A bicycle odometer (which measuuqrv(bg 2w .3k-ekh7 pres distance traveled) is attached near the wheel hub a(qh7.w u2vp-rkkeb 3 gnd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim n, s*dl3a9a/zbt *ltw5ksqiifx / 5*j have radial and/or tangential acceleration? If the disk’s angular velocity increas*/ f5q,tnaw a *isj 5s b9dx*tzll/3kies uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocitmj*7 f o wub75l qdis;*fw:rv8y $\omega$ Explain.

Can a small force ever exert a greater torque than a larger fp8pgae c)y +si)8ha q7:v)oo zorce? 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 yo,g rjv 3awt1g2ur hands behind your head than when your arms are stretched out in front of you?vtg 1jw,g ra23 A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel achk 8 b2kp.,vhg8g*hqnd three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprock8hgv2 hkp*g8k ,.h bcqet? Why?

Mammals that depend on being ablezc .xr*4ob,m7 c1oiztjjx5)anaj f+. to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribu,ab7n5)jzt.rc i omxf4xz jcj1ao*+. tion of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carru4ud2jtlq 0c q8x(y,m y a long, narrow beam?

If the net force on a syshmj7 ;)7icp lhtem is zero, is the net torque also zero? If the net torque on a system is zero, is the net fc)l hmj p7h7i;orce zero?

Two inclines have the same height but make different vwe1;,w7rchi0r ki /f angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be great7iw1rrw/0 ckifhv ;e ,er? Explain.

Two solid spheres simultan2lpf hb g,2 a w+6(zjq6us+dwu;m)u a+wvxyr6 eously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic e (2;b26avd wq )+a6+y6l sfw+rhpm,u xzuujgwnergy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They .q3(m(3v dbav mhu iv8start from rest at the top of an incline. Wh3huqbvd.(m(vim83v a ich 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 moving or ro z,2 gi(l)s)-bq 3puwe*vyiihtating objects eventually slow down and stop. Eu)hv g y3*bziw)2ps-lqe (i,ixplain.

If there were a great migration of people towar6tp9 b hr,z/j7 wimxz4d the Earth’s equator, how would this affect the rb4 zihzmj/w76x,p 9t length of the day?

Can the diver of Fig. 8–29 do a somersault without havinw b,h6mlr*o02bpq.jng any initial rotation when she *2qhpoj,wlr0bmbn6 . leaves the board?

The moment of inertia of a rotating +i w g:0xjo p g(gdvbv;x*s:)gsolid disk about an axis through its center of mass iss:g)j* xg+ow; 0d gx bvgv(i:p $\frac{1}{2}WR^2$ (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 oukov pewt 7i5*g,55mns tstretched h w,*i t5 gn57skvmep5oand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical andw3w1/ale ka f0k0l( dr have the same mass. However, one is hollow and the k /3rw0elw(l0fk ad1aother is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pox 4lc8c(q8 wiuints 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 tang4i(uc qw lc88xential 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 larq 26n9dsik6q sllw h2ij) 05gmge freely rotating turntable. What happens if you walk toward the ci2kli9ws qs)60mngh 5q dlj26enter?

A shortstop may leap into the air to catch a ball and throw it qu/*02i,k.*57 j irwh5y txaifomhnvvj ickly. As he throws the ball, the upper part of his body r,*ao mk r2iw*70.yti/hjxn5jhif5 vvotates. 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 angular momentum, discuss why a helic:i 7;asv t,yiwqlccu8 ckue/ lq 22y73opter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to2u i7 37;qv8:ykl,l c/csc2eiqatwuy keep the helicopter stable.

  Express the following angles in radians: c g-n .;hyymp/m7oo1r(a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using th9xgl- vl9y;4 gqz1yece information inside the Front Cover,v9x q4 l1 c;z-ygl9yeg the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km br- vql6i)i1wfrom Earth. The beam diverges at an angv1rii - b)qlw6le $\theta$ (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 650aa1es50js5vh 0 rpm. When the motor is turned o5a 0evss5 jha1ff during 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 a2gu( 4 apbt;yu level floor 3.5 m to another child. If the ball makes 15.0 revolutions, u4y2b(ga;t puwhat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hvay5n17t-.tptby b 9 aow many revolutions do the whee. 79p5an1tbyttab-yv ls make?    $rev$

  (a) A grinding wheel 0.35 m in diamet-,r;hct m v- hb2e:-n 3k(vsllkf2stmer rotates at 2500 rpm. Calculate its angular velocityl-2, r-kc hmemvv :-lh(3fss btkn;t2 in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one co5hgr6p9v6 2u:le q dqv1m)z abmplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sm )egpv6q :zq5v2uhb961ld areated 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 (1we5lyc2qk* cir 0d9na) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pom8y0oj0af/7liuuo 5 tint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm fr -qnrc4h :1v5fq;u unrom the axis of rotanr4uhnc q f1ru-:qv;5tion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm tskwoq6dtk/3w 2f0cp5 af r,.jo 280 rpm in 4.0 fq pd6c5ar0sf/k.,t3j wo k2 ws. 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 radiui5l9 srbz89vxo n :kf )dk44gfs $R_1$ 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 thev.;emsjr 7uc-b*uj2d 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 t b-7.2vs* cju d;ermjuo 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm inbbv97lfl*s 8u 220 s. Through how many revolutions did flbuls 8 b*7v9it turn in this time?    $rev$

  An automobile engine slows down ftp ;iwa4mnla x:3o-uvf *q1 /orom 4500 rpm 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 tht 9nd -8;ymz4a/aqxc2skej q ;vp9jp:e stresses of flying highspeed jets in a whirling “human centrifuge,” wp4zn- jkeqax2yt ;d cm;:vpq/ 98a 9jshich 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 360 rkipd g+,8ic.upm in 6.5 s. How far gcduk8,+pii . 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 at 850rev/min It turns 1500*jr+u-(r: i .nvgt-ltlqn y9m revolutions before it comes to a .uj(q--g+n*lrtn imrl9 : yvtstop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fobzt;,g vada0x:ej*k,cp7f 5o,lg1jrolgx o,g ,ba ojfek7z5* p;dj1tvca,:0m 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

  The tires of a car make 65 revoluile qb o9nr-vu.b ,1,( gdq*twtions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8(ob*nr-wv 1t,ud bq q9 i.lge,0 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cliv p5vxgd3n3ug2 p8 rd+mbing a hill. The gdv83p ug+2n5rd v 3xppedals 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 tmkcodo (r 4)p*t mk0w(he magnitude of the torqu(w(4 dm c0 tpmorkk*o)e 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 whakfk4tb5rv - 8qp/fvfrz,zh 6x.-1hweel shown in Fig. 8–39. Assume that a friction torque v-rv5qxf- z1 48fhk/ twzb.akr pf ,h6of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m7:.nxzzih kk ., are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude andzk..znikh x7: direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteni9t/ex-kwgs5dm;ont:l+e p- rt +- 0vtm8xv kong to a torque of 38 -m en0;lvsgv+ret- odkt m+-x9/ wott5kpx8:$m \cdot N$ 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 wes 7xpbk6e, tuc;p67 7uk1jtchen the axis of rotation is throughj p6; ue k7e,cbcxus1p7t7kt 6 its center.    $kg \cdot m^2$

  Calculate the moment of inos7y7j*erh0m zzk*ct 71 tlv:ertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. thze77ky07ot 1rz *mscj : *vlThe 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 horit hk*ldc7 w3j6e .xh9lzontal circle of 6 lc7el. hjd9hwt3 *kxradius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular spek, (iy: f:ofhir5 i4jmj51jxxed (Fig. 8–42). The friction force between her hao1ikhxjmy 5ij4:fjr( 5f,xi :nds and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown ypb2ue0k1qy+6ut0khpl+ fd 2 in Fig. 8–43 about pdu2k 6 yut+efqy0p0kl+h21b
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass tft(b3go w)4 l0 +ryuhis $5.3 \times10^{ -26}$ 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, unifo9 u- jk7nuomzgfa)68d rm cylindrical satellite spinning 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 of each rocket if the sateldmua 8ugn6z 7o9 f)jk-lite 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 5f, fsx psw6ahvr4.4mmass of 0.58a,sfhv64f wx4r.sp m50 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rtfkxh1 2 fikq;58yta w6 4bepe;zfs;(est 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-go-round andgo32)wy*wif (l o nwy bhh:f69 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) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictiooio 2*h3fy (fw:ybnlwhg) 9 6wnal torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is she62 p xcxzd9c5ut off and is eventually brought uniformly to rest by a frictioe9p5x z62 dxccnal 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 7 1ehlogij s(58$m/s^2$ 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 sol:v1) gpcbnwe 7g. e:r/cq:yj lely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The bapylg7 /cbr: nqcgj:e w :.1v)ell 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 thinopq 1u:, p5ebmh5omfkb/+f q 4 rod, as shown in Fig. 8–4q5qpbb+4 ,k 5e1/u foophfm:m6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two massesm4zij6o wjt-s.5j7 g w, $m_1$ 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 turnsuq87w vyot/9 h (revolutions) and releases it at a speed 7/yt8h 9ouvwq 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 in3/q9o e unx. 5avmpj6gertia 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 de 9.pgteqm5b +y) g0kmcvelops 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 radiu4y 8hx lzvif, a k,6s4fj0au w.ny-,hms 9.0 cm rolls without slipping down a lavyk0 ha hnf mxu 4y4,f a.w,-8,zs6ljine at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy o7u8(y nv wd 6gl48 golofs/mb:qb b0o;f the Earth with respect to the Sun as the sum of two ter bo6w8/vfo:lqm8g 0b 7nls(uygdo; b4ms,
(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 1640x( n 104z,4uuytcr5pc/ nhi; lgtq g/m kg and a radius of 7.50 m. How much net work i5h04 u,myugnpz gx4r(/iqc;tt c 1 n/ls required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest ;szrot(qjl y3ox ;.2tftt 4e/and rolls without sli/r;z yqot2e;j(ttx4 ltf3.sopping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It q7t3 g4z0khp2ynx :e r jucf(+7vte/e starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just b3k+ ent7 0txjv7:cq2g(ruz/fe y4h e pefore 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 endt1v ea9 3b(pna of a thin string in a circle of radius 1.10 m at an angular speed of 1p9a3v n(a ebt10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding kel 0m lkc) ds15)f)kwwheel of c)wlfe5d)m1kk l0ks )radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform thkq- s728rxu c4)9ik(j lbqqloat is rotating at a rate of 1.oku -rq)qj k( 42sl7xcbi9q l83rev/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 in Fig. 8–29) can rg0f j.ywm2ib *raxpi8.-,.s kyo smv2educe her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes yks v2ia8s.,ybf. *-2pji0mrmgxwo. 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from pv soudfj8cwv-(+qv7 s0m,k + ntq3yass7 6i8an initial rate of 1.0 rev every 2.0 s tosqomjqkv,360sn(cv siya7vpfd 8+u+-ws 8t 7 a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at p*xv:ou cm)n s cufq10*5 6e.eqzif9ua frequency of 1.5rev/s The wheel can beifqm6 0.cop*n cs) u5v9z e fux1*u:eq considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure ska34aaj (dlj,n ater 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 q ml*fdb)7dwz,ybt*- ,j7rng1cc bm: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 d+oc/r q l:hun;isk of moment of inertia I is dropped onto an identical disk rotatnc:uq lhro+/ ;ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atwoby v6jj.v5y0 24zem 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 center of a rotating merry-go-round qrs8cy7ayt y07r vo3wme8kz 53j4 q*dplatform of radius 3.0 m and moment of inerwo45qskr8 vy y*707q ajtc 8m3r3zdey tia 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-ro 7z04e evz -(llndj5msund is rotating freely with an angular velocity of 0.8 e znle5sj -4ldv0z7m( $rad/s$ 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, losing about half itfgn:lab dke 80h/,gv* s mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest il0,8de a/n:bhk v g*gfnteger)$\approx$    $rad/s$ (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 mw6 pk8vxt +hiup/wy/crm:2s: z /vu0;fl ao)winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass -csr x4az sbtk3u :,v:$ 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 resnqiby 6fe4t*f3g cbn4t fep770 k2g)dt, that can rotate freely without friction. bfq3ty7f0 e ke7f)in24gtd bp 6g*nc4The 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 edgev9rzqha;02e thjr .p 0 of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170j ;. v2tr0hh0q9r eapz0 $kg \cdot m^2$ 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 th(soel0q i i.ih 4-tdir /o(bp0e end of the rope lying on the top edge of thehro(t0 b(ds i.- eiii40qolp/ spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth suc: sykqs5ftwp3 n(iez r;zy (3*)ald+th that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital anguzz*pw sd iels3+ t3(r:a n;5) y(ftkqylar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerateshd15u wdsq)s , from 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 cylindftij8+k4c /bper of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration.+ 84kfcipjbt / Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindriqb 5czr:1ex3/i /xf cmcal disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reachecz:c /mq3ibf r xx5e1/s 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 smmwz tl,v2 7qik6 sa,3peed 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 k t48+md7 bbhrmass 8.0 times as great, were rotating at a speeddkm+h 8br4b 7t of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it tovtve htw33e7 vi3 t79qb5f+c5*qqyyo use energy stored in a heavy rotating flywheel. Suppose such a car has a total ot *vf+ ey595q7 beqy73 vc3w hivt3tqmass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a. ork m7/zhh3- 8r5d toquueztbi;+j 4n $H_2O$ 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 horizon mzn /hztc).hu/xva;(tal 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., we iaw/cpl vt(6vqmynv5 2k u79* ggnore 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 acc7(q lm6 /v9ptkv*5vw2gacu nyeleration 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 radizdf qjhj (vm(y,hz3v7,; mrp3 us R. It is standing vertically on 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 h(7v jpzmhrmf3j zqy;,v,(3hd as height h, where h

  A bicyclist traveling with speed b,) 3 0ekxs29uhwej jtv=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclisj9e,2s3eu khw x0) bjtt 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 1d0 eq 6jdz92fcw v:xhm and begins 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 this f:xj9eqhd10z fwv2cd6 eat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms a411fy ymbe 6ypmx0z4xs thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and1ypfym x14myzbxe 406 with arms held tightly against her body.   

  You are designing a clutch assembly whl.jlu6t ;:quk ich consists of two cylindrical plates, of mjukt;.6l uq: lass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r7+kqqb jw1*m 7u-y;cld32y n oacnf.cough 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 the track? Assume m+uacdf qc j ;o*.12y3 q-ky cwnb7n7l$r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu/z.ldjmq9d1z t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniform8ezmjdzkx 727+ia u7/kef9ecbah (3k ly from 90km/h to 60km/h The tires have a diameter of 0.90zaih9 8med ac77ej (7b kf/+eku kz2x3 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