A bicycle odometer (wh)7bs07h*6kefka dexw;ta fs3ich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-is6d*ke3baw 0 7h ;t a)x7sekffnch wheels?

Suppose a disk rotates at constant an:49xmann e e kqs2qn0*gular 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 a e2q0xn:easnmqnk 4*9cceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by o ryn3(vkg;+y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gre9t .e+oxaxy89 kugh h qvj:..b 40 qswasxw6;aater 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 vzuw2 /cs8tb: head than when your arms are stretched out in front of you? A diagram may help you to answerswcz2u b8v/:t this.

A 21-speed bicycle has seven sprockets at the rear whee9typk gl/q f2tsx /ap()zo.- ,6raprv l and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocketxsyf lpp2t vk-zo ,/t/ ar6.p9a qgr)(? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being ablc6runb ;8+8(pw: 1svyttfj s fe to run fast have slender lower legs with flesh and musclu sb:6fc nyst+rpv8t;(8fw j1e concentrated high, close to the 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 longoammed*.h-+ 35ikxo oc v1s t3, narrow beam?

If the net force on a system is zero, isd)rfc pny;t *8b)oid/ the net torque also zero? If the net torque on a system is zero, is the net for);) ybtdfd*/rc8 oipnce zero?

Two inclines have the same height but makeum t5pw6oac4 y ,5mjss)0+yw s different angles with the horizontal. Thes) ac6s4+mystw5jmp 0 , 5oyuw same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolli ofb5 go62rmr5ng (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reac6b 25orrgfm5ohes the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radii gw*w7js33gy us and the same mass. They start from rest at the top of an incline. Which reacy37g s*j igww3hes 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. Y-pa rx.iggdvl*x wh,cq d :obt/ )-::jet most moving or rotating objects eventually ac/:jdg,ph br-tq:x* l).g :oxw-iv d slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator,tla vv x h ob0rf4+:b2,ny/1ze how would this affect the +rl2 ztvv4 xyb0a/ ,en1f hbo:length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotg(i eim5 .ibnr).zm 8sg.tk :w+9z njxation when she leaves th) e:gjtmbi z9rz8iw+gs..ik x(n n5.me board?

The moment of inertia of a rotating solid disk about an axis through its centp 0;kue ds o9.r4n;iqno35b wwer of mass piwkdnru4;95qo. nowe3s0b;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 mass in each outs-sh . 3v xagj7wxmdqmmtpe.rx 9+e6;vmzi (39tretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or sw(im;. mvx.x3 3hm7 vt+-9zexg 69srmqjpdeatay the same? Explain.

Two spheres look identical and have the same mass. However, one iv,h-5heg:spzqj*w (s s hollow and the other is solid. Describe an experiment to determin sj*-5sh w:pvq(,gzeh e which is which.

The angular velocity of a wheel rotating on a horizontal axle pointwa)qfk. q6yu,x1 .)na ks: f,oo t-wnvs west. In what direction is the linear velocity of .k,)) ,k-wto:. 6fsa nuqafwoxn y q1va 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 myv+e5 7os 4mhon the edge of a large freely rotating turntable. What happens shm4me5ov y+7 if you walk toward the center?

A shortstop may leap into the air to 5t c.)brtgy/ ws2x mv): g xykm9xpb-,catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quictp)9x gmy2wb, )m5/g:yvb-xstr k cx.kly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatio 7x.sfwt6k8hon of angular momentum, discuss why a helicopter must havh6.w8ko7 f xtse more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following atg3vd05 x8oxk4bmf4 dhg:ce8ngles 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 cmxwju5s:w f;2-kp k9coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen pk-u:xw w92mj;ck 5sfon Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ea- feo 6mxn/rn9rth. The beam diverges at anrxm oe6n9f- /n 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 ark: o2l 0,w5bga mkr*n rate of 6500 rpm. When the motor is turned off dk lw borm nk0gr:a,*25uring operation, the blades 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 the ball maef,sn1 deo n8j1cz4(ukes 15.0 revolutions, what is its diameter? (csu 8f1e4e1n, onzd j   m

  A bicycle with tires 68 cm in diamp0jwsv4 u -3rneter travels 8.0 km. How many revolutions do the wheels make? j3u-wsp4r 0v n   $rev$

  (a) A grinding wheel 0.s:po14vi2hh/ t ,gumf35 m in diameter rotates at 2500 rpm. Calculate its angular velocity ii/mfptgvh :o1 u4,2sh 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 in 4.0f6ekkeq f7.2ilgco ws9vj:)( s (Fig. 8–38). (a) What is the linear speed of a child e.27oskqgwi6j: )c efk(f9l v 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 n8yua+ o(s99k x/blmpthe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofq1zoi0;8suga5 jr qi0iem4i , 9;e/6 amyc bmo 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 iizt+;7 :yjn1o z kfzo,f a particle 7.0 cm from the axis of rotation is to experience an accelyoz:zft n1z7+koj i;, eration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from xm7 k7qh3gz u q7b*rh9:c75yoo0 s won130 rpm to 280 rpmhq7h7qomoy wg 7cn7br5u*o :3 s0xkz 9 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 radius kxt ; 0qh4h,- (d(q5vxhckqgq$R_1$ 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 Moonxjj78pnkp(s 5w )9vh kuer);h, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from 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. Determ )97sj hrw;hj pn8vp)5(euxkkine
(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,bxqju0o8w *b u+h;q0q 000 rpm in 220 s. Through how many revolutions did it turn in *qqb+0jw0b8 ohu ;qu xthis time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula 8 u9hmo3xrypb qa--:fte
(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 (a3p;0hocu+yjr+8* fob wa qh x-jlr)a whirling “human centrihbq3ar j +frp0cl 8+wu ;xya(oj-)oh*fuge,” 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 from 240 rpm tou ij4:p 8 qjc-.ogjx;s 360 rpm in 6.5 s.8qg4-xj ;.:pjjcus i o How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running aahu1wr)9,7 fnj ir;3 ez tzjp(t 850rev/min It turns 1500 revolution;n) i ezh71 rp( wjaf,rztuj39s 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 unil;qd opr crwpf9+srg5l 0*eca.)oc65 formly from 95km/h to 45km/h The tirec9r p+; odlq )oe6 0c5 fsrpg.wa5*lcrs have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car res wmbn* iee/-9duces its speed uniformly from 95kmwn is/ee*9bm- /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 ridin9v3gij:q i0fg b 8n6zn(vwzd 4g a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius i(8v w9:jnfg6 v3bzgdzi 04nq17 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 7426 zfylq z)5sv cm wide. What is the magnitude o 5lzf 26s)zvqyf the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel showv*w:jlo/;9y1l vj .wjrq es3in in Fig. 8–39. Assume that a friction torque of 0.. 9vws/ j1ojriyqll:wj;* 3e v4 $m \cdot N$ 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 whvc0ig9 u;a1ivich pivots as shown i0 ;iav 1iuc9vgn Fig. 8–40. Initially the rod is held in the horizontal 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 require j:h -vio,(jzr tightening to a torque of 38jr,jz: ih -(vo $m \cdot N$ 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 ofqn :*2rrn jtt1 a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its centt2:qn1n*rtj r er.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm fodl 2;;ant we7h8eu-ls85 3 ,elvtrkin diameter. The rim and3kofel tws t;2l;8-ea ldev85h7rn,u tire have 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 is rotated in a horizonta8 y46nrtsigeb tk0l6(l circleyl bte06g486 nrsikt( 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 ppk,f41:k q rdk wg6u7 0s/zkja potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the claypdkk g,4zkpw70qk:6s j/ u1 fr 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 b+)oonth (u(k Fig. 8–43 abounto +ho((kb )ut
(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 ak/lar9 ni9 /xtb20s1 lvds 4fwhose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, eng7j*x4yxw/g 3hxy;8lkhcu1ykyj w 6u3ineers 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 of each r3cgu8wj y /ykh7 xyxhky*x3lw4ju 61 ;ocket 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 rlbnkf+a(u m8 dq ghf e92./j0and a mass of 0.580 kg. Calculau f/k0g(bnr9mh+fj a8l. d 2eqte
(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 reem97z 1-f lq5 yxhzbi/st to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-lrp f)c8p2 f; ;ac6volgo-round and is able to ac lf pcolv28pa)f 6;;rccelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite 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 h-6u5k sx*dr wrpm is shut off and is eventually brought uniformly to rest by a frictional torque od5w x*s-6hrukf 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.ujpetk y r.qo4r *i1.,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 a -pfw+)ge) 7weej7:k sth:hqoction of the forearm, which rotates about the el ftwskj7p hw)e+:- e:oq)eh 7gbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade caw-i, x.seul v9n be considered a long thin rod, as shown in Fig. 8–4ixw, v9e-.sl u6.
(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 cons7cvz u8cm5 eup3 c+)t0ej et;bists 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 accelerates the hammer from rest within four full 3uxngm0 ihagh3dt19 9turns (revod3gm hn1a9hi0gtx39 ulutions) 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 moment of inyou( mat349c5xpo ixi+ 7hm; pertia 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 dxog 0it r,5h7k,g mz-apa:c4 tevelops 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*jli pibkt h0;cl a/;: cm rolls without slipping down a lane at 3.3b;*ialh0ji p ; ctlk/: $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sugu- npntfz-z):w:2 ).ha dwaj m of two tu-agdfz): )njw z tw.2pnh-:aerms,
(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 ma-vkk8+j-cuj m :xojz5ss of 1640 kg and a radius 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 it is a solid cylind zo8vk:-j x5jk+m-jucer.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wit -ri; v)wrmglpqa0,o7hout slipping down a 30.0alo7qmi ;-rr ,)g0vwp $^{\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 sta4 sqohh7rhx *3u ski10 xe9r2srts to fall and its lower end does not slip. What will be the speed of tr9i32shx4o*1 skr7us0xeh q hhe upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on q gfwzcch0hzh.du9,4 zk)(d-the end of a thin string in a circle of radius 1.10 m at an angular speec .zw4q,c k0)-hdhfgzdzh( u9d of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular vpo4 q0ep***vlyfsm t691 zv gmomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? m1fvo*04pg v*szp* l9vy teq6   $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of be yn* (kzh1c1e(t 8cd e*/zcj1.3rev/s If hej1(ce kzhdnbc*1t(8* zec y/ e raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of6gnx:, h .ymzxcjlzhv48o63 f inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2nol.: hz 3cgv4, 6f zy6x8mhxj.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 r:kh2j s ac1l2te) g(opate of 1.0 rev every 2.0 s 1 pl(:cs) kghaj22 oteto 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 vertim-*xa xt)r7lwcal axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a*x -x)mtalrw7 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5r5f6 s dj 4wtl(tun743l3 erx8lpqi f( $rev/s$ 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 Eatbfcex1vo5v riio0+i0w7suh (3(b(irth
(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 d- r1qgp:)hc1en9w gmcisk of moment of inertia I is dropped onto an identical disk rota 1r1)c-genmcp w:h q9gting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4o ,tz9sfha kae o77x59 $rev/s$ 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-goi:s ao1v(x+6 xrc)p 5pl4hxnm -round platform of radius 3.0 )ns h+mr:lx vo(xpc6xiap54 1 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating f9ok igh1 v9 wnvxxg4,3reely with an angular velocity of 0.8 ik4xv ,99hg vx1o3n wg$rad/s$ 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,zihf2ezl r)/- losing about half its mass in the process, and win h/)zil-f zer2ding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest 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 ni w lip)2ncl9- s(7qh120 $km/h$ 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 ptmbc 4q5j1u9f) du7-* j qf c 9pwpl5dkrz56h$ 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, ini) i -hvsw-fwqs.(n pzvn47i(rtially at rest, that can rotate freely without friction. The moment of inertia)7(. vwz-fpin wvsqr -i4 sh(n 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 at the edge of ako a*fen;ltmx-hv190 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700axnkv0mohe1 *-9; ftl $kg \cdot m^2$ 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 tdm-(bq; mv4t6q oamen- pxc2*w uh*o9he end of the rope lying on the top edge of the spool. A person grabs the ehx(mmq42udt vm-*c 9*-awoq b6p; on end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Detewzyu vp*ceit v+eeu k53w3g 9o0,w *)wrmine the ratio of the Moon’sw**3g) eowywu95+30k tzvevec ,puwi spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frotgl zf5 gft5,h;v6be 6m 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 0wz7b 9f2 6iwep.tz/rtp; +atg eu,sh( .20 m acquires a rotational rate of from rest over a 6.0ph/;r92(,be fag.t +puetzww it 6sz7-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg ,* naj0,1bj hon okc;vrtm08i. s-vs uand diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear spe,h; *jus80 1cosrmnk-tbvv nj0a,i.oed 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 4fmi jl9- 8krl0pg5nw1sw :lvis 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 2ch2 r9+nvo92sx h qhtmass 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? Ashvc t2nh9s2 q+r92xho sume 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 itj98wg 37zf pogp+qjc : to use energy stored in a heavy rotating flywheel. Suppose such a car has a totalpgpj+fz7:q93w co8 gj 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 illustratv(p 9::hrb2 gp/kak tues 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 horizonta -by:yryjc( * ff27pdll surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e/v0f xtiw/fza+ma-s8sc 91bv., 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 zs9t0xwm-aa+ s8f /1/v ic bfvlinear 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 standing vertically o p/h* t0;cblzyk):zvfnx 3+te n the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has hpclfh)*nyzv/ ;te bt: z3+x0k eight h, where h

  A bicyclist traveling /bjf)f dy/6mw l1v(gf with speed v=4.2m/s on a flat road is making a turn with a radiyg(bdl /jfw1v)/fmf6us The forces acting on 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 bebk10i 5c27(9y-chcy4phaixuzc :,j.tie q l zgins whirling the rock in a nearly horizontalluc27c 4chy1z:i(0yc kxzbia5p,.t j- hi eq9 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 this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armmn+1wuv5 ys6js as thin rods, making reas5wmu6j1+vny sonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical uhxicw rv4+0.2p6.bwctfyw 0 plates, of vw62.xhf0y0 ui tp+ 4 c.bwwrcmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 of Fig. 8–58w8w8kcma.tql wqqd/e;xa: ,i a ;h2xb)g so 04. What is the minimum value of thw;)b2 xt ,ckiq:8;ha gm ew 8q x4lsa0do/aq.we vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as2(dsksr4:r ll0bw xm5i) ,xzb sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces itv i ;iwh8*ntwns3og 03s speed uniformly from 90km/h to 60km/h 33 s;*ntnw 80wgi hvioThe tires 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