A bicycle odometer (which measures distance travele zq;(pu8nzt b,lm0zs,7cwdq7 aonr 57d) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use i dm,0z77z;z7 nq ubrw( o n5a8cpqts,lt on a bicycle with 24-inch wheels?

Suppose a disk rotate*htl(lnn) 7 a6q5o8 9qs ujm1yjtm vm(s at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? (m6 8qn1tyn 5 u*)som7t( 9lljjqvham For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocity yi6wpq .lw-,qa0hm7zuas :1 r$\omega$ Explain.

Can a small force ever exert a greatepb hzn ; - 4a;8ia.+opmqp-sqxr torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than e ;zpkdjp(+avh ny3te +9t;r7when your arms are stretched out in front of you? Av p e;k +y;pz7trd9j3eah nt+( diagram may help you to answer this.

A 21-speed bicycle has seveny8b epw q3ye-86vn-xf sprockets at the rear wheel and three at the pedal by3-fxnqwe6vy8 -pe 8cranks. 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 or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs w8155glizy zfeith flesh and muscle concentrated high, close to the body (Fig. 8–3 8 1e55gfzzily4). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.i f+cu 3a)0lse 8–35) carry a long, narrow beam?

If the net force on a system 2/ax2kdyw5yfa 0h f: dzfs39ris zero, is the net torque also zero? If the net torque on a system is zero, is the net forchdyf2 /sxw3fk:95 raad f0 2zye zero?

Two inclines have the same6 epj;,wkqeb)m3ou d7 height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of th u3dm)webokp,7 eqj6 ;e ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an ydflt0 i cu:b*8o,se* g) 4ublincline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the iny, *bloudcb *t8silf 4 eu:)0gcline 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 st )owu+ 9qg qhyg0-kf7ts zm,/vart from rest at the top of an incline. Whiczg s 0wq-)t9 +/fvkhmgq7u,oyh reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most j0 lxsdn: )o 3)qzebk6re*m0u moving or rotating objects eventually slow down uk3:x l o*)nzmjqs)ee0rb 6d0and stop. Explain.

If there were a great wrd6pn-; w)g88w tl frmigration of people toward the Earth’s equator, how would this affect the l); flw8rgt6-w 8wndrp ength of the day?

Can the diver of Fig. 8–29uud,a mg2lily),n+lv:n 0 us 8 do a somersault without having any initial rotation when she yn:0,sl,g8ulivu )+adn u2m l leaves the board?

The moment of inertia of a rotating solid disk about an axis tho /yz 0 otm ui k:6dls)7z-u:jeqi6o5srough its center of ma:u)o d5t: o zlo6juqzykis67/mies 0-ss 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 masw-ala +17,f -6x9o/wdthfwdhm* wy qis in each outstretched hand. If w/m+la 7 ow-9qxff w,at ih-dhd 6yw1*you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Hou+ dxs/5vdan05; sqnlwever, one is hollow and thds05qav;nu+ d/sl 5 nxe other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheeg gonf,d5k6s) e dr)u3l rotating on a horizontal axle points west. In what direction is the linear e6du,rf )onk )gs53dg velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating :s xzt 66d dwfnc;w*3hturntable. What happens ifw6f :hw; cz* x3dd6tns you walk toward the center?

A shortstop may leap rlnozk/(ea4 tz.2+ zu into 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 oppositl.t 2azn 4u/zr+z(eok e direction (Fig. 8–36). Explain.

On the basis of the law of c4 ;;.xiqdnx8 ,i+x szccd:b ftonservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the hs bqx +x;f:ncd8;idz.xic4 ,telicopter stable.

  Express the following angless/k85k v 2fa /iah0wgxo)tiq) in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincides(f2ckt2a d 7snce. Calculate, using the information inside the Front Cover, the angular diameters st2 7kdaf c(2s(in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam dxkm96 yy0zw /miverges at an angle /x 9m6y kyzwm0$\theta$ (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 mop.r mi(f;u :mftor is turned off during operation, the blades slow to rest in 3.0 s. What is the angularrf.m:i ufm(p; acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5lqq 5 v6/0eiucq y.-g:ph9f0 iygo pi; m to another child. If the ball makes 15.0 revolutions, what is -:glgyio;iiqf euq/pcq0059. pyh v6 its diameter?    m

  A bicycle with tires 6bi3 u70r)frst 8 cm in diameter travels 8.0 km. How many revolutions do the wheer)3tis bu fr70ls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atu g zv) +(uk4+4kdlgge 2500 rpm. Calculate its angular velocity idv+lgezkg u ) kg44(u+n $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution rl*(bkzkbb; 3cfdh qr8 7dcs2o)c-/a in 4.0 s (Fig. 8–38). (a)( dkqbcczob */;) r 3af kh8brl7dsc2- What is the linear 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 *j36drf8yl 42zn 9wurk)gspnEarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed6jrdnb9-vb1 c.a d5 qkyb 7sp9 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 x+k horr 0fsfk.k17 g2bu3 lk5if a particle 7.0 cm from the axis of rotation is to experience an accelerati57uskkklbro20 k.f r1gh3f+x on of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its t 64:ui dw oa( jt+(+xy/f3f t(blfesdcenter from 130 rpm to 280 rpm inofd /l(ed(f tjws3bfau(4+:t yi6x+t 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 radius v. :lgo7wwt ida+1c ff pt(w)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, astronauts aboard the Apollo spacecraft put themselve(s0 cs75 k i3pr:vdrkvs 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 minu( s0krk7pvvsdicr :3 5te 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 2/z kt0 m f1:8xgghoq/a20 s. Through how many revolutions did it g/gzoqtm8x :0/fa 1khturn in this time?    $rev$

  An automobile engine slows down from 4500 rpm j(fuce23h3:ng nt *yu 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 flo,2+try tw.x 4pzq 3ig-mtq /zying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finali 2wqmptt-y/ xzzg. ,43+oqtr 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 t*)9 7r7ui4pnb goftm ho 360 rpm in 6.5 s. How far will a point on the edge ofp 4hi 79mrufn to7b*)g the wheel have traveled in this time?    m

  A cooling fan is turned off when it is b-f)p auie a;xk:o3n*+bhs,u running at 850rev/min It turns 1500 revolutions:x ;+)abhsfb*3nupeo,ku-ia before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uity, )f,q-dkq niformly from 95km/h to 45km/h The tires haveqdy k,t)iq f,- 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 reduces its speed uni0o j7 e s.njyel929nqpformly fro9e n70j9 njp2.qeosly m 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all he6c-2mghxf i9 sy0, wfhxqz(0 fr weight on each pedal when climbing a hill. The pedals rotate in a ci9, i(cg6f0 zq xy0-hfsx2hw fmrcle 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 of a door 2-qf5 o jclioryz5 17q74 cm wide. What is the magnitude of the torque if the fqo- fco57 zlrq1j25 yiorce 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 thhks:s:l11ev6m-ov : 3 bhlgnhe axle of the wheel shown in Fig. 8–39. Assume that a friction torque of m 3khv:::6slnshvgl -1 e 1bho0.4 $m \cdot N$ 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 which pc hjo 5(92jfwo/c.vko ivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitudc 2c(o9ok/ f .vw5jhjoe and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require 3jbvo 3fg/ )if-y,hj wtightening to a torque of 38 gijhojw,v3 3-/y bf)f$m \cdot N$ 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 in jv.0rnj nmpoyn-;.7sertia of a 10.8-kg sphere of radius 0.648 m when the axis o7v.j ry-n0o jnp s;nm.f rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bip,9d3m s xo+s ys3cyc+cycle wheel 66.7 cm in diameter. The rim and tire have a combined mass3dm93 o++xc css,ys py 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 5qesi7-e zuve:f6,i rrod is rotated in a horizontal6qe5siei,f zv-:7r eu 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 consdjvh,y, -*h5 de2srar(2dfc0yx jh - xtant angular speedh x5fr c(2yd,-hh,sy*2jd 0de-xra jv (Fig. 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 0b7lq s9mjei7 of point objects shown in Fig. 8–43 a9lbse0m7j 7qibout
(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 whosev0d 63tm*cxkf 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 spin4z(yjryw9c3yqs4u 3x y0h t r,ning 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 steady force o 4yr0qh9r 4yt(3,s xyw3jzc yuf 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 withhp v:gj(gcc95 a radius of 8.50 cm and a mass of 0.580 kg.h 5jp v(c9gg:c 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( *kyj(2:qoq ozy l79ynxj8 ( ytrwe5f, accelerating 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 tangentially on a small h,uh6favq 4)cb shp:4w(( tivhand-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 two children ( vi fhhv64w:(s hqt), pa4ubc(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 rot-jr0by-gfpy2k(b*en, x*p1 so, gs n dating at 10,300 rpm is shut off and is eventually brought uniformly 1 roj0y-p n-skgy2fb gd(pebns*x ,,*to rest by a frictional 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 at miyu lss;h tcq9p-1c7dp7-d. 2 i8kfc7 $m/s^2$ 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, whichh,7fp wt ;po,l rotates about the elbow joint under the acti;tolpph,w,f7 on 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 twn g4 kd/+ q31cr)/ b-hglcixdhin rod, as shown in Fig. 8–4n cqdx//hk 3g-l g c)bri1d4+w6.
(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 ce;lwi s4ux;. a,p 5oxronsists 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 accequol1x +lydrh *abgr vlx4 :3rf::71ylerates the hammer from rest within four full turns (revolutions) and releasesx7o* hxrrllr1f:y::+v y1 blud34ag q it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inej m6i s*6gu2;k1a/rxg0neb oartia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque og1d *t*oc4wtnf2a)fq f 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping do f 3kdckjq4g4.wn a lane 4qd4.k3 g fckjat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun asp*c8v2;dx lwa the sum of two terms,dw2 cp*l ;xa8v
(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. Howt,)sr8muo*yk fq 8 b2t-t8erv much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it isfvm,r)ub -rt 8q8ty*t2 8 eoks a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls witbsplpxa .-4 ,thout slipping down,plp-s4 tbax . 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 balancek k-qopic p32pa -du58d vertically on its tip. It starts to fall and its lower end i8kq- -pp5uap ocd2k3does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum o7if8 xz7h2 b yptv,h0bw-az/af a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angula,0p8 z2b- tazw/yh7hixva fb7 r speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical g5/;y*w5 / yxbbhxyw bbt:n(ulrinding wheel of radius 18 cm when rotating at 1500 rb b;x5/tub 5* xyby:(/w hylnwpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hispa z)e*i id,-f side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontaef d,)ipz *ia-l 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 reduceq3gi lnfw6o) l5c*w s/s jmr2; 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 1.5 s when in the tuck pow6l;/m2*sgofjc nsw i3lr)5 qsition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increass:zyq a qc,d/i,hag e40mwf x 7122v:t szllm2e her spin rotation rate from an initial rate of 1.0 rev every 2.0 s a:h1sla: /y x z02 4qm,v tqlim2z,7s2cgedwfto 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 aoai(d 37g2 gz9 3imcxst a frequency of 1.5rev/s The wheel can be m329 sg(7x3 g iadcoizconsidered 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 angular momm1a z ruh:n*;jentum 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 momel 26blsh+ v9r+dr,jfx6 7nuj yntum 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 ofvu*coq8kalh5 /6al 1t inertia I is dropped onto an identical disk rotating at angular spetko1luq8l* 6h/5acav ed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 r8h:e6te.;xbkw we) q$rev/s$ 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 pla4nc ;/ rpkc9zg7rq nt-tform of radius 3.0 m and moment of inz 7nr cn-49cqp/gk;tr ertia 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 abq/x+izh) 0 ,s:za guv9tvg6velocity of 0.8ghaz, +ibt 9uq za0: )vgsx6/v $rad/s$ 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 j)o;v*)i gp(crszo5re-g* yr2 9etpf 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 5rv9g- *) sg yip)jt(po*ocr eerz; f2carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exc51fdhg*c)+w5ghl1iai znmmky a6 ;x9ess 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 gz6klcp 8x x:uli our,0g5 y;+$ 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 rotateutx uu19 f;:w+2nmiii freely without friction. The moment of inertia of the person plus the platform is 1u ufntui29:;wiim+x $I_P$ 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 th )w4byyl+da.c e edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of iny.ly a4+bdwc )ertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyib 8va1 1payk-qng 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 rb 1q8vpyaa k-1olls 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. Deti(j-tdx9j gzhy 4.yteq62 o1 hermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momei hjxqyj t9yo-dgt 4(e61z. 2hntum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a8qf 1 nqkc1j9; /ywlyr 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 un0vpf 6 82jzm8chk8hc giform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delimv 6z2kc g80ph88hf cjvered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.05hj0 z82jpbx8a+i/ 1r0kvz xjp0 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of zzh08p a2jxkx jj8/+pbv ir 01energy 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 ous 5g;c+3 xbeun5x +lr )sga.mf 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 Sunfckq 2dcb aokfm 9417z s78tg4, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that84gc z7 mst49dfqc27f ok1b ka the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energ hq 1fhh4:opl4y stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km wi:qp 1 4hh4hflothout 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 anjtrfs n 9e7+27swun8h $H_2O$ 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 sur de7t79d9 dm.v (jrl8m 8fwoxoface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictctx)t0 d ws:eis8,i q/v:hhh .ion) 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 lin0) ,/:tdhtev8iiqx schw. : shear 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 R. It is standipsw2vc zk 4 bva5(b2. s-7scneng vertically on the floor, and we want to exert a horizontap aw sb2.v-s4vb 5k7ze(n c2scl 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 road *mm/zb off90mrl nwh-n(w *z.is making a turn with a radius The forces acting on the cyclist and cycle are the normal fzho/ *mwmwnz0.*rn-( fmb9 lforce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 beg0o32. zsavl;.a qbdrbins 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 torque required to achieve thib.z vlasqr oa2d;30.bs feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a sh 8xztb 4gr.,uolid 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 4huzg,t.br x8held tightly against her body.   

  You are designing a clutch assembly wi ,2kr)h. b m6rlgk3gmhich consists of two cylindrical plates, of k g.bhmkm263r, l)irg 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 lbm2b0c94zr7 7ty:,b w h uoteiooped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving b47tur 2:tmb9wc0 oey ,hz7bithe track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a37dveh uq)l0essume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as t (aj7,ajlev 8)sgkt.ca ran 81he car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eachcaav .set ral8n(j1 8j)a7kg, 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