A bicycle odometer (which measures distance traveled) is attached near /fp7w il6w04fap 0 l i pzhcs0ah7pz/+n4k;yythe wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicw p7p/nfk a;psf izc04 /iaywl7zp hy4l00h6+ycle with 24-inch wheels?

Suppose a disk rotates at constant angular vblq1*m j+qzr g)3he/aelocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point hrzh+meabq* g3j 1/ql) ave radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid bodybbb,/bv8lfkd6i h l9 hz: ;+ip be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greafo8fjjvw4*fc26sdk r en ;n05ter torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your headpf0wo i qup w,7*sue-wdgp3 )ackv.e074j5 j e than when your arms are stretched out in front of you? A diagram may help you to answdwi)vcj7 ,pse5eg pj qw.foa7 0u*-k3 u0 ewp4er this.

A 21-speed bicycle has seven sprockets at the rear wheel ao*9wf6m, pqddnd 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,qf *ddomwp96 small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have sl99v:wpizxy z+n9-/ljni(;n4pa q y e8nzhn 6 uender lower legs with flesh and musnn; 4- u ajzxw969:l8p +/yzpy9hzivie nn( nqcle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narromyhavp5 so*af 3 0/jw,w beam?

If the net force on a system is zero, is thb+q )am88nizs6 s:upp e net torque also zero? If the net torque p)pnsqbi688u az :m+son a system is zero, is the net force zero?

Two inclines have the same height but make different angles withn4 ir 7 obe;g(*x3x3rthu)dts the horizontal. The same steel ball is rolled down each incline. On which incline will xr )*3se(xtduontr b3gi74;hthe speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from randj.xy hmc2**jy 7;k:3seixest) down an incline. One sphere has twice the radius and twice the mass of the other. Which rek.73cxh;*jej dyinmx :* s 2ayaches 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 radiusqocubpdz9m jk0 rw61db1)7k /+eo ci, and the same mass. They start 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 total kinetic energy at the bottom? Whicmjeqwuod), 7rc zk /icdb1+6pko0b 91 h has the greater rotational KE?

We claim that momentum and angular momentum are conserv i9zyqr 0/+cpdk(hdkg3o ns,1 ed. Yet most moving or rotating objects evh1 kg,zdyc0d (s9 qr/ nop+i3kentually slow down and stop. Explain.

If there were a great migration of people toward the Earth’fr6cvik+o lmsc+y69/r;-as4x+peo 7iz 4pxts equator, how would this ip +cmp6x/l+9oz4cfs xrvr;-kyo a 7+6itse4 affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without8s8zb8lacn48l da oy: having any initial rotation when she leaves t8 cs8a8dz8la: bl 4ynohe board?

The moment of inertia of a rotating solid disk about an apm-x kn1;rz +xc ba9r:xis through its center of max;rnx +zac:m b- r9p1kss 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 sittfx 0htago.x5i cp.:h syr8+ 5wing on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? y 5rh0fig ha+px ox..tw8s:c5Explain.

Two spheres look identical and have the same mass. How+9(tgxxat n o,ever, one is hollow and the other is solid. Describe an o ,tt(a9xx+ngexperiment to determine which is which.

The angular velocity of a wheelcex;z p 0w4b:ou83nqb rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east,8bw;xz 3e4:q0uco pn b 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 the2-2rq2i9 bxvtyk-kyg edge of a large freely rotating turntable. What happens if y2rxkiq-b9 vy 22kgt- you walk toward the center?

A shortstop may leap into the air to ((mve 1lk0 q8viwn98 5skawzicatch a ball and throw it quickly. As he throws the ball, the upper part of his bodyl1vev 95snaw(m88i z0(iwk kq rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatim4lms7(h. 9 gbzh/tmy huw) j56puw6 oon of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to kee ysom7t6w5 .gmj4whzuhb6u(p9 )mhl /p the helicopter stable.

  Express the following angles in radians: (a) 3uu 1 kxt dv1km -he (-p;ddh0u-t3.vmk0 $^{\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 berq 8rb l5j)b5k ,rlbpdala(a2; n8hk6cause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Ea rb5(r8l8ajk,n bda 5)r;llp b k2a6hqrth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divadie*v +y2. / (v2lwjh,mecro erges at mh+yd,rv(jow2* ece/ a.iv 2l an 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 blendetm148ealu n- dr rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angula ld1en-4atm 8ur acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes ya(p bmgi. ,/b15.0 b,gp .mybai (/revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travel*djr m9u b,l2os 8.0 km. How many revolutions do the wheels mak lr*m,2j9buo de?    $rev$

  (a) A grinding wheel 0.35 m in diamu.w 2, w .ymqj 3ebwqt9sje-a0eter rotates at 2500 rpm. Calculate its angular3wq, mwe29t.0s -yu. wajebqj velocity 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-rounw2s yw gn346jdd makes 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 ny gdjw2w4s3 6center?    $m/s$
(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(fewjaui5 y89up r/y+ its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi1 edqavec6 (t9bh w:3ont
(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 7i7fdyz q:b ns-5d5p 4p.0 cm from the axis of rotation is to experience an acceleration of 100,000s4pfyn-d5 q5d p:z 7bi $g’s$?    $rpm$

  A 70-cm-diameter wheel ac*vnyq 5dt mp(xo.+-kycelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Dete.5y tq+(n d okxmvyp*-rmine
(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 radiu amio3x-xxy4)s $R_1$ 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, astronas1qb1 0vyxvs(c r+vz, xd,+ukuts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Deter0(+1sz+kvvuv ,1x,x r qybcdsmine
(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 l8lzk dv : gb;2v8/lgfvt2k+9qtu0m yto 15,000 rpm in 220 s. Through how many revlt u ;g/f8dv+vtlv gq:92bk0 kzlm28yolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. ;3i0xi .y bhr+ajlq )gCalculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling “hu5t gnrk7+ztvy jkf g3mb.tk.p0c:7 j) man centrifuge,” which k:7tbf5yj0kgtk . .jvr+)gm ctnp73z 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 uniform. t eh/vp4;vllt3mk9wzg5 - hgly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this timk-v5hp ghl9/.emlt 3;zvwt 4ge?    m

  A cooling fan is turned off when it is running at 850rev b 1n+cc8i7wawk, 2bzg/min It turns 1500 revolutions before it co2z7gca wnbkiw +81cb,mes 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 cfya*.8(6 jbh6 hgho8vctkbd wa) c9(kar reduces its speed uniformly from 95km/h to 45km/h The tires havet ga a)b hhb *d.v8co69fcyjkh6w(8 (k 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 speed uniformly fr+ka tu8*t.v g; kak9d b,9fknrom 95km/h to 45km/h The tires have a , k9;t+k9d raukgt8vbf.kan *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 puts all her weight on each peda o6;+ltriwtv b36uzs ,l when climbing a hill. The luis6 , 3bo+;rttvwz6 pedals 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 /rbupt-vo( v6 pko9ubm 7f 46i+.n fpqof 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if thev/ui7b n-vqo f. mf ob9(r4t66pku+p p 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 rkxxfw3+ v68; zquq/zu(h/fjx9k)+d *tsgoz the wheel shown in Fig. 8–39. Assume that a friction tof/z sq* 96d8xk)+fuuzx3gtk;qov /+jwh(r zx rque 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 qjh)0q pwdw85the 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 t5dp0h8 qjq)wwhe magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a2 cl: rqpmsmf(a h80w6 torque of 38hr:cpa 0mfm (w6 lsq28 $m \cdot N$ 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 radi 0- /tqpu3 z2jd6q9aj,mipkpdus 0.648 m when the axis of rotation is6z 2p0kjj -tp/9q mqu a3di,pd through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in +fcj djm qc9q6:e j)q7g2rqh6diameter. The rim and tire have a combined mass of 1.25 +7c6 fqemchr)gq qjj6j q 2:d9kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thi( 14 uu f9m,vej 7sblxtlx;mch.x3 (v. qopmo3n, light rod is rotated in a horizontal circle of radius 9j,3.muh(l.x(l1eqv; bxvfmo7o ms43uxc pt 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 whm:dg;6e6c j 2 umtfyocg8o *8heel rotating at constant angular speed (Fig. 8–42). The friction force betwee6heym ogg 6 c;tuocm8j2:d8*fn 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 poin sr2vppgxz mjv xw2m,/)/n3f-t objects shown in Fig. 8–43 abo)/gs p zv wmp-2,n/xrfvx23mjut
(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 ofq;sqb nbd0x7chqp73r1 7w)s l two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite sp(fm rxw gzc/ 4i :w8y)pbxy7)cinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If thep)bx(8:icy4rwy cm)7 fg/zxw 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 satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is eg4b/w9kdsho+2,2kv u duw)qa uniform cylinder with a radius of 8.50 cm and a mass of 0.580 k/d2gd wbqk+u 2 wv94),u oekshg. 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, accelebe+/ 78xawn7iom e xn6rating 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 /fgg.pnm3a5 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 opp35 gmfpag ./naosite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventfxc(,6lx mj. oually brought uniformly to rest by a frictional torque ( fmoc,. lxxj6of 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 ball9r4dfge9ne /h 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 s0gv+jb ).d jq5n-lctg olely by the action of the forearm, which rotates about the elbow joint under the action of the triceps vg0nj.j-bc ) g+ qtd5lmuscle, 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 consid0 e2+ +p;tbbh. yenbkrered a long thin rod, as shown in Fig. 8–4 +rpe+tb.02;bkb ynh e6.
(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 twotz) v )t4eajc1 masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest withu 8 k8r0fpskgc 0qe/s5ha:1l rin four full turns (revolutions) and releases it at a speed of 80r h5sl:p08ceksqgk1 uraf/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 inertia+ fs+f6uiyiqis 5v )/i :;mltp 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 280 4aqrjhjzm2zf gwti ) :1( u/mc194yge$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass .pundr;gryzxb: vd.c8g3 p 0. 7.3 kg and radius 9.0 cm rolls without slipping down a lane av.cznxgd bg.pu0r. : d; yr38pt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of h8md( h dm0e,ktwok8( m0m h,hedd terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50z7lwvbn7 u 8l4 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.08v47zl blnw u70 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kgtl d-4;xfcy/-fn d v)k starts from rest and rolls without slipping down a 30.n-;ydx l /-kcfvt4f) d0 $^{\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 starts tvdo)( 5ycfs8hk2 .iua8d 7ujyo fall and its lf8 7ciud)uydyaojs5v k.8 2h(ower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball roje5r rj zocd)ms3 41f0tating on the end of a thin string in a circle of radius 1.10 m at an angular speed zdro34cs)j51fe0m j rof 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momenevieqrf+0 :l1tum of a 2.8-kg uniform cylindrical grinding wheel of efvrq +0li e1: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 that is rotating at a ttr. bt7xc1j4fu/p4 zrate of 1.3rev/s I4ft u xct.z/4b7tr1jpf 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 reduce her momentla:g:5taahhzg ) 0e:k(.6l w qr*ffmo of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If qo:k:fftg. :0zaa hhl*gr)5( 6lmeaw 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 ratnefl ijmu 4y0)i;,kc/b:g ll8(cak4lch.:c ee from an initial rate of 1.0 rev every 2.0 s to a final rate of/kf icuji,)4:ml8n; 4 hy.acellec g(l0kb: c 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 yd ;ted, 4ulo/re/ pr8at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0,t u/de;ld4orp /ry8e .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 + 3dbj6ebecsl -8/ bfmmomentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the 4.:z634:hhqyoub yd/ hk thd bEarth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I isg.g -ncj :yo9(yd:gsd48is21p vog qb dropped onto an identical disk rotating at angular sp:g8v2 gg-d9bsy(d4i jg :ns.co1 oq pyeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2hi 0wy+p tle6/xbbo66m2ae / p.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 centieoa8jm+9a19 *rtjs 9 6h mtqker of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 929j*q1 ta6ehm o +i9akr89stjm0 $kg \cdot m^2$ 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 freel1th yaxj a,n/*y with an angular velocity of 0.8yt /a*n1hj,xa $rad/s$ 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 itsba5e-ok/)wt hlbvq3 8 mass in the process, and winding up with a radius 1.0% of its existing radit-hbqv /o lk3b) ea85wus. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120g9qetz /t f6alxzvf0ucj 9 on )o79-5r $km/h$ 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 8 spfn7+g g9tz$ 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;4(k8s e)tr0) wvaom-czqbj r rest, that can rotate freely without friction. The mo-()t wvo mq)ra0s ;e8crjb 4kzment 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 standsv.it . 39v)dhqzlt8ms at the edge of a 6.5-m diameter merry-go-round turntable that is mounted onzsv .q3 ml9tdv.i )ht8 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 ropjvjtw*;b:c 0 ae rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope *a :w;j0 vtbcjand walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the s 1rjhe0dbz./ xame side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Mr1j.hx /e0 zbdoon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frw2l 31 vtrr6ykv4 z+lkaku.p. om rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m a t t0xxzn+p/ krn4t1+vcquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor0x /+ ptn+rtz4tv x1kn.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg g6hf8: 2lnjdq*m wi,q(qz r7uand 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 reaches the end of its 1.0-qfq8gq u,hz i *jl7 n6:dm(wr2m-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 speed of the reare;ad1uhab q5l0(yl( w 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 8.0 times as great, werebo t7 *g(0nkde 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(td7 no*0begk 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 for0y+s j:5v f znxy krmqi-3xz5,y9so2ow1c6 bv it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diametksv9:+36fyxzx52 i mo y,bcys-z 01owrn5vq jer 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(. aa vua7b)3*mqa efo 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 speed v=bkhks9v(n7) m /r.qs:qk ,- fra8glci3.3 $m/s$ 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 ignore f9.g/zm 4tm zhkriction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of /km.4hzgz t9 mrelease, 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. It is standing vertically on taz c;z5yk sy /x.kjoql 1hn-28he floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where 8-xkc1qs/y aoynj2l.z5hzk;h

  A bicyclist travelin9gnpv4d 6fy .kg with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the kd6pv.94gfy ncyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirlggro j:k/u.ejz)3jv:ing the rock in a nearly horizontal circle above his head, accelerating it from rest to a rat:jojr.z/ v3k e: uggj)e 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 as:w(ffy1 p1p x+msvk:n bzf66c thin rods, making reasonable estimates for the dimensions. Then calculate the ratio o1p6y s1f6m:z wxf+ :pb nkcfv(f the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whichq: cb ;9e6 2wekh5mv r:hyct8a consists of two cylindrical plates, of 6kwe m2 cerch;8aqhb9:y t:5 vmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ra /7itnrf , x7)s)rlgngdius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop ntf7lrs)rg ng ,i7 )x/if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume o o0 vpd7-g07nlp) ouwya1fv4$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the cat w za7- o85hbv)5gebk/kpd ,xr reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a)zh-bw a5vok/g 8)7 pkb,dxte5 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