A bicycle odometer (which measures distance traveled) isc vn,8i3:+snkxa n0la attached near the wheel hub and is designed for 27-inch wheels. What happens if y ci +ns8va:3na,0x nklou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the ripj2x e3:stsz 2tp),oa+qov0 xm have radial and/or tangential acceleration? If the disk’s2x3v 0t ppjta:qs+,sxze 2)oo 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 i-l ;(m iln.e3vucp)ydescribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greae8ig7ert w*ygr.95 d9 orze1lm+/tf pter 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 vn7+d 9guuh(k sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer th+7 hgdu(9vn kuis.

A 21-speed bicycle has seven sprockets at the rearkve/)5/hzy96de( c3n me ns tvvdvk2) wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprock3ks56nez vtv)ke) h//ye9mv d(2vd c net? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bebz-r o7d8i ) opi- 0xb6ab4okging able to run fast have slender lower legs with flesh and muscle concentrated high, closbo4ozg8xk 0)ii 7ra dbbp--6 oe to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (F0 bco5sm*uot(: 3cud yig. 8–35) carry a long, narrow beam?

If the net force on a sy9 cwu*bv /f4a.:-ox8 fv+9lvxy vkflostem is zero, is the net torque also zero? If the net torque on a systemvf b 9vo+x f.-4uk:*v/cy98 llwfvxao is zero, is the net force zero?

Two inclines have the same height but make different angles with the zo.zx/ee f60 al t+9phhorizontal. The same st h/o0epz6al.xz+ft e 9eel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dow.u fd rn(ax3d(n an incline. One sphere has twice the rad rnd x3ufd.(a(ius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start .-br,phcg 8r+(o0:n zfw6zyio b e4vc 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 ener8 -(pbz:+zcwv e, fgorrc4bni.6h0yo gy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movin+t bk ntt7ai4k;gl8-c g or rotating objects eventuatiankt 48g;lcb -tk7+ lly slow down and stop. Explain.

If there were a great migration of people t 8co1 mkre.7z7nm9nmkoward the Earth’s equator, how would this affect the length of the mnkz8me7kcr o9 mn71.day?

Can the diver of Fig. 8–29 do a somersault without having 7fi/cui49* ikf(oqw dany initial rotation when she lw qi(4d ckui7*ffi/9o eaves the board?

The moment of inertia of a rotatit)ng9p 9xm.z, r *2t9 kl5qz.z mef55t rhefnang solid disk about an axis through its center ofzr 9 enlq 5g),9ft.a.z tp2tk x fenmmr5*9zh5 mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sittics t1p: t7q4 wyvf ot;*ic-di6ng on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will w4 qc :t;yv1-6i tsti7cod*fpyour angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havem8h k xzqx )0t(;6xppz the same mass. However, one is hollow and the othth0x ;m 8q 6zz)ppx(kxer is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poisn/wai5gxmxn8ey , u*+-v mx3nts west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration x,xwxisg v-/uanm 3nm5e+*y8points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge f(neb tr z9j.lb)uc 4/of a large freely rotating turntable. What happens if yotl4r9zebj(bc.nu ) / fu walk toward the center?

A shortstop may leap into the air to catch a ball and i91za- mhrv0vthrow it quickly. As he throws the ball, 90v1vhz a-rm ithe upper part 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 angular momentum, djs8 jl8u(5bm :yx +zbj8k ka*jiscuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopterjjxlb8 m:*(8z bsj85jk yku +a stable.

  Express the following angles in radians: (a) 3m(v+ c ox- 0ndvm 5q5baw8nm0z0 $^{\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 amazingq6y u 056i(rdxevsv b( coincidence. Calculate, using the information inside the Front Cover, the angul6v(u0y5(vxed i rb6s qar 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, 36jjjrp p4e pyj l03p2c m/s(o280,000 km from Earth. The beam diverges at an(0rp2 3jl sj4jpo6/ec 2yjmpp angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned to6 ;izkd8 5b io/pn1soff during operation, the blades slow to rest in 3.0 6pdzbtk ; on5 ii/1o8ss. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level;i (pf2lzg q9r 9de(ylksd1:w floor 3.5 m to another child. If the ball makes 15.0 revolutz2:ky fli9gw(q (e9l dd1rps;ions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8d1f q.8,ypmtwno8x * c.0 km. How many revolutions do the dtx 8wyn, omf. 1*p8cqwheels make?    $rev$

  (a) A grinding wheel 0.35 k6ya4af3sr s.i ch p9:m in diameter rotates at 2500 rpm. Calculate its angular velocitskirscf 9a43.h ayp:6y 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-rounz0, ,kqhod*soe6l(u e d makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center? ukd oe,h0(, qsoze*6l   $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocit*szra 9t5a8mx0d +cskvq55 dki* 9jwyy of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a99 pdchjxsn.m h0o+2x 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 rotat 46k.esebo; gfe if a particle 7.0 cm from the axis of rotation eo;46 ekb.fgsis to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates un rv4, 2:y)oe z fn1/do *: /whvgzvjbcoq-ie2fiformly about its center from 130 rpm to 280 rpm in 4.vz e4zqfy2o*:dh1ebn/fogcow vv r) ij2,-/: 0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius imxj0shb 8*dw/d bb 1+$R_1$ 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, astrona(c h0 wxp4hihi(p.a0eb x)m)i uts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from noxhh00 w4 hiiep)pxmc b)(.i(a rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,u c:8 * vd8dtjbxvj+9x000 rpm in 220 s. Through how many revolut dj+vcxv ju xbd9t8:8*ions did it turn in this time?    $rev$

  An automobile engine slows down frd3h/ 5ys-.2atrg jf4 xh7+q kl eps2iqom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of r,f g8505 fi rch.5xubqrq/kgflying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachigcgfufq b5x. 5/r0hr8r , 5ikqng 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 uniqj0f44;h+67q blc +hyzpea ff formly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the;h0czf4ff6hal j+ qy47e qp +b wheel have traveled in this time?    m

  A cooling fan is turned off when it is runniccy8+c u ;0j2t i3j.ddfrpa2nng at 850rev/min It turns 1500 revolutions before it comncd.pr2c2tyjf;0jc3 a u +id8es 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 itz wby wdojs95f 6s5.-us speed uniformly from 95km/h to 45km/s65wj 9dfsyb-z5. ouwh 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 reducbllrzq2 3ld 96es its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0. 23lz6q llbdr980 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 puts8o8 ky5fwl, ev all her weight on each pedal when climbing a hill. The pedals rotate in a circle of wl,koeyv85f 8 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 td,+p v7j*nktuvu7sp e* ;l)c xhe end of a door 74 cm wide. What is the magn7 jsluvc,k); 7upn+*dxv p *etitude of 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 the wheel s 1 ;hl,i*zhl9goqn kl gf4g7-vhown in Fig. 8–39. Assume th -4i*lzfo l 1hqgglv 7h9;gk,nat 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 at8a *cp2 mn5t*nvh3qu mtached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizonta cuvn8tm5qp3 2h*a *nml position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinderfnk 2mc-pzhy9:wl bx*x ;w0-bk vi;i: head of an engine require tightening to a torque of 38 h-pmkb ;:k2y zwcl09f *i-:nvxib;xw $m \cdot N$ 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 b,o4ook4 wj ; r.-dut8;zrh,e 0zvcpz 10.8-kg sphere of radius 0.648 m when the axis of rz jz 4r h40- czwp;ob. dok8te,,;vurootation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 ca 09gin3 q0itzm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?) 903intzqg0ai .    $kg \cdot m^2$

  A small 650-gram ball - 3docrreq..tj4a.t ;d r )*bjmor/jnon the end of a thin, light rod is rotated in a horizontal circle of raqtdja)3* -r;. .m.ootbedcj r/jnr r 4dius 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 atu0yvb, or9/ rk constant angular speed (Fig. 8–42). The friction force betweenv0 /bkoryu ,9r her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objecmmf0h.u)q8 eats shown in Fig. 8–43 muf hqe0.am)8about
(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 oxygogip +quiq ,,0q j3;xr5 dfj4cen 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 ,wm)-ghrpi7ospinning at the correct rate, engineers fire frph - wm,)7goiour tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinden ei.12 j dg-rrb)zwt;r with a radius of 8.50 cm and a mass of 0.580 kg.d - jnw2t;r gb1)zer.i Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acce7ujmhu)b,8:f71a f aj ra)badlerating 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 tangentxbt. 1.0 rbog6uf wo,mf(x+ (xpa x1tyseq,/ tially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and trt,(+ sy., a/f6 tguobfpe(.w1obqx1 m 0txxxwo 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 rpm is shut off and is eventually brot,.v81udg /r cj3;pb ;qc datqught uniformly to rest by a fri;8v/u.qapb ct1tj;dc rgdq 3,ctional 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 ball y6 nr,lc:8wdaat 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 s 1 xhnv07be:4mck8lyball is thrown solely by the action of the forearm, which roex841 lhmyvnksb7 c :0tates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as s .zzm qe;(uf4ux9 rch7hown in Fig. 8–46 uxmz(4eh79c. uzf qr;.
(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 consir co0 so4.m 8*tcz0qkvsts 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 accel8bm4etbdrb e0 kcz9- )erates the hammer from rest within four full turns (revolutions) and releases it at a sptmee80)b cz b-b4kdr 9eed 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 in+8 pcis2bkf ji;nfr rylrwt-6g(1,6a ertia 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 torqpmfv2 ;*dokb w( jg m6cm) g;),9kcmbaue 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 s.u v-t/k,jfo2yl 41jpd (t cghlipping down a lane at 3. hlp2tj-d1g jf,ku/4otv.c(y3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of th+atg3 5o7xxm rey*v c0e Earth with respect to the Sun as the sum of two mr+0c xx y5og3*et7 vaterms,
(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 much nx3kycnr +uc kho64o /,,t(nfe et work is4nuokfrht6 3ce/ o,x(kync +, required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without s2-xn5 j-ois,cdefb: tc z0a:c lipping do:e5-:t jc, c-ab2fx0 zsicn 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-nh2cj2a1ya 8y ckgt9 balanced vertically on its tip. It starts to fall and its lower end does nykycjaga28 9 tc-h 2n1ot 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 rotatinku lu ma--g*,xg on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.ulak* g,-xm -u4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum * +p6xla +wxm11k/c cwei+vv aof a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at m1+ w+l*c61 waxaicpv /ve xk+1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of) hn) k+sdqep 82afwq* 1.3rev/s If he anq) 2 ks8qe fd)p*h+wraises 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 momentsj khv2civ82;r(: fja of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in tjv:ks28cr j;v fha(i2he 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 of 19eolz 56c xby* ir-ifvtn+f19.0 rev every 2.0 i- 1yr6*b ftevlxicn9zfo+9 5s 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 rotatia 3dgp5*cn1evcg 6jxxt2m8q9 ng around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.02x1tg 6dvnj59qcp*a mxg3ec8 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure sbu ;s0a ydbi w3xcj8b82g09b u3k/ iozkater 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 momeyov sthk,ve: 6v1pj--fx6y f9ntum 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 ofp)hs oi7sz h/ki)tm4 4v+ddm9 inertia I is dropped onto an identicm )i/ 4d9 +vd)m7ih4hzsotpskal disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns nd a)7qvp.czhw *h+:qm jf;i3at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the mrsvspq1f ww g(3k0vb i14z(.center of a rotating merry-go-round platform of vb3sqk 1mv4rz(1ps(w gif0. w 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-g ggi6- a,:98pjxesm fhftn2o- byyd -ap0j/, wo-round is rotating freely with an angular velocity of 0.8 n6 -,x wg29y,-js ai-gp/b8efy odj0h mt fp:a$rad/s$ 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 de 0qeudwz6ra21twp5. 91 ybkowarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integerzd1qy0tw bw61k.r ee 52aup9o)$\approx$    $rad/s$ (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 cz.sw99vt8 nayy v:a 6winds 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 91n0opx6 wx lqg2sse5$ 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 fr)6 a bz6je9rqs bk2j2ueely without friction. The moment of inertia of the person plus the pla2s2u 66 k9b ajjrz)ebqtform 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 edgexqr-ce, 6oh*vh* ti(d of a 6.5-m diameter merry-go-round turntable that is mounted on frictionlehxr d,v* qhco*(i-6e tss 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 tm-abpr7s w/i9 he 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 of m-/as9 pi7w mrbass move?

  The Moon orbits the Earth such that the same side alwa*,cxw vjee(9 bys faces the Earth. Determine the ratio of the Moo bj(cw,*x 9even’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at ak) uxikqqgti.9p3+ t/64ncpi 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 unifotn+7 d nkng)w7rm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acctw7nd)n nk+7geleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyda).eo;h+ h7,ncc a +cnhzzm.lindrical disks, each of mass 0.050 kg and diameter 0.075+m 7 h.d znao,)h+zcc;cae.nh 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 released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angue 5hfn0bw z4nf3eib q++4uxurpb53j +lar 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 si v -2otreu(/6wc o7cwtze of our Sun, 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 neuc(62 /o eu t-wrtvcw7otron 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 lv;w::8 ads*hy0guo v 0e p(zotow-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 mads0; etu vyv 8:wa*0p(ohz:ogss 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 illustrat.ah jy7 voej 60q+ kz(opn.c0qes 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 horizontal surface at speed v= 0b+y mbia2x9nom*hjnmr y4q+)v+ *rd3.3 $m/s$ 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 c- iv1wk:oy , .:qmx2uid/ys eM and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, 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. 8–2yi: m2.-okseuv,1 wd:x/yqic 1g.]

A wheel of mass M has radius R. It is standi xh022/ko sbghn 4x9mzgnwsnm302z c6ng vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it restw0n os2x34h2 n6nhgb2kzx0zm s9m cg/s (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2csan- 1d 3p6wcm/s on a flat road is making a turn with a radius The forces acting on the cyclist a 6c-psn1wc3 dand cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a a)*7eal 6 hfie:bgim(pz- el e,i 4)ex0.50-kg rock into a sling of length 1.5 m and begins 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 torqpex-)e7:) i4 zfhii(,l6eeam*ae gl b ue required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s4 1 v175g mtngvjopaf; body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against hegpa1f g;4tvo5jmv17n r body.   

  You are designing a clutch assembly which consists of two cylindrical pla 7 qwyan*e.sd ,p.5f.o* pcrkrtes, of ma fqprdnws .5*epcy o.r,a.7k *ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped roucllidp b1s:+ v1dqpw(;uj 6 6jgh track of Fig. 8–58. What is the minimum value of the vertical height h that the marblbuvi1+d wlsql 6 1(:jdp;pjc6e 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 2ehnvw05-6eobty f l 38pxh2m :1 u5viu9kpkkassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed un+woe7 pf(/k qt luesw( g,h:m.iformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each ti/7wekpf:hlesw+mq(ut(,g. o re? $\approx$    $rad/s^2$(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