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frict,clk/zt r oj05ion) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of/5lrt,cojz0k mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is stafeijy55a9 xcpb;: 4xhnding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step againsxehpc: 9x54fi5; yabjt 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 roa oxr4+ pjiwpq 4,61-zn*xh,j5dsrgzd d is making a turn with a radius The forces acting on the cyclist and cycle are the noj4w,zdg,ozq i5p* 4xrsh j n 1r-+6xdprmal 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 lengthck; mdu5s1xxpz; v*8o 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelezdx;sm1 x5ou;cv*8k p rating 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 -sd7yz73 iiirsolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched -iizs7 y i37drarms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cy.ztrdn p-z da)2g5b* lbv-sei) 1uc;s lindrical plates, of 5) -tbe2a sgrlbv;d.z *p1-i )zud sncmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 thep): atjifkinmfgli c2g l.h6,5;b yc9u4b a ,, looped rough track of Fig. 8–58. What is the minimum value of the vertical heighuni ,g,.4tfgi bic 9,k aab:jl2f6m5;l chy)pt h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume osxk62zbq(ps0x8. c8 9/vzjg ax- ukz$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces ilxys9/1y,p+ppa+nv i1un hm 3ts speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eachx3p +h1am/,y+l9yni nvpsp1u 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