A bicycle odometer (which measures distance traveled) is attached near the w55m es5 1op8yxvnw8eywr .a9 wky)rh/ heel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 2m k rh5y wy81sv)y9orewp5/.5 eax8wn 4-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim b glm,sqc/,c vx7w0u*have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration?g/,*cmxq0usb lv,7 cw For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described bt5hxhvpt;-6w.r;z ; vhtwgr . y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?) l.a2 nry2 kfe5c/:j8vgk ixy 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/es6,p4 ddsth your head than when your arms are stretched out in front of yo sh6dts,pd /e4u? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three lk+x c.0 01q n -ytvwux-ldi. i-nnk;oat the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large ck0.0;kld-u no nxlnqvy-.i+ xwt i-1rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on ql xca0f7ra5ye/1 u ,sbd1 *njbeing able 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 distribution of mass luxf 1 der0cjb q/sn5 y*7,1aais advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lont 6s dw+ma-7kvg, narrow beam?

If the net force on a system is *gow fom4b..b; bplo 4zero, is the net torque also zero? If the net torque on a sy4fp.wb ;bogo mo.4* lbstem is zero, is the net force zero?

Two inclines have the same height but make di)/rlc m;pt7tmcym;;kos* pn4 fferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ballkm*4/yc;c7tmorn; ;)l st mpp at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)6k,qk45, lrvqv aer f)m .g2iv 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 hrvqeq, .rm)a igl52vf 46 vkk,as the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the s 4iiubs :f:b1q,7ljetame radius and the same mass. They start from rest at the top of an incqe 4litj71bf ,:us:biline. Which 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 mwt ,y(c )4 blo1rmd0+zzr;qc mvsh1o8 oving or rotating objects eventually slow down and stop. Explain.y zomsr0wo )z1tc l,vc 4;q(d+hb1r8m

If there were a great mit2s6d yf7neymap3q9d26 bsm:gration of people toward the Earth’s equator, how would thi6f 2tbd7nm qpy2 6mas39dsy:es affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having anb3 uaws: dy7+oy initial rotation when she leaveowayusbd +37:s the board?

The moment of inertia of a r9d8acm8 rpe g /mfvd*(otating solid disk about an axis through its center ofed rg8f9 vmcm (8*d/ap 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 a 2-kg mass in each outs7- yrzgd 49p91 g. kivzzfoxi7tretched hand. If you suddenly drop the masses, will7zf.9i 4kio vg xz-dgy7p91 zr your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow o/dg 5 yanr,waa2vqb838ue:ip7vu4g and the other iu3:7/v agy5 b odg2qri8, pae 48avwnus solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In5llqq 4p6aw 5 y8ntuv7:axk. a what direction is the lineak :lal4a.xtwu5aq76qv py58n r velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely r4xs0xacsm:w;(c ds,f3 a7 d tjej8vu/ nwp4;notating turntable. What happens if you walk toward the center?;3pw0x7(u,dacc :4e8 v/j tds mx;snsj4fnaw

A shortstop may leap into the air to catch a ball and throw it quickly. As h* ewkc4f 3qtr1e throws the ball, the upper part of his body rotac 43rf 1twkqe*tes. 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 anguldwu3pn vnn-igm6*; li tz8v,*ar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep w;6dntignv i-pm,z8 u3v *n*lthe helicopter stable.

  Express the following0 .x pk8;vm t.3jqybrm angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth bgmm onwfai: y3kzcap5)d7 9.7ecause of an amazing coincidence. Calculate, using the information inside the n kzi57)w.g mad9 3y fp:mac7oFront 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 Moo a pg,ftg(-.+uahb. xkn, 380,000 km from Earth. The beam diverges at a+ak(-tgp. ,f bx.hg aun angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motu)tlnrw+ mdkoh) e*3- 4k; ih8.qmmlkor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration ad) l 83 timqek;mmu*k+h4h.ro k)n wl-s the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chil8wagp6-q pf g/d. If the ball makes 15.0 revolutio/wpg qagp6 f-8ns, what is its diameter?    m

  A bicycle with tires 68 cm infzgv.n2i8 mu mpv w76, kihe26 diameter travels 8.0 km. How many revolutions do the wheels mak2kn7hg2zv ii6u6, 8fepmw .mve?    $rev$

  (a) A grinding wheel 0.35 m in diameter roth e1l2azk17a3n s+x rpates at 2500 rpm. Calculate its angular vea+1 sp1kea7x3n2hz rllocity 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 cssq8d 6as-: b ts.)pzlv;s 7mpomplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child s -s.dslb86z:qppa; stm 7v)sseated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in ih rfrp5w7ph7bmr3q j8(n1h 6 nts orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe4g 4bi dzp+l/4ws0qmp ed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centk-2walc-uaw8 tyi-y** (is zurifuge rotate if a particle 7.0 cm from the axis of rotation is to expes w8iy*tz- ly ci*2-k-au(wuarience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ceni q5cgdf*e+8( c tl:fzter from 130 rpm to 280 rpm in 4.0 s. Determine+ fedgfcc:5l q(t8*iz
(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 9n2s );klcz5loes kc+$R_1$ 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 Moonejh;sm fj g6st16xa:7 , astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute th7:ssfgh;ea1tj mjx6 6 e Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. 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 fa *4uf5+v wr -3cv4iingo5a j15,000 rpm in 220 s. Through how many revolutions did it turn in this tiv5fv-r g+454foic3naj uia*w me?    $rev$

  An automobile engine slows down frozg h .ff+z9frgnaf3v.5: )qi gm 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 the stresses of fyd,) q8i)0gnnovit h 4lying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20o0ig8dh q vt, nny4i)) 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 rpm in 6.5vhw l pf0d5+ m3(ob5cl s. How far will a point on the edge odvb lwf moc+5p3hl05(f the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 8la/s m,b0pj c-50rev/min It turns 1500 revolutions before it comes b/ -a,p0lcsjm to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed 9hm;gxx 55v iwuniformly from 95km/h 5vmh5; i g9wxxto 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 as the car reduces its speee lky0iv9 7 9ii)(5ggb .fq4eu hro3wwd uniformly from 95km/h to 45km/h The tires have a diameter w7kl99 bq3v irug .(iw fhey5e)g04io of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbix( 9ak+5jdjb ong a hill. The pedals rotate in a circle of aojb+ (dkx9j 5radius 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 ofb3 x4iwa ru- jux/j5d( a door 74 cm wide. What is the magnitude of the torque if the forcr4ja/j(xw uxdb 5-i3ue is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel 0qmeqeuc pq8o s0a 5o,uf/2l/ shown in Fig. 8–39. Assume that a friction torqu0o5uo8es2e/f qp0 cqla /u,qme 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 the ends of a mausm y46 bm 6slkm ;m0lcnq6ek+x(v8 3bssless rod which pivots as shown in Fig. 8–40. Initially the rod is68;elbsv0cx 6bsmm(uy3m l km+q6kn4 held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylil+5m2 vae;sr (uq ht0 n8cltd/3rg)eender head of an engine require tightening to a torque of 38 t ed25;chu3t geemnr( sra8)l +0qvl/$m \cdot N$ 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.;3jzxv4rawz 9 8-kg sphere of radius 0.648 m when the axis ofr3z w9jxv z4a; rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diakxap4, 1ytbuti -dtc b9kdq56)g qf;/9g9bpq meter. The rim and tire u6 ga)9tt 9qk dc9t ppbkxq,yf1 45g/qd-ib;b have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the vap4e6n5fvspk (bz;l shcal4 3w 0 r0ej0 1y;yend of a thin, light rod is rotated in a horizontal w1eyb sk4y 0vn(4ejrah06vl ;3 z;0lpc s5pfacircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating a g**ek+i)sgo;8:t f 3 hsmay zr/.wpqzt constant angular speed (Fig. 8–42). The friction force between her hands and the*t*szie+g py38;mqfagok wh/ : )z rs. 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 in *2 ,apmwer t*zct9mua.5 aj9 ayp:.omertia of the array of point objects shown in Fig. 8–43 abouap5t*:ma2 r 9., 9awu pjcyz mte*.amot
(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 wh, v0mb wr.qxk+ose 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 7bj d3ry npg*.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 rg n3j* yp7.rbdocket 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 cyliu:pb0 2dbab(yh wt3 ,bnder with a radius of 8.50 cm and a mass of 0.580 kg2bywb0b 3ph(at ,:u bd. 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,9d,vlno 1k)ob n6arx8 )0adhz accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drivde3k*ak 7)q)ci :zoshen merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm c: i7sza*kk)3 eoqdh)in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and isz34 mmqk7c/eb eventually brought uniformly to47/m3 zkeqm bc rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 udiy6w ap:.c-accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the act*ydxcwno 9j /yf0/,nc ion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is/j9 c,xcdw/0*nfyn yo 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 r 0mvk 0 gxwmj7oy9 b)xusm1:6vod, as shown in Fig. 8–46.b71ymsj v 6wxum0komv0 9g: x)
(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 umg, xf/ 8;k8layofo. two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates n2pc2xek kr, +d f3r7bthe hammer from rest within four full turns (revolutions) and releases it rb2pcnx2+rk7f d,ek3 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 in:z(mnsrb/v :c hk oux,bt),(tertia 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 vr8*f/ aub v+o*vq qw,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 ra+z.ku8p1jp*teal i;d dius 9.0 cm rolls without slipping down a lanjzd tu. 1a;*8i+ekp lpe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eart8q*.hv,o1dxe xdhg; lh/c/ukj):xqz h with respect to the Sun as the sum of two term1 *;zx:qlcjo h8uddvex ,. hgh/qk/x )s,
(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 rad)s+ci8n*jan uius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.sa )nni* ucj8+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 withouof/5w+i ya-nqcwj po l -7ppz28e*w,hx)kiw2t slipping dow i 2 fwwpo)zwy2j-7q+wcnpk* l-xa8/hoi 5,epn 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 qejz01qs;(* 1qcwf n b1rn(kftip. It starts to fall and its lower end does not slip. What will be the speed of the upper end ofqqzr kj e(n01f(c ;b11nq*s wf 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 th09.w3hyote+owg w; axio0r+tin string in a circle of radius 1.10 m a0xaryo+w ;+ .wt o0eg39hi twot an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cyli.kvtdo1j+ c sggko50 c,sl0d2 ndrical grinding wheel of radius 18 cm when rotating at 1500 s v kcj5gk2to cd.1god, +s0l0rpm?    $kg \cdot m^2$
(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 i3rk/ b xv-sbb0af ,qv;s rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48a,3xb;b/ kqv0sr b -vf, 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– fy d)1kmd*vu/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 in1kfm* )u dyd/v the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spinprt fc.m7s ,a- rotation rate from an initial rate of 1.0 rev everyrsa7 -tm.c ,pf 2.0 s to 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 ( -e x2wi.utdsrotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considerdesi(2 - .utxwed 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 /,f *qezb07fec js4,mvc d/bga 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 mom.qh;5i5,kqzx :vz cf gentum 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 cylindrica h)k;: smw)hhm +tw3scl disk of moment of inertia I is dropped onto an identical dishs km+h cwht):w)s;3m k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsoz p h6 gor--w5lh *bxoo+r(l8 at 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 ss7 0p (,wxutai(nuv4itw9hn oxc-q03 tands at the center of a rotating merry-go-round platform pvt0 n3ix 07(nw,i4uwtq-9 c(uxsohaof 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 g75nh1; ly xvirotating freely with an angular velocity of 01 v5x; 7hnilyg.8 $rad/s$ 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 collc n1ars h+1 qjydm8,iay/s3aputarn 7w8) f.-apses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rot-imr)1 adyj8t/uf.n8sa3q7 nwsh p1 c,+ aryaation 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 windz4wx9)kyorwo4t ;m 6 os 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 h q9mt vc-k8e wrxf1k* plq-/;$ 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 u-ftjr+ 7i:nvcan rotate freely without friction. The m+vnfij:7r- t uoment 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 persoyeg/2j)syqkn jq( 2: mn stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings)k:yj e 2s(n/gjq y2qm 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 end of the rope - ;blui*m ,+hoa bf.guoj n 1 xod/o*icr94yb*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. Whaoc -oua+m *db ; /fr*iuxb1*oybh9gno i4.jl,t 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 alwayotoi7 wn7xme): ;1n brx5f:lzhtrq-0s faces the Earth. Determine the ratio of the Moon’s spin angular moment:-n ) ib:otq7rh;nrxw5xz0 ft1el7om um (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a raqe4:ng , ymj 8 1y(jya+0 fs2lgg+zsrnte 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 un om4x*exhz rbg. di56:8 dy ct7fl6)mziform cylinder of radius 0.20 m acquires a rotational rate ofd.hmx:i clzm6 xfr856z7b4 y dtgoe*) from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyhdc0n0 8 5k tfen qp9 ::x2;5rj uji6hs2woieilindrical disks, each of mass 0.050 kg and diameter 0.0c0 dqurp56e95e i 20iift2nh jkn x:wo8hj:s;75 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 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 the3y ( 0zayge6t g;du6zbyi)*rz angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0fv(s j*w-:gk x times as great, were rotating at a s*s :-kfjwv( gxpeed 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 aui .: eiilea) 8n+(g2jynbon1o i6)crrtomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a toa6ir1n ii.o )ebl2 n r:j8(icn)y+ge otal mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrateo+ zxdwpxdh/eq3(*0n s 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 horizontal surface at ssqjyesef21) k b954jq4 u qwi(peed 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., w+ f2l.vwyoew; e ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontwe volf .+;2wyally and then released. At the moment of release, determine (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. Is lba05 v/em9nt is standing vertically on the floor, and we want to exert a horizont/vemn 5lsa 90bal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=s7nl5yu i*-iysbj7*2 sf t;bd4.2m/s on a flat road is making a turn with a radius The forn2s byt s;*yilsfj7*7db -ui 5ces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1./rxx c1 b0uy;a7 fm6o,1zdycfi ph (j+5 m 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 feat, anddyr7(p6,ocf/m xaihc + 0bxjf 11uy;z where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a sol;*pz/1ghm4-so:iklrg h g s *wid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and wis - *mosghi4:pg/r ghz*1 wl;kth arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindriculo0p bws7d2t:pf20:fi t-leal plates, of maso lfipt27 :0ul sf 0:t-wpedb2s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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:6xb) 7 vkrl,dj-pf sn track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to rearj-snvl) x7pf6kb ,d: ch the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but doi,sfq/*l-sl8* k oypg not assume $r\ll R$ h =    (R-r)

  The tires of a car mak x2v8v-y+key ord48fm-;a j exe 85 revolutions as the car reduces its speed uniformly from 90a8 dr4eof2x+xk yy 8;vevj--mkm/h to 60km/h 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