A bicycle odometer (which measures distance traveled) is attached bcb 1)ne (q7,1r:h iea hstu* (lg3ieknear the wheel hub and is designed for 2 gher(ia) u1,3b1l b*(e:tec7q sh nki7-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular o3j1 /q h46 ;p;yt9 of,frezzq:lcglb velocity. Does a point on the rim have radial j ;,o9ft1hez:yqobrzgfc /lql36 p4;and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veckobd ) 8i -lf*dhq0/elocity $\omega$ Explain.

Can a small force ever exert a greater torque thkur36 ct1 gdcx27 1jlqan 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 paj/xs)dh( 71ylm j1jdo a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may helpjdm7)(1ax 1lh yjj/sp you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three qrxl7: xv+5z4hdsj qzitsjp9 ;8*7kr(ks ,swat the pedal cranks. In which gear is it harder to pedal, a small rear sprocket j 97k4zsq+rhp;kqwsz7sij ld(8*xr5sx :vt , or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast hvww: oqk .,h -1j6iexvave slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain whkvo wv- i. ,e1wxq:j6hy this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) ca -*wm:c*zemranku +avw .a1lv2 4ikc.rry a long, narrow beam?

If the net force on a system is zero, is the net to vjvm e/il8,6:wwibs(rque also zero? If the net torque on a system is zero, is the neijv slv( :/8iw6w mb,et force zero?

Two inclines have the same height but make different angles with the horizoh6p 17m c 0vlcgmvd.78*hhcgi ntal. The same steel ball is rolled down each incline. m7c0.ph1 m*cvgv76il hgch8 dOn which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fromdi ,x a.d e0l:b*p6 *fiv0mhus rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which .*:pisl*dhdbi ae,u0 vm6 f0xreaches 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 radius and the same mass. They n0vx2ku3jai lphbrd e8(z6/ 6start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater to0/ 3j(u nlherpk 2v8zb66idaxtal kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conseczqg8(h,30 miux; v dfrved. Yet most moving or rotat0vg ,(ihd;qx 3f mz8ucing objects eventually slow down and stop. Explain.

If there were a great migratw.bz/ x-j4u4joru8x cion of people toward the Earth’s equator, how would this affect8wu.ro4 - j/cbzuxj4x the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initis )tclv-f a7-hal rotation when she leav-f -lv)tah7 sces the board?

The moment of inertia of 2:4 -ud kcr b;lldsqi:a rotating solid disk about an axis through its center o d-4l b;rdu:qk2si:lc f mass 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 c-:mu y 1k+/29cmgv gi ecuc6wa 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay thu+c wv2g/ yck9ce6 cim m-:ug1e same? Explain.

Two spheres look identical and have the same mass. However, one is h/i--9zsjj ,wbv 1f pg:hxny9bo -x: jiollow and the other is solid. Describe a9nhs j9fjip y/b 1,j:-bizvwog-x-x: n experiment to determine which is which.

The angular velocity of a wheel rotatingzr6 1kg54x pfc 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 of this point at t5 x4fr zk6cg1phe top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large fre 6nnk)93vwq )jx ;mtjyely rotating turntable. What hap knm9jt qxjw)3)nyv;6pens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it a bvu -j:o86lfcvl4- squickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will nvbo v8lsc46:l- jau- fotice 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, discuzl8-g f, arzr)ss why a helicopter must have more than one rotor (or propell- zfar,rg8z)l er). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglne*+rta- biw0cx) w:tes 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 Earhz3 )lns+ya bvr 8 ;juc0lgc7)th because of an amazing coincidence. Calculate, using the information insi7)jrc vzy0b+u8)hg;c la n sl3de the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam j (by/ml7u5o6c4u yeediverges at an an7e4j ou b( /u56mceylygle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. Whix/k8h/)agk7wd4+s m (e j jtcen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as t d+m/)/a4j7ijw kt ghexks8(che blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ba) l t:o3na b4est5i-g5ftrz.lll makes 15.0 revolutions, what iszt elgs5i3): 4l.bra tnotf-5 its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoluti*re0 ien; edg:ons do th 0; en*r:diegee wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. z 2 esj;lfv:v 24bogfbt 15/yoCalculate its angular velocity : v/2fygfboo1be5;s4 j2zlvtin $rad/s$ $\omega$ =    $rad/sec$
(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 makesocqlulm e 3 2:m10fy0f one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cen:fql o03flm mue02y c1ter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular v: j31aj;8p8td, 2ephunr copbelocity of the Earth (a) 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 pointb 4q05 zadc+dqy:o: be
(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 from tzz/sowe- psbbo7zn8c0 ;v.)m he axis of rotation is to experience an accelcwp-.bso )zonz 0me8v; /s7zberation of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accysfs-bbhl7:gh /h( w0elerates uniformly about its center from 130 rpm to 28sh h0w7b-h/ b:gfyls(0 rpm 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:x uiy0mg2a m9 $R_1$ 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 4 ;ew f/,r 9aq/qnfhczthemselves 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 b /rf,w/ch4;eqa9zq nfe thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 st*wmnwz s90i (o1 s6qy. Through how man *o 10m6tynqswsiw9z( y revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. y85- 4y:ch b h hd+v,hgcr,uokCalculate
(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 fog56 d n;yahksmj3zf44a(hwu 5r the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rmy435 w jsg(dh 5n;ka4zhfu6a eaching 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 rp0t8a:qaf (q-,rp jr5 fxs(xfg m in 6.5 sg:xp(fxjasrq -(8 frt5 0fq a,. How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off whl)(n tgedb3b9vmh6)e; qfj*s c,2 wggen it is running at 850rev/min It turns 1500 revolutions before it comes to a stbe9 *c wstb jf 2g,)q3m( egl)6hvg;ndop.
(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 revji(3rjy gfamx01x*25loy ano-8 q t-p olutions as the car reduces its speed uniformly from 95km/h to 4gxoq8jo 0j15ry-naai3mft ply* 2-(x5km/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 ) p-a1q uh2w/r3tavefrevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires havewhu/a )atr2 fq1evp3- a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike pd 69o nswt4b- nb6v2bputs all her weight on each pedal when climbing a hill. The pedals rotadt vo 94s6bn6bw2n-b pte 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, q8+o nsatsh-r:;ife door 74 cm wide. What is the magnitude of the torque ifiaeon; h srt,:+-8 qsf 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 tonc- f:rw 2eiu1rque about the axle of the wheel shown in Fig. 8–39. Assume that a friction torquei c2e:nw r-uf1 of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of masstyf(5 v 3/fhyp m, 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 magnitud3 /f5(vpfhtyy e and direction of the net torque on this system.

  The bolts on the cylinder 2s8qp/6l-pg 3 mzd iqyhead of an engine require tightening to a torque of 38 3q/pipy lm6q2gsz d8- $m \cdot N$ 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 -mw5rkc fiqh. .g9va;of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its c gc9.iwfkv;ma5.-rhqenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle rvi6s9s)f.c*,k;+ q cj7oxzz ;zj xedwheel 66.7 cm in diameter. The rim and tire have a com,kf*jd s7)zcx9v c;ozrsx+;i eqj .z6bined 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 ist62st;rmn5 l z0,z*d 8rf8lkl h m/pa horizontal circle of radius 1.2 m. Ca;25ti f m l0 z8rs6kzll,ms hrnt*/d8plculate
(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;ba n, gp5visf3yn0i a-:oa-m on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The fi s-i;o:a,yb -p3avmgf05nnariction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of poi:pf4 s2tu yo6n fxio64nt objects shown in Fig. 8–43f4s:2y6 ox ot46pfi un about
(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 mase t+ni hxz8,npr8d4o+ s 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, uniformd9ab liv7n u1 0o9tn:k 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 361nbu9n70: adki o9tl v00 kg and a radius of 4.0 m, what 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 cylind4ojb9xet6- q mer with a radius of 8.50 cm and a mass of 0.580 kt9 m-ebx joq46g. 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 fr(eghb 3 /2l5jautrd *yhk1u4tom 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 onslo (3djtj6y5 a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of 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 fri5 so(l6dy3jjtctional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off andeo tde*a66u/khrf *a/ is eventually brought uniformly to re /*k6te d /oufrea*ha6st by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball- wse*ujmf-t - 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 a*0hru *etvsy 1sc6m+a hv75zby the action of the forearm, which rotates about the elbow msh*a1evytavh 7+cusr0 6*z5 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 can be considered a long thin rod, as shown inwd2p(bd:jhr p/ic6ub z.h h5 1*vy7hm Fig. 8–46.7 i.chhpu6jr b1myhvb5 /zw (hp d:2d*
(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 masses, mkn;.bu13ggyk;t wg5$m_1$ 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 accelemd3 nkx2dck w ;xa.(4*hh+*gk /nn6v /waryz trates the hammer from rest within four full turns (revolutih.y k*k maa(n4;dnkz+hr wwdv2 xn x36c/ */gtons) 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 isc ui/+0 r, ae6fts0oenertia 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 280o3 -5ycldf:uc $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and 50c bitb*nw1vradius 9.0 cm rolls without slipping down a lane at 3.in 5c*wbbt0v 13 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as tpf0xf7ly7,a/d l/5fx m g7kqqhe sum of two terms/ f q5qflx7m7k,gyfa pldx07/,
(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 anz*b6 gpc,c r: qt9r:vipw/pgh3.ie) ad a radius of 7.50 m. How much net work is required to accelerategbra vip )p/3c6rt:qze 9* cp.w,ig:h 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 qii/gm 4hie6(xu o4 9icm and mass 1.80 kg starts from rest and rolls without slipping doigh(o xiu9im64i/ q e4wn 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 staew f;ep+ bw+9xrts to fall and its lower end does not slip. What will be the speed +eb f;p+wew9xof 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-mfd//jj n-yxhmk(m5 ap66 cy0kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 15h60a/myjk6c/x -p(myfj n md0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of x;750cmh h.vxt4jr d y1le u.ta 2.8-kg uniform cylindrical grinding wheel of radius 18 cm t0h4 d mrev;.7.ct1jxxh 5ylu 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 thaz(nil) no; cb 9u2a1ilt is rotating at a rate of 1.3rev/s If he raises his arms t)nilinl( z219uao c b;o 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 of i xnz so109m* d)yhbba r,z49p7gal8bynertia by a factor of about 3.5 when changing from th79ma lhosb *zr ax)1bp,bz0d4 89ngyye straight position to the tuck position. If she makes 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 an initial rat- 7dwr hmz.ae:o8xd u)e of 1.0 rev every 2.0 s to a final rate of 3 8d zuew-r .amo7x)d:h$rev/s$ 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 *iksmw+-s 9dwsf q78wa 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 diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, a-qi* 89 kwsfw7ssw+dmpproximately 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 w5lo--p xjjfsp e/hh 9tg:/8wat 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 mobk cm9a sbk3qb8i4 c41m*ww i8mentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk ;pvykd m;d sj)h24vr3of moment of inertia I is dropped onto an identical disk rotatingh )k24;3jrmy;s vdv dp at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2dk*di gtq r,o ydqg+664 ehz(5.4 $rev/s$ 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 ckq/8 + bi- eq,lgrrxl6v91vbdm 7+jy venter of a rotating merry-go-round platform of radius 3.0 m and moment of inerr+8mdij1kbqe/vbly - x 7glq, rv6+9vtia 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 an0aranz 52m5;1uivb pagular velocity of 0.8 0uanp55 vbma;r2iz1 a $rad/s$ 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, lov83moo/osvn4 ;top6(7odv ci sing about half its mass in the process, and winding up with a radius 1mv d6v3isntvo / op47 (o8oc;o.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 involv;x uiawlgz2:tao::lz d0 77 hke 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 m o,(5z jdpfl)k21s 3t9iv9hik wu*jh $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freel:tjcphh7c oro9o g(3)y without friction. o o:j ()3ch9h7grpct oThe 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 ;vg.ygd25,g 77 -q b0uaxetqq ja2l skthe edge of a 6.5-m diameter merry-go-round turntable that is mou tqk;2daylsqg2bg,x0j .g-q e77vau 5nted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the enkay6-o /g mts4d of the 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. The spool rolls behind the person without slipping. What length of rope unw6tm 4os-kayg/inds 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 Earth0vqm0)bp. uvpb v:xr.. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the . 0xbb0:)v quppvv mr.latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/a+(dj7u*zb6knk) zrf $s^2$ 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 0.20 q/dk2p- cfxp;zl r()lk +m1ia m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration fkckli qa;/-ppm1rl+ dx 2(z). 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 dljkfu4(4,*j99v nyhsy rv e-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 whsl9-ey*,9vk hj f n4ju d4y(rven 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 l)u )+m bc6eguspeed 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 wh3 hj y0tjux4*ith 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 mhjty4 *0jhu3xass 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 itzki-fzu i5;,ea 6ys n,hpd1:g31x d di to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg,3ki,6z nda1 zd g dei5pyifx:h u-s;,1 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 illustratestd8mr2qa*p9y rttwn .i 371 vq 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 hor5 zt3 gas/i/w roewn /vakd +0f-/b:svizontal 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 ufv;6znf p d .i,2i*cf(i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the ti6nf v *ff d,2.zipuc;ip 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, :zi p*asywj2jo3-4. nl )xmrjklal 7 8and we want to exert a horizontal force F at its axle so that it will climb a step 7k82np3x rmzlj 4* so. i-wjal)lj:ayagainst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4bg6sq+;ty -/ 7f; u;os jksjclxo7/ 9+nbyjut.2m/s on a flat road is making a turn with a radius The forces l9 q+/ubfs-;;xst 76k +oc jotjn y7/yg;bsujacting 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 at-lfknd o698l sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle 9- tnd86flo klabove 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 arms a8l -oxu in/bsi* 6x:bfs thin rods, maki8bln sub:of /-i6 *xxing reasonable 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 p w)pb( l1h(bqe17ucuk5g ;bb cylindrical plates, of ma) wg7(bh 1bpbep;u1 qk5 lu(cbss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roug):.pgxh)d afgh track of Fig. 8–58. What is the minimum value of the vertical height ggpxhf:a.) )dh 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, b66jab4u gcy0e5/5xonn0cm u/y xq c,lut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolr 1m+ukqsg7 z7ndgf9: utions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire?kf7 md+r7:g sgnzqu91 $\approx$    $rad/s^2$(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