A bicycle odometer (which measm :pczg)h,2ca ) vpyf/ures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on c:c m pg2yv) a/pz),fha bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does 1lv37lgxat0, f 8y1)oaa vvoda point on the rim have rad)1oto8a l1lfgvva 0,d37yavx ial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a singlso26uuh,0., i qrj x.qrqq k/he value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torquih 4(;dfru l;qpz3p q7+t+v mue than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your he)s hpr4q txk4k79j28m g; eqgfad than when your arms are stretched out in front of you? A diagram may help yo 4fqk7t92;msg8 phgjxk)4r qeu to answer this.

A 21-speed bicycle has seven sprockets at the reaqp/v5wx c1b 9cr wheel and three at the pedal cranks. Invwq p59c/bxc 1 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 or a large front sprocket? Why?

Mammals that depend on being abv,:mkcyw(/sz7t p y0 mle to run fast have slender lower legs with 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 ptw: 0y z ,m(/ms7ckyvadvantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrohz*hjdxw0a6 p9/y 1c lw beam?

If the net force on a system is zero, is the net torque also zero? If thq1 )o2nvky5 m:eg m+dm g;s6use net torque on a qm:v2u es)m+;1 go ms6d5ny gksystem is zero, is the net force zero?

Two inclines have the same height but make differfn7. ywwo7j7u ent angles with the horizontal. The same steel ball is rolled down u7jf. oyw 7wn7each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) d1 s)iv eo,a7n3 adngi6own an incline. One sphere has twice the radius a a,3ea761i digsn)von nd 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 cylinder7pr /icfbjoc70 l)e( h have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom?c0p/brl7fj(7 hei )oc Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and j b/hos z)8)i0k50hft)kc:a-wsfy0. opzecn angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explaop/c i8) )ncf5)wz0ksteoj 0.ayfhsh k :zb -0in.

If there were a great migration of people toward the Earth’s eq*myz 8+ k8etxzuator, how would this affect they8k+t *mzxz 8e length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rot;tb+h j44-p+ek yicgj ation when she lec thg-p44b kj;++iejyaves the board?

The moment of inertia of a rotating solid disk about an axis through itsr( xdy+x-(uj /iw xc5q center of masiq(xdjcy uw(/xx 5-+rs is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg masc(zj: ,pum pa7 3 6byhcxeur dk*h;y ,2x5o(pcs in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Expm cya c7phdezkb6 uy,j(2h xr:5op,(ucp*x 3 ;lain.

Two spheres look identical and have the same mas/:iy( ew1vn oys. However, one is hollow and the other is solid. Describe an experiment to determv1in (/:eyowy ine which is which.

The angular velocity of a wheel rotating on a horizontal aubscy 6:/(wy .d8j/gfva j ;nbxle 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 accelerati6:.bfabwg;/dv jc nj(8s/y uy on 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 frl-fxc1q 6mdwzco 5 /x2i,iwkrns10v,eely rotating turntable. What happens if you walk tosnkl cf5mwxxd1qi-2,1cor iw0/ 6vz, ward the center?

A shortstop may leap into the air to catch a ba(qup c lvg7 -yf9rz,s)ll and throw it quickly. As he throws the ball, the upper part of his body rotates. If yo,sglc (q -9uf)z7vpyr u 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 momentaxhn3p. 6prk xy85vk6 um, discuss why a helicopter must have more than one rotor (or p8y 66vak xprxp3hnk 5.ropeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) (5ffxy6 m -otr8nsntk-yk 7 z,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 coin* ,y tvo,bo lr.h(ivp-cidence. Calculate, using the information inside thl brt-*,p. h,vo oyiv(e Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kmab7vg pa5jzx 9ja9-t( from Earth. The beam diverges at an p- b(a95g7v9ztaajxj 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 qf-p kvqi,3io;gcd,. off during operation, the bladgv3-ki ,;qcdo , iq.pfes slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If thev3 g4md2ozp 9)wk0b zo ball makes 15.0 revolutions, what is its dom0 z bwg92 o43)kzpvdiameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revo5f; qkpp 4i: 5ejmg,4xsod/l ylutions do the wheelpdm oxsieg p 4yk:j 55;/4lf,qs make?    $rev$

  (a) A grinding wheel 0.35 m in/szq(7 t xwkf- diameter rotates at 2500 rpm. Calculate its angular velocity 7 s-wx/ ktzfq(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 re ggp0bg/ jw7n.ykrwf5g5 y)e: volution in 4.0 s (Fig. 8–38). (a) What is the lineajg p 0kwfwg5yr 5)egg. by:/7nr speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun u*+g hjda9 vg-   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe:.p):qom, e la.iln tlcs+7knl.9 kfved of 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 froe pg fud10fc+) 0rzj.xm the axis of rotation is to experience an acc0pgxd c u+1)e fr0jf.zeleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly lu val.*c8zd6about its center from 130 rpm to 280 rpm in 4.0 s. Determvlu 8*.czd6a line
(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 radiuw;j .bnt86 l fpzcj3w5s $R_1$ 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 ab-+qp6v4rfs m gmuop85 oard 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 no rotation to 1.0 revolution every minute during a 12-m58-+p4gumq6pov s fr 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 unifo4kop/)ir oth9 rmly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in 4i o9hrt) pk/othis time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate06uxs zd) ecw0q-wf h-m,/q gn
(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 whirling f m+j4-48ojiyp y;fsf“human centrifuge,” which takes 1.0 ms4fi +8fj yp4- oy;fjmin 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 from 240 rpm to 360 rpm in 6h -iabr67ngz6.5 s. How far will a point on the edge of the wheel have traveled iba ni6-gr76zh n this time?    m

  A cooling fan is turned off when it is running at .ds+*evp5cnq - gtwk-850rev/min It turns 1500 revolutions before it comes to a stopd- k-sngwv et5q+.* cp.
(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 speed uniformly frx.2v7 ws vqi3 a1vox;vom 95km/h to 45km/h The tires have a diamvi.1xv v o327vqsx;aweter 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 reduces its speed cwo5+8-5sglz v0v 8m8, ejbejxavw4duniformly from 95km/h to 45km/h The tires have a diameter of 0. ,a4j8c elm5wvvx wgj+d0 vb5z8oes-880 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 8jyj bkg:n, 3tbike puts all her weight on each pedal when climbing a hill. The pedals rot38j ykgb,t nj:ate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end ofoxz+hhhm w1.a1ec; u * a door 74 cm wide. What is the magnitude of the torque if the force is ecmh h1*. awxe hz+1;ouxerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assum * x2 2lau 9f7xznsmdrk;r.an;e that a fricti mrnaar s fdl 2*xu;;xn297zk.on 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;ysl 2h penp+:, v.uut rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizo e :u.u2,plnv+ shy;tpntal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requixo:c7k x5ywh8 - ln3abre tightening to a torque of 38ax 5l 8xwhknb7:y3-oc $m \cdot N$ 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 bklv7p ; 8pr3ro5aen0a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. 0arpo8kp lr75 ;vb3ne   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Thoo3(/.za(d t ahp6gbhh0on ,re rim and tire hartp3 .6no a/hg,bhoda (o0zh (ve a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod isao7c3m-l5nci+bsqz + rotated in a horizontala7co+ ib lcz+n3q-m 5s circle 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 on a potter’s wheel rotating at constant angular 0ckk8 d8d480rlq n9 f2hoc v1pwcka7pspeed (Fig. 8–42). The friction force between hen2 7vkck481olh8pr0ddk 9w8 cf pc0qar 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 Fl o+p-7kgny/rig. 8–43 ao7n-/y lgr+ kpbout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass ff5jha/67bn+ 9g9 /h8gpdb 8b pvl rokis $5.3 \times10^{ -26}$ 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 spiz j,ou4zz/t3cn+ .c uonning at the correct 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 required sn uoc,zc u .o4+3/jztzteady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass ;l6c lkd:3k7y56rq/wffycg fof 0.580 kg. Calc5:lykl/;qc cf 7k63wg 6dryffulate
(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 t) (5wz upg5a b/fhi9jro 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 sfvu*s4u r8j 13w4 nwhimall 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 radiu* j3iv4w1wun8 s f4ruhs 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 eventually brou6 ;9 fubumcmz3ght uniformly to rest by a friction cf3mu6m9b u;zal 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– nk8ziq92nmwfyn+l w,v4 h/y745 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which o/rj2+yqvs ( qrotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fr2j yo(svr/+q qom 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 lonk 4qma/,wkug iv m1.k.:yn/c ng thin rod, as shown in Fig. 8–46 aikwk:/gn.ukm .n/,mc4y qv1.
(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 4)gg-a +k rrhy syv63sof 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 accelerates the hammer from rest within four full turns (r5g1. 4 e(dj1rf djjenhrtj3z ;evolu3.nh r1gzj1rde4 d5(fj te;j jtions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moal*t tg82g 4htakd ;v7ment 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 torqjd9b h6+lkfbb/r(g*hue 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 slj; lqmgz8f 75dipping down a lane at 3.3gz jf7 8dl5;mq $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thexvu9 sgmg. hxr8.o39f Earth with respect to the Sun as the sum of two terms,x8.s3m9h gor xgu9f.v
(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 ra bi5s0jj1rk i5dius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume itk1 b5r i5sjij0 is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass.ai m33rbj9j,4:tnav)5ggr qv. ua d tuf,-uh 1.80 kg starts from rest and rolls without slipvj,a.aftt5a3r.3)r q9,hvujg: i u4dumng- b ping 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 fabep46it v/ ;bzll and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of ;ep6 /zbib v4tenergy.]    $m/s$

  What is the angular momentum of a 0.210-kg baffv 5ndb* *ikbpq+3 /5i9gsbl ll rotating on the end of a thin string in a circle of radius 1.10qlfd9fi3*pn+i bbk5b g / *v5s m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu+pl8iofs9zo 53/wv7j uc jyb 6lar momentum of a 2.8-kg uniform cylindrical grinding wheel of radilub vj+8s/6f3 975pcoioy jz wus 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 t .ibghv8edsqbhe/+w, 0u2lc)hat is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rol,beh h)i+ 8cbqwv0 2eg sd/.utation 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 rem-i m sg) :ww+gaq;/cdduce her moment 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 +s;qmwgm gdc/: -w)ia 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 ah8oip..b4l7 0 pnicoit bc r//n initial rate of 1.0 rev everppihi07bo n.rloc/8 c i/4t.by 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 rotating around a vertical axis through its center at a pnqx.1nht q- lw,.yy ,frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What,1ly tx. w.n,qnyq- ph is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular moments 9oy 04ftgin w1disp58+ :mnwum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eawtwq+ bq7wk 08rth
(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 cylindricalw gvo o/pumui8:; ht63 disk of moment of inertia I is dropped onto an identical disk 8h3:ouv6 m; ug/wiptorotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ues,k4wgcg a+ 1ntf b4 k399yp2.4 $rev/s$ 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 rotating merry-go-round platfort*i:63v mumx pm of radius 3.0 m and moment of umi3xmv* t6:pinertia 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-rounk y7/m+o, e5k sht;ykad is rotating freely with an angular velocity of 0.ykeaykt5 ks7o,m /+h; 8 $rad/s$ 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, l j8 ijpj+lxep;7ie+y2 9 pvh6uosing 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 wo pyjjvej89;h2ilieu+6 p+xp7uld 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 excess of 99y.lx3mdgi er5z zx5 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 u3248v qif bw8xg,wbn$ 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 freec 0/5b8k-r)/rkvly qh q: imdmly without friction. The moment of inertia of the person plusd r lb-mc 8h0k:r 5vq)/imy/kq the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the 893io lxtey )wedge of a 6.5-m diameter merry-go-round turntable that ytlx93oe8w)i is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the ropeox;h kl-+ hag p,;)k lk3tb:sa 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. T3lk tg)pb,sk-ohhl; x ; :a+akhe spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Det4feqz.y3 ) ;qes(4u i1xp epzuermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. ( yi3 .e)pu4;p (zef 1suqxqez4In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest jt fcrr7+binz 7 j.-;)x l1xjkat 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 cylifaywv )8yu-wh)5xyd7 dy). fcnder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelerati5dw)7xd .y f-)wy8ah) fyuyvcon. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of twok1 yva k4hcmo4;ij: 4n solid cylindrical 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 of4 1k 4mch iav:jk4ony; 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 v8l;t buzlg izq6a(w,+zl7she *;m+dis the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were ulc 65fkh:pwqd)ug 95v 9c0ny 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 losw 9dfkuhny9:qvg pc 506ul)5c t mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fo2aywb8zn: * zwr it to use energy stored in a heavy rotating flywheel. S: 82*ww zbynazuppose 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 without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a hys. qwk:r5w/n $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) j*u246tciewn z /;pphis rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freelyekpj + tc:n;vhql4 84o (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 mcl:+ov4hp48;kjn t e qass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we 0b5l qdx1ctw1l3cnnw z:6lf9want to exert a horizontal force F at its axle so that it will climb t9xw1nn c1d3lw zf6:qb5l0 cla step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v*k6jxb cg5)bzbi4n70 uf ncp 2=4.2m/s on a flat road is making a turn with a radius The forces acting okb0)zc2b * 6ginb5 4xfcnpj7un the cyclist and 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 0.50-kg rock into a sling of length 1.5 m and pvomo:2(6xwaod:m w3u4b iw;begins whirling the rock in a nearly horizontal circle above his head, accele6m ; oi uwwo3v:42dwoa :mbp(xrating 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 adc6s)4r b qt7qs 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 skate7bqs4t)6d qr cr with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which conspj8c tk ;9 k:6ztudel.ists of two cylindrical plates, of mass ek6zc 8t;k9.d :jutpl $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rough track m xq qd.58lhm)1 idwbqtf;:n1b4tm7sof Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest poi.qnhx d qi5 s8mm m4d11b:btw)f7q;t lnt of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dvk ct74ldb )6uo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlyderr/gq1/ +w1ite p u- from 90km/h to 60km/h The tires //wpgqr1tei-+r1 ude have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s