A bicycle odometer (which measures distance traveled) is attached nea r+x o8ktbu;v.g5h( orr the wheel hub and is designed for 27-inch wheels. What happens if you use i k+ o(o r85g;turv.bhxt on a bicycle with 24-inch wheels?

Suppose a disk rotates at consti+ ybsg8a php)5 39 (cl5ftpyixqfmc,)u*jb 6ant 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? For which cases would the magnitude of either component *tb,q)+ilufmjh5p gy x3y5 )bfscp(ip98a c6 of linear acceleration change?

Could a nonrigid boddgtot)p 9: p9o3nu/ie y be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greate8i 5zvtr /hxm9r 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 her:z+4+lajn/gq f bse+ad than when your arms are stretched out in front of : nz4 fl+++g sj/qbreayou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at theh8m voxj+kt3fq pav+mdy /,3/ pedal cranks. In which gear is it harder to p3 vj+f3+/,m/m qthokv x yd8paedal, 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 runrci.rxand h8ew4 7( w/ fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis hr8a7/ inwc(. x4wredof rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8cwi/ tnq.xh py96kkjzh1 (y /3d7-fj i–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If a 41e7api0 h+9vm;l rbmzi2outhe net tor;+bvi719 ma eup4roz l hmia02que on a system is zero, is the net force zero?

Two inclines have the same 4zq0bzzj7 n)a 19ckarheight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the )n07b k9 jca zrq1a4zzbottom be greater? Explain.

Two solid spheres simultaneously s9pvg7.q y,uiptart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Whi7.9p gyqi,v puch 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 sa6ywek) y57l n70zhmmdme mass. They start from rest at the top of an incline. Which reaches the bottom first? Whizmh0wkl7 ym ye d5n6)7ch 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 acog4qtyr x2:eyz 6n:0ngular momentum are conserved. Yet most moving or rotating objects eventu r:zt4 :0yo6g xe2cyqnally slow down and stop. Explain.

If there were a great migration of people toward the Eazhqb v1v-;00 ze*sm krrth’s equator, how would this affect the length oz; kbzvr hs*1v-e0mq 0f the day?

Can the diver of Fig. 8–29 do a somersault without having any initi :gan94tvnhzmbs(x2r vmm-: h:w w7mv 5+ge4s al rotation when she leaves tvvm(h:wmb v +mnsx4g-tm7e:z:2r a h9nwg54 s he board?

The moment of inertia of a rotating solid disk about an axis througher o j3g*vgn8u+6-jn7+vwvdn its center of m-j+v v n +e6rd3ou7*vw8gjn gnass 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 u+wpjmr)2xxk v3*l)rd/ 4wn ooutstretched hand. If you suddenly drop the masses, wiopx)xj kwv n4rud +m*/2w3lr)ll your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,qqjq5/+g..tk z3w+dva.lvk v one is hollow and the other is solid. Descrijltqk+q g. vdz.5q v3/k+. wvabe an experiment to determine which is which.

The angular velocity 5aa4 i5;j qb0e( omj(imac uv4of a wheel rotating on a horizontal axle 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 acceleration ofa(i4q bma5uaoc j;vi 4 me(j50 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 frt-hn6 k)v,* i3xux;; bcvpb njeely rotating turntable. What happens if y jv6ti;b3x -pn *ukvhx )cn;b,ou walk toward the center?

A shortstop may leap into the air to catch au1k fg+ j4p0mn ball and throw it quickly. As he throws the ball, the upper part of his bmn0 puf1gjk 4+ody rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angu2i aj -+tdehh)lar momentum, discuss why a helicopter mj+2 hae h)d-itust have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radih(- 4(i toyxqidy eyn52 :r*avans: (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 becaus8nxi7 .x 5 ihufhl,uf8bd i:rvp8i 1.de of an amazing coincidence. Calculate, using the information inside the Front C v7f,.hxrb8.d1lupdf8i: 8i xni h5iuover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed amy/5+r0a.od swpkv o- t the Moon, 380,000 km from Earth. The beam diverges at an ang.sovro+w /kdym 50p a-le $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotat eb.cz:4 -8c 2kvdqguje at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades sloz.j:-4 kd vcq2 egc8buw down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball make1:)xv itvrrav7xpn0 *s 15.0 revolutions, what is itsx )v7:apr0 vvi1n*trx diameter?    m

  A bicycle with tires 68 cm in diameter travn:yd *6lc x/j(nt)9s ku9qzy d(kp :uxels 8.0 km. How many revolutions do the kck/t)u6*lnj9q(:n(ysz dd xy xup:9wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate +v- oqkn y(u oybkb;51omi6+z its angular vkmybob qu+yz1(;ok n-5o v+ 6ielocity 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 mapja-gu n; xhh/;zpu 1.kes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from th h- u/.auxn;pghjz1p ;e 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 arou*p /pleq2z eo1nd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a g*k8r 2zj f h9-r ox)*q1ikvbnou -ta;-hyf:fpoint
(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 r l7: g6gtmmz/zotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 1lzzg:t6m/ mg700,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center friyur5d-vps-8 /f z .atom 130 rpm to 280 rpm in 4.0 s. Determinr.5 s8t /vzi--pdfya ue
(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 radiusj.mz uj 0a7 7v5niitm3 $R_1$ 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 themselv :y rz 2kbt-i+zl*)e+dkd;tn ges into a slow rotation to distribute the idtk+r lb k-+ng) *;:yze 2tzdSun’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. 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 unifo bmuw0.f 2wr.zrmly from rest to 15,000 rpm in 220 s. Through how many mzr. w 20u.wbfrevolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200zwd3 rc v*zn 6 u.xw5f0l(bs8r 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 flying highspeed jets *m 8/+t2:nhpuwzfnlyin a whirling “human centrifzul :/wtm+ *np28y hfnuge,” 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 to 3f ecasb ,,0,ap60 rpm in 6.5 s. How far will a point on the edgespc,0 f a,ab,e of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850 , gydz/)hwaz:1+k ngc6tfx2wrev/min It turns 1500 revolutions xa :1) g+6/ tw d2nh,ywzcgfzkbefore 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 red2c*2rsz- ho zay;td l+g/fyyq8mi5 9n uces its speed uniformly from c amz rqy ly92/f+yo2gn5-;ih *tzd8 s95km/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

  The tires of a car make 65 revolutions : fwrgda66* oxas the car reduces its speed uniformly from 95km/6o xf:rw a*6dgh 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 bikepkk g4k /v2)0;ioyini puts all her weight on each pedal when climbing a hill. The pedals)2p04i;kkygik in o/ v rotate 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 of a door 74 cm wide. What is the ma/x zmz3p2w-q lgnitude of the torque if the force is exelp/-2 wzmq3 xzrted
(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. Assu2qqg c8s b +5c+lh/egdme that a friction +cb g/c g85eshldq q+2torque 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 ts5tti 3zeq9sfwr+ddy 5o a*fs49 w*- hhe ends of a massless rod which pivots as shown in Ft*+wzsysi te9d*5 f h5q-4 so9rd f3awig. 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 ofjiz.7hd4rq rupg*7:i an engine require tightening to a torque of 38 7j.qidhuzp*4g:7 rr i$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m 9yxif/bl7u z ys9, ,nkwhen the axis of rotation i7ifl,u ,/ysxky9 zb9ns through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diam8l1p 8 *0uqxl 1y;yr/jvivvjmeter. The rim and tire have a combined mass i8vjxqv m8pj 1l *lv;u0/1ryyof 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 f23d.rin fl*orotated in a horizontal circle of raf fo3*r.iln2ddius 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 rta+, +o jwcj.zujp7:; k nsx6notating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.jzx wnjsukjc,7.:apo+6 t n ;+
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–4.bbz6bbd h l4:3 .:6dzhb4 bblbabout
(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 consis7 9s 7h16hfk ) q9qtws)aizqxzts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, engvsd9p) 6 upk3gineers 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, wskg 3udp)pv69 hat is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a cvf0*8 n2 kufomass of 0.580 kg0 *vku nfc2f8o. 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, accfsa ;znv y1b )zzjf**.elerating 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 hand-drk)g,.2c0w vv-rjyzzkiven 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 (each with a mass of 25 kg) si) r g zzkjvvcwky,-2.0t 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 brought bkju iikj3b3610i8k runiformly to rest by a frictional torque of bj6k8jui b k13riki30 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 ba+eat: s8tld f8ll 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 th l;xqo;.wo3 h6 va2m.+qg *bc)h liqbrrown solely by the action of the forearm, which rotates about the elbow joint undwlvqq. balbr*3x+h;)h.ocqg im6o; 2 er the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can bc ;wqdrepn +0 ;0dxxq1e considered a long thin rod, as shown in Fig. 8–46.q0d;rw;n + qdp 0cxx1e
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of twob1jcev; 4*p1e (v65hnvt t iz l7hoodz1h5ur8 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 (re- fsoi-,l y 9weyvevozj90 ;)gvolutions) and revli -j9 0oefv9,yg)-so;ywzeleases 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 hhyzv *yd 2vlu5x r:t/*v)7bnb as a moment 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 torque of 28oous u-f.bh6 sfr/ 8.k0 $m \cdot N$ 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 slippingpsa71nqp*cvt;rc7 .3 b0yr-v 5tfcv is9 k7jg down a lane at 3.3- ysvc 70tq5s;aip3rk7gv. 9bcfn1pvrt jc* 7 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Su,ixf4e 2c2u(an ,h5jguzv u v7n as the sum of two5( uav7 uhu2,icgzvj4xenf2, terms,
(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 cs ctz8ip5(50mrndf;hp8e 4j 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1;cc58s8ipnpfdtj r(hez540 m .00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wita k ;uf;k2 b,vb;zmm54/nnb:ciznw:l hout slippm;;n lbkbnc:2;i v5auz:b,f /z m4 wkning 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 verti;u3y.qdp a)- p( u e8db b-im/di)gkvvcally on its tip. It starts to fall and its lower end does not slip. What will be the speeddg 3 y b).mbi-vi8dv e-( du)ppk/q;ua of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thij1gh- .kx zx3rn string in a circle of radius 1.10 m at an angular speed of . r 3x1jkxhgz-10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whektkqg z pfwl*z,/,/k en k68j,el of radius 18 cm when rotating at 1500 k , /ntjkkfqgepl,8z/w*, 6kzrpm?    $kg \cdot m^2$
(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 atbhkasy g8 mr)+q4 hf1: a rate of 1.3ry4)qfar 1hh:s+gm 8bk ev/s If he 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 y 7htlny4/0gq2z xqor --9 ecpin Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck pos2o7xhng / 9cr-el0tq-zq 4yy pition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her sp/b4yp*h7g,io yx/b nq8+ rbkoin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate ox y*b4kbop g yo7qibhr+n/8/,f 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 rotatinkpjv -b)71*p0bb kthb g around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diamete p *pbk)1- bvtj07hbkbr 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum o nqx **5kl5rxrf 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 E7 k,dg l(kp8wharth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertiark q.-w.j40ckbn dsxk(6kq2 j I is dropped onto an identical disk rotating a sbk402kjdj- wx.q 6 kk.(rcqnt angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at k ipcs2c0yfnei15x8sz32bd ; 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the a,0)dbv7pc6upp yn8m , xtu8 r/cx2ps center of a rotating merry-go-round platform ox rpnv68pcuxy/p7 u d,b28m pc)a0s,tf radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angugy0 vlxit fm*sr3j;: ;lar velocity of 0.8 ;fil ;sv0jmtr* 3gyx:$rad/s$ 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 col r50frzs g*iam2 5d/ )xtucf)olapses 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 thetc5g2zrf/50xfm)*sa ui) dr o 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 u ;v: ,e d(iusxacko0/in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 1qi4 alz yvffi))i)l;esqu1k ,a q.x9$ 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 p s*v *7 ) g+fd0amarn,jnsb7z*eufet1latform, initially at rest, that can rotate freely without frf1 b an+7r*0d zsegm7)s *ujat,f vn*eiction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameteri(:vmbld w+yl( f64 i/txara : merry-go-round turntable that is mounted on frictionless bearings and has a moment of ity+rm(lwd :iai/bv :(4 xaf6 lnertia 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 thetz;k ) h )4ph92qluvcr rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. ))q ;pk hvlucrh2tz94 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. Deter;s+- c h*anadnmine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a chs*-a+dn; na particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest xwunva7)5p5d 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 nmc0barsl (y:17+arg a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s intervcb a 7+s :rymg0l(a1nral 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 ; w4 zc+am0j1(ytvj ,nt 1xfdaof mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of ma;14cwyttma , z n1vj+j0xd(afss 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular spwrcs:vog(c8 g8xsl+;k yt 0h, eed 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 +bka)2dvo:rh6oo r8f Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assum od r)2ao8o rkbv+f6h:e that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored * f+;+iz/7mx9w cdjl f1yg azwin a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50*zwm+la/cz;y1+d7fj 9 gw ifx 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 ane 6sbq b(fd gk ;/w-p(fs8epc)+k,hbp $H_2O$ 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 sutc- vl qnl.ib0*q1t,hrface 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., (/ rq*eslc8cu 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, determc8esc/uqr* (l ine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and-qz.1)sibc1 wn6y pyo we want to exert a horizontal force F at its axle so that it will )i yb-ocsyzw.n qp6 11climb 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 is making a turn withe/wdcg z q+ w5ysav3eg3j165r a radius The forces acting on ee vcr+d6jzwwg 355y3qsag/1 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 begins whigt,7tq4 4d ucrbvkj9qg.1-k7ad 7s cbrling 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 trjtc74q9,u gts7k1b4 gq.cvddk -b7ao achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s0+,uabz w *ucbg6h0 jq body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculatec* 60h bbu,0a+jg wuqz 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 plates, 6ep8o-.rlozcx14 qn tdr/*fgof mass nl rtfgcoorxp.*4 qd 68 -ez/1$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tdi*18nzp v1c gncf)g( rack of Fig. 8–58. What is the (n* d)f1c pigvng8z1c 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 the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assum7+k .k3gp.p8rruujsme $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its spe:m3 .at9dwhl w(x1h hf:mf.htry0;ew ed uniformly from 90km/h to 60km/0r9m.m::d hh1wf. t3wht ax(he wlf; yh The 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