A bicycle odometer (which measures distance trav/4 egi:buo iru+0gm 2ieled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if yb 2 g0iu:i+oe 4girm/uou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates ghk+) t gv6e4iat constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, doe)kg6iet hv4g +s 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 , h:xz.q,pjnogs, )motfj r.0be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?/(avs4 d;di 5u2 hg2m xbthry, 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,y/gcx 6ybzdo- 8kg(q do a sit-up with your hands behind your head than when your arms are stretch , 6kdx(z-/bcyoy qg8ged out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wih08z pd*dk7t 9 8azu.pufld4heel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket oru.d8 k4dh*ztf adp7plu8 0i z9 a large front sprocket? Why?

Mammals that depend on being able to4qx gh 6vmd13yz80ydw run fast have slender lower legs with flesh and muscle concentrated high, close to thgzq m0 y w4186dyv3hdxe 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 sg5/qgdd),walqx 3y tur) 1t (zybb2:, narrow beam?

If the net force on a system is zero, is the net torque also zero? If turqny.q :7 be1kgaj, :he net torque on1g,7::ueqb. a y jkqrn a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the*x-zfmrxwa:9) amf0 i horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the mfxmx0r :)a9ai * -zfwball at the bottom be greater? Explain.

Two solid spheres si txqw0p8 qf(fpw6sa2(multaneously start rolling (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 gre0f 8pa qs2w((q6xf tpwater 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 stk81srd :tndf,;wvcpd aw ,5f6 art 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 the vns8cfkp dd,t5fw: a,61w;r d bottom? Which has the greater rotational KE?

We claim that momentum and 5uk0h ks4cajpe( p2z *angular momentum are conserved. Yet most moving or rotating objects eventually slow down and pz2 u*ashpk0e jk(54cstop. Explain.

If there were a great migration of people toward tw a ;s8iwx+ u*z6,q8/he ummsu6lwiq)yc5s ;khe Earth’s equator, how would this affect the len+ws) suweiu5q z6w ac* 6/8slkh iy,m8m;xq; ugth of the day?

Can the diver of Fig. 8–29 do a someri,l ).tgb,afa sault without having any initial rotation when she leaves the boar,,) b gtfa.liad?

The moment of inertia of a rotating so p,cxc u1fpl9qxb9z 6v m5gq1m/,0korlid disk about an axis through its center of mar 1qu1kp,6cmqovb,l x g/5zc90f 9x pmss 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 holdingodac*-k e:(b .ljil 0m a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease,(ea:o.imbjc- l* 0lkd or stay the same? Explain.

Two spheres look identical and have the same mass. However, 7l40uceq a.vmv v*l,5ygetp fm)/-k hone is hollow and the other is solid. Describe an experiment to d.e/)hylg ,0*vfk avcl q7mpvtme5u 4- etermine which is which.

The angular velocity of a wheel rotating on a horizontdmxywspbix l0fo -i8-t/0v.-al axle points west. In what direction is the linear velocity of a point of/y- mx0 bsi .v0dtxo--8lwi pn 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 increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatg( d z2g82hqofing turntable. What happens if you walk toward the gdh22( fqzgo8center?

A shortstop may leap jcij82b/ +veqs2lu x:j*aq c d2p.(jninto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). E+: *2ju/q 8cc xjs2dbjipnl .e2qjav (xplain.

On the basis of the law of conservation of angudy9f lmf) 7tq9lar momentum, discuss why a helicopter must have more than one rotor (o)9ydq 79tmff lr propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles ig gaf,: p76u8yax7o( (e bwwqcn 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. Calculwz:ox4ah- e qs* 5rha9c3 hr5mate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.re* s 3m4qw:5 x rzh-aho9a5ch
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 381d m t.d2nghng2ns i:+0,000 km from Earth. The beam diverges at an igdmtg2 d.2n1s:+nhn 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 0shm ng,l )t5zcn2r,p motor is turned off during operation, the blades slow to reng ,,0rhznsm t5p l2c)st in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on g (pa2nnq;+gj a level floor 3.5 m to another child. If the ball makes 15q ng2p;jga (+n.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter traver(3q) otgv*wy0 y7 /heo 6yjmtls 8.0 km. How many revolutions do the wheels ma oq3j06mthg( ytw/y*)veyro7ke?    $rev$

  (a) A grinding wheel 0.35 mf8,fe cu-suk7aq ,6sbz1g l: raw+lu; in diameter rotates at 2500 rpm. Calculate its angular velocity inu wlr 8f- kbu6 q7asguas+, ce;z1:f,l $rad/s$ $\omega$ =    $rad/sec$
(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 h e:njf 45d8sgrevolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child gn48ehdjf :5s seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun9qqb gwn0c2 e 9f0eo-k    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin1reh+y05 lwz ht
(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 i;pt/ ) wjuu7ljf a particle 7.0 cm from the axis of rotation is to experience atju7 u/ljpw;)n acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratesa1ua3 4tg10ci-*xq tej u* :q5c iyxga uniformly about its center from 130 rpm to 280 rtc1u ga* q*q5aiatueci3-:4 x1j gyx 0pm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusv6ns7aesdw 8 9 $R_1$ 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 s/jh2ztw a 108d sn2lszvnb:l/ pacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelwz8 0/ d2 vbnslz21h/tal:nj serated 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 uniformly from rest to 15,000 rpm in 220 s. Through nj:oqzg/e.3) .-ilu p 2ou4rlz o f3qthow ma/nf4o.)3 jl orqt -lg2z:.z3i eupuoqny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 .5j lqwg58 ruarpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirlis4 3rfi.ryeg/ pjf (4vng “human centrifugee.rsi /j gp(f344r vyf,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly fu owbm5a3zr5d2u2; hll- 9 wedrom 240 rpm to 360 rpm in 6.5 s. How far will a point ruwhoadumzd93b e ;2255llw- on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 15182uz 9( q6i;pmjlu2 cmaaopj 00 revolutions beforejoul(c2pai9 z1um ;2mpj 68aq 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 r 2nrex 8pu6aw-c. 8zuievolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.8-uz6i 2.a8cpwxnur e
(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 carq++*ju4 i id7 cj0rtluo 5dv;ihnr 4u2a4c7ro reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8o4+ +d5clr*2i;ioir7qth un7vu j4 4c 0jarud0 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 eag: sykjxl 3.,5b5kmswch pedal when climbing a hill. The pedals rotate in a circle of ras.5x:ymk s5,b 3lgkjw dius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Wp9cl+4.h/anre6 .z2y s zvqkm hat is the magnitude o2n.z/ m kc4slre6va z.yq+p9h f 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 mklm 3 nx8ht,8d5 i)*d8w irpxabout the axle of the wheel shown in Fig. 8–39. Assume t8pki8x xi nlm5 dwr d8hm3*t),hat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod 4/-wv)pg muubpa2 pmn 0j4 8skwhich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the m n8m4 gv 24mujp-p0spaubw)/ kagnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening m l) l/e:t2)sdf,qbveto a torque of 38/fm2lv):dt,es b )qel $m \cdot N$ 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 sph*a:dqoh*1 (mp-ddqm x ere of radius 0.648 m when the axis of rotation is through m xpdd**(hq1qa : m-odits center.    $kg \cdot m^2$

  Calculate the moment of z +u9 n*sb:m,0 :dzl3omaxqx hinertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can beus:a* bo,m+n0 zq xmz93:hl dx ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a th lo7d;a;ydjm8 6d rpp+in, light rod is rotated in a horizontal circ lpjd8 +6 dy;;mpoa7rdle of 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 o ih9 l5 0rtnk5bj0x*xrn a potter’s wheel rotating at constant angular speed (Fijxbr5 5*k xni0 trhl90g. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–lbuxbp3cz:7oj3bnb0cf:/ e 1 43 aboutjo3c7fbu0b:x1 cb l zpe3nb/:
(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 atom )x7f-nvl:(t5z ( f,chxt51wxx elawfs 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 sa ha9u u5 c),7af,adlbb f;fly5za3lk6tellite spinning at the correct rate, engineers fire four tangential rockets as shy d,a9;fa 735llf,k a5uc)bl b u6zfahown 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 cylinder wi(9cl2 umk + l6gi x3ziixua-b2irw;;ith a radius of 8.50 cm and a mass of 0.;l9( ;u6bi wga+-rkmlzc22x xi 3uii i580 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3.)iaurx nolr0 6v 1/zn $rev/s$ 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-rou src;/q)fuuv9 nd and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume t/u;r fqcv9) ushe 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 rpm is shut off and is e7:uj4fdbest w85i2 t pventually brought uniformly to rest by a frictional torquebif8pw:t tsj eu5 d742 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--7eaw 6bgq atvjyhysi(k 4(te6c 6-1rkg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action oadtbv q9ij1 v/xqh.7ld 8 /,odf the forearm, which rotates about the elbow joint under the actio /dod .tabqq91jxi,vldv78 h /n 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, cnq+t-(m 0uy yas shown in Fig. 8–4 +u0c y-nqt(ym6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses, vji(s8ou0 q7u $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turll/hhup c1v : pgai+w/ij-/ i0ns (revolutions) and releases it at a speed of/+:l viia1chjp-up li 0/h g/w 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 me7cn n/ rf7i4eoment 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 devd,6h/zxxhz+b elops 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 roll d qd3s.yfo98x4r8:dtaw kf) as without slipping down a lane at xwq9aff 3s dkrd.:ad) 4 yo8t83.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum 4d.3usa u 9bd/nveaxd7:/xr w of two terms, uxdde/xr9 vsaa4b3dw7u:n /.
(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 8 lmd 08l4xak-xna0 q7(k96npdaqpul much net work is required to accelerate it from rest to a rotation r 89a60q0adql(-xk lnunaklm87pp4 xd ate 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 o vl:+wd l:i:hstarts from rest and rolls without slipping dolh:+w lv ::odiwn 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 stafc k:aaz t(-yc*abh ::rts to fall and its lower end does not slip. What will be the speed of the upper*y:-a bfzctac:kh : (a 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 ob/-z;y(nrbw 2k+hds : +llb dzn the end of a thin string in a circle of radius 1.10 m at an anguld hzz wr(+b2nkbsly;+:d/lb-ar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8,h2sdvy h0v6e -kg uniform cylindrical grinding wheel odhh,y 6se v20vf radius 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 tha*sx,27 qceu3 cbf:4bcqh3jxq t is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed ofe273 f3 cqq4*ccjx, bxhq:sub rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the owoy 0l(d6/ m-eig wgn):gq;odne shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If sh(gdm)idg;w6/w0y:e oq- og lne makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 - cjow q3ok z(fn/ss7p09-aun rev every 2.0 su a3n0fcjo(pz- /9n-kq 7wsos to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throughnbb p)(v 8tbcq rae0xz nh;7 ;c 4c(7o+v6xyxn its center at a frequency of 1.5rev/s The wheel can be con( yb7ve) oxb0pb7;rc;q+ncn t8xvz nhx (4c6asidered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 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 angularz- mq-zm,wr+ f 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 m c83i+kmtw7aw*a n/ apomentum 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 i4yu9qcc8af b,s dropped onto an identical disk rota 8y,49qbcaufc ting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at pbs8yxg97iwe,0z 3;mfl zzrto i ;m*72.4 $rev/s$ 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 du5 bn cg0052ccmw4 pq6i ubu-rotating merry-go-round platform of radius 3.0 m and moment2-b04i5cuu5ubd6p 0cwqcngm 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 angular velocity opy hfh-,v2 y8of 0.8 pohvh- y28, yf$rad/s$ 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 eventual9y-cd*k.+ifgsal-4 g0e s ajgly collapses into a white dwarf, 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?a ek.9a0gys 4-* s-+ljcifggd(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 invol- +y 5vvw)nhcw8mq8y yve winds 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 bb)bj-n+lan3 $ 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 plat(:kg(noa0th r form, initially at rest, that can rotate freely without friction. The momen :oahg0(kt r(nt of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands a9 ets2 x-r*pfpt the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless be2spt 9x r*f-pearings 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 grow6e*.vyjnm+ ct ks vb3).g4jl und 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 doe.ws3ekljvnv b 4tc ymj6.g*) +s the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwa4 fl6x3qvakvrn,0 ,u cys 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 the Moon as a partic,nqu0c3 k afrlv vx46,le orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from 8d,ugea-huyc);q wsv1h( q* d 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 radius /(8 n6mut/xgfc7 7 no iodwkd40.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant 4nu oi w7gc 8(o/dtxd7k6m/fn angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 z tf b*iff.k2/:tcl;6.tkry :0vlffsl, um* v 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 cz*. :6tftylskk :frmf.*lvf0,v l bc2f/;ti uonservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear whercvd. osyi db1g. 7g9h38l 3xiel ($\omega_R$) 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 ob 6kf5t 6wc mn-rxwk;5f 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 neutron star of radius 11 km, losing three-quarter n5kc;6tw-f 5xw 6bkrms of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy * g,yk9hyuai5stored 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 kg, and should be able to travel 350 km withau9*ykgih5 y, out 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 illustra(g og4g6m ww9 x:r ,sw9eb:beftes 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 hori:ksgq .h5 .nzxzontal 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 ca k f7iz43giz6gn 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) thz k 7gii4f3gz6e angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius)k/ wsews rfzf q)y7+u,1y y0s R. It is standing vertically on the floor, and we want to exer/0y7+1f sf )us)ys,r z kyweqwt a horizontal force F at its axle so that it will climb a step against 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 6abcj/ t7 y20pqn:wobroad is making a turn with a radius The forces acting on the cyclist and cycle areyojn20 bpq6c:ab 7t/w the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of lengthar()y) ynx rk* 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceleratirnk))yy( ax r*ng 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 soli.d-xphkti31 vli i57j d 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 agaij.h5 xi pliidt13k7-vnst her body.   

  You are designing a clutch assembly jj4zk klh+9wkr)b. mz8bp+ d1 which consists of two cylindrical plates, of blkj 9mzk8p)+wzbk d.4 rh1+j mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 s;o n1f/,ijcirough track of Fig. 8–58. What is the minimum value of the vertical height h toci nj f;/,1sihat 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 9 ;;/t .p ejv+amlxotxnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformz/df4 l5mwz 99jt60,wtskjtxly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each 95jmwfz9w4kxjzstt d6/0,lt 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