A bicycle odometer (which measures distance traveled) is attached near the wheejxp 18lvvba0 i1q6;n dl hub and is designed for 27-inch wheels. What happens if y1 0iv8 qjbvx ;np1dla6ou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constan+fg qwb4195 6w3v 4zsoermbmr t angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases unifb9frz5bq 13w4m rm s+w4evog6ormly, 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 ofxs934(a vuq nk the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forcb nw ctjjr ,;+ci:mk4eq (;0zee? 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 handocxj9 h iq,0d2mawov(1 *uk+fs behind your head than when your arms are stretched out in front of you? A diagram may help youc* 2(fdwuxo9i0jqvmk1h oa +, to answer this.

A 21-speed bicycle has seven sprocke* tn: 7ft)yhegts at the rear wheel and 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 gty t:gh)e fn*7ear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend +nk f65ha bsy49i5m*m j6i bkyon being 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,5m6km* 4jh9n5b + iyfsabi y6k explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrsaqrf:* t k,2 4oc07fafilip0ow beam?

If the net force on a system is zero,-c:f dqll) 6lx is the net torque also zero? If the net torque oncl:f6 l -dql)x a system is zero, is the net force zero?

Two inclines have the same height butsx-rk-r psvc+vek(*4 1lh9w c make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be g vsv ( 1k9w hc-rrkpcsx+l*e4-reater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incs gq zzfh*u+,*line. One sphere has twice the radius and twice the mass of the*g+sh,*uzqzf other. Which reaches 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 cyling3 jtes:f kf;, bb37xlder have the same radius 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 enexgffsk7 b33 lj :;bet,rgy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mol q: jolhd;u o2t/xgj:8cw 838 r8fhmfst moving or rotating objects eventually slow down and st / rdgul8 o;:8m88 jjc3hl hoqx2:ffwtop. Explain.

If there were a great migration of people toward the Earth3w:w g2,z:vsao r9t*p kxoa9p ’s equator, how would th9saw r9kv :,t*ozg23wa :xpp ois affect the length of the day?

Can the diver of Fig. 8–29 do a somersault wktw,- ehl o hh: xwy0(gp1m**without having any initial rotation wh yk 0-,x :hlw1h*h*p(mgwtwoeen she leaves the board?

The moment of inertia of a rotati ut+uj1+4-ret ksy0e 81lllkn ng solid disk about an axis through its center of mass jk0l 1l et 1nu84rlu+-set +ykis $\frac{1}{2}WR^2$ (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 ohtt,a3/ 1l n/rdl6wlmass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? E l/ h/lt,ndtoa wr1l63xplain.

Two spheres look identical and have the same mass. However, one is hollow anx 5t4dgu:)wd qd the other is solid. Describe an experiment to determine which i x)tud: gdqw54s which.

The angular velocity os3i1y4t82tv+uyzrco) qtqhe moa w)s -.o7s9f a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on thoqsmwryh8 2qusv9a3 y7-+)c.ze1ot)t 4os ite 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 on1q:g t8m 5mhxhu2v rg.6br.nu the edge of a large freely rotating turntable. What happens if you walk toward the centerrqbvm5hh268n g g 1 rux.:t.mu?

A shortstop may leap into the air to catc*wn: 9avntk-u h a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you witu*a-9 :nv wnkll notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helickj 82o4 e9hnpsopter must have more than one rotor (or propeller). k h8s29jo4e npDiscuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesv.qnnx +:kg ojx70v) t 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 because of an amazing coincidence. Calculate, using ntg 5:3ng:rm r;fke,x the information inside the Front Cover, the angular diameters (in radians) of the Sun and then ert;xfn 3rk :g5,:mg Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moo fg1xqr5ew 62zn, 380,000 km from Earth. The beam diverges at an angle zqxwg1 6e52fr$\theta$ (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 motor is turned ,hja7*n+ u id0 39u1bodgixbt off during operation, the blades slow to rest in 3.0 s. What is t bdu 3a*hjoi+tud0bi xgn9,1 7he angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m tk)1s8amhh rx)wb 70nro another child. If the ball makes 15.0 revolutions, what is its diameter?s0)h) rx7kwrba nh1m8    m

  A bicycle with tires 68 cxd; 4sz an9.eenn0g k1m in diameter travels 8.0 km. How many revolutions do the whee 1skne0 egdnz49x;a .nls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Ca3zj u l3.)mtxplculate its angular velp .3)x3utljm zocity 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 complete revolution in 4.0 s (y7lkl hi8x 3q5Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cenyxq l7h3 i8l5kter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of wl m; , *vvmlgh0:z+prthe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speewp5w 70 b*g8sw1l loiwd 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 centrifugeuc7 9du9m5l:e1dlfv .3 gzzfb rotate if a particle 7.0 cm from the axis of rotationf g dzmfb.3c 7euuz:v5l19dl9 is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm ti5t/hp t amerdsb/.5)o 280 rpm in 4.0 s. Determinem)est5prt bhi5 ./da/
(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 +m*u9gu u,oz//thfzb $R_1$ 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, astronauo16,j :hvgg- kfmqi1btu6 x6yts aboard the Apollo spacecraft put themselves into a slo 6 b11 v yokux,ifg6qmjg-6:thw 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. 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 s. Thrk n6 q.;v kflls(;m8ruzcx30mough how many revolutions did it tumx;kl ( .8fz0kqvmcunr;3s6 lrn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 120o5vvzo3 xd lfs wt88i /clm6*,0 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 flyinkv*ajy17 6eu0 7i7 lzkjwhjg( g highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its 7j 7liz61j7ay0vk*( eghjuwk 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 frx+cm q/p 9/i b -bbu0hpd.rf5tom 240 rpm to 360 rpm in 6.5 s. Hou5 pp+/m hbbtf i9c-bdr./ 0xqw far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned )+aomig2.2d)nfp c*t r6tlg r off when it is running at 850rev/min It turns 1500 revolutimcg6 *.t)p)nrl rad22f t+g ioons before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolmrh20e0o ++e ,7(fdz o mhjftrutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0dr zm+eh,(efr20fojh t 07+o m.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 aszd wr0f(-2 s arg; c+3cmr)rwt the car reduces its speed uniformly from 95km/h to 45km/rrztws mac)cwrfr ; 2g 30+d(-h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each peda 0 (gw(tqnjz8sl when climbing a hill. The pedals wngq(0( 8sjztrotate 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 ok r6v mmo,q*lec)588j/gvn luf a door 74 cm wide. What is the magnitude of the torque if the force is exertl 8uejm)vo*g n m/6qr c,85kvled
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–s* -k99y+(y fns cgyzu39. Assume that a friction tyuynsy*f9 g sc(zk+-9 orque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached tox +lq9k:pp2glvm2t4s h po;1 x 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 msq2 pl1klo xvp:+h24; xtp9mgagnitude and direction of the net torque on this system.

  The bolts on the cylinder head9xfkj q +4ff5c of an engine require tightening to a torque of 38 9cxffk5j4 f+ q$m \cdot N$ 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 radiuxxt5*cc: q+q0e nz/ k)g-agnuf w)ml6 s 0.648 m when the axi)-cwmz*0:/lnu5tgan6) qxcfe+ kq gxs of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertii5kuok 0*z:zhd :u jw-vb+em+a+2u uqa of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kk i++eq:u0*-ouuhm jkv wb2d 5azuz +:g. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on toph.g(x 0l9j shqj4n+ he end of a thin, light rod is rotated in a horizontal circle of radius 1.+ osxngj 04qlj9p(. hh2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant anlw34o 7 ngd,q:nu61wt4 dxndx gular speed (Fig. 8–4u7w43 ld n,6nx4ngoddtw x q:12). The friction 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 tzme*zz(o fh5bs v m :.uiyz+**he array of point objects shown in Fig. 8–43 abs+(. my5:e*o*mzbu*zzfh iz v out
(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 h-7q7t;,qp c nx w*aggvbv z3;tw5 z)9db1pyeatoms 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 cipzr w3lb - v*kc+cvjz s9qwx3t0:m*2ylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellit *p0+lc-zc9 zw 2 3qi:bmtrv*3kv jxswe 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 a uniform cylinder with a radius of 8.50 0 zv+i.0.;r*h e1jmligsdbdycm and a mass of 0.580 kg. Cz1 g*yjmiibsdhdv 0lr;+.e 0.alculate
(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 ba8f3uzus2 3b dvnwo 8up1 fn3e*t, 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-driven merry-go-rouq*ao2s+fe y)l6a) mrqnd and is able to accelerate it from rest to a frequency of 15 rpm smo aye)+qqa2l rf *6)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 is evenkl0fe7uq7 9cii pq29 ntually brought uniformly to rest by f7li920icp 9nq7euk q 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 qw5xrtb o /g13m h4wk9vcy5ag/jdfq+ +bkv90accelerates 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 action of the for u0or q6)u,hydearm, which rotate,u ) 6dh0uqoyrs about the elbow 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 lx *d ley):.en3tpy; nlong thin rod, as shown in Fig. 8–46yyn)ll*;t:ex3n epd . .
(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 masses6 6 mhl.1ffxtg, $m_1$ 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 accelerat*llo)qtlg9/n-d 5vd w es the hammer from rest within four full turns (revolutions) dlv/nd9) *tw5g ql lo-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 inertia h6fk )qmwpm ioq. 4jd+3vq 32jof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a t q2jfbma he;-9r3 p+zvorque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass cq1sxgdz o 739jm2wl* 7.3 kg and radius 9.0 cm rolls without slipping down a laq3*m21w7c odx jglsz9 ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with rublowu *mmz v7i/ (1,z.r*3yxxi 9esaespect to the Sun as the sum of two taum(ze rxs3.,**l 9/u 1woy 7zivixbmerms,
(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 m) b/, /i/l+rvx bxpkmjass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinrl/ix +vxp)b ,mbj //kder.    J

  A sphere of radius 20.0 cm and *bs nui/m3i: vzky35omass 1.80 kg starts from rest and rolls without slippio ny3z:k5iu i/*sbm 3vng down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced verts8 .a.tqim*c fically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before .ta. q ifmsc*8it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momen 4m0abp3ba7 gn0b7yadm + q2mftum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed m+ g0 f23pn77qmd40ambba ybaof 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2)x: yqg6 pfpk7r5+de v.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatdx: ypefr+) 5q67kvgping 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 ay z wx0;r(hls5 platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal posi yrw ;xh50l(zstion, 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.7ga6 -o bdxzhm re4;s her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position -s;d4beomr6 xzgh7.a. 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 in u5uref3 *1ajk2erit- itial rate of 1.0 rev every 2.0 s to a final rate ot1je5e rk- 3rfui2*au f 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical w6oa5 - w,+y(7e kmcn x5zctksaxis 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, approximately shaped as a flat disk of radius 8.0 6k7sw+ te(cko xwa5,- m5c ynzcm, 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 spinnin99ydr ln: *apwg 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 momentu49jw/nhc fe 79 8vhiyfm 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 cylindril ja:b+d5qhonyt u-mja(4 3vr5c /ly:cal disk of moment of inertia I is dropped onto an identical disk rotating at angulaabaom (q4c-l:j vn yl3d:+t5 u /r5jyhr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.2iey)t 4z;6k c tcabzdxo6+j4 6 no/qo4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rd64v w(w) )t(pfgofnd/o/i i0okmp r8 otating merry-go-round platform of radius 3.0 m and moment of inor/8( n4ifd 6gwokpm v(0/ dw)fot )piertia 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 frx kin6sxg(e8kw3j41ioo khiu) 4-s c,eely with an angular velocity of 0.)6gk(x xjowki1 s ku i-8 one3h4sc,i48 $rad/s$ 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 aboutdx2q +4 ifnsqat wu4jcg24 n vn;*u3,d half its mass in the process, and windi,4nfjg4u sx3 nv2qq2d;* cd+t a4inu wng up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest 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 vwl+j/gwc eq0q 8 xs-5120 $km/h$ 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 +kgiy/f lp 7x5cip*+9uqq y+r$ 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, thatqa3a+ysgdrehf(ohaz 84*5c3 can rotate freely without d qa y*+er h33aa8 ohcs5(zg4ffriction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-v+kq3 bp n/g zi8wu3a*m diameter merry-go-round turntable that is mounted onbqkzn w 3g +83*i/avpu 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 rot1 y, dz3ndu.xhthg-:7iu7o ills 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 and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwindogd ,3h.d7tt:ihn1 iuuz7y- xs from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Ecipgjyhc u ty8+6+ +r:arth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particl y+crg+iu6+ch jytp 8:e orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rqf05c p x yj98 qflir05x:e.cuest 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 b42d6 :rqc b0xlw6 4cau cabv;cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the t;644bc wv c:bc2ax0aqdb6ur lorque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mai)d i (gs5;avdss 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-la5 d(g)d;v isiong 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 sp 4nesql7,u:lhiz z62py. t1t/ /qmv bkeed 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, /ny.y )ypes x4*qz6us but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were usyx .yy*p)szn4e 6q /to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use e)w 8 .5ktpgnjw)dt7u3h sdmv7nergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrupsdj5)v377dn h 8g mk.)wtwtical 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 illustrathb-d-;m;n(4g poz vcf oap*2mn : ik(bes 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 spee j:2frpidf 4( o-0wjhw7k9devd 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 l5 :mio)c9-v sf6fdb tfength L can pivot freely (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 tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. 6f :cfodb m95- )vsfit[Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertic:kos 5j +2vsu+rq* gbdally on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where hsus *j r5g+:2b +qkovd

  A bicyclist traveling with speed v=4.2m/s on a flat road is maka;nus*w6pdty3 7ys *d ume37wing a turn with a radius The forces acting on the cycli ty 3dsm7ns7e 6uyp3wwu*;d*a st 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 beginsx2 ,//,dd jlk 0oa4el y3mwvov 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, and where does the tor wvk0x/,dl2/v mja ledo3y4 ,oque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and7ck3lqpca03p p5d ( so 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,k(plpqc o ad35c7p 03s and with arms held tightly against her body.   

  You are designing a clutch asserd7v;xlpjo:(ph.gv . mbly which consists of two cylindrical plates, of mass ;l:pvpo7(hj.d v.grx $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of F*4n03fmmohwf a3m+5mjc(w r *;wivty ig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leavi+m (0fh wni*mw5oyrct*;f3mmv4w3a j ng the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but doy ; 68skmuqyi t hlk)vt.0p2g( not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly +zviwzkk1 w+ ,f2.,mmft 5sg-amt es1 froafkt,5me-w2 +gzw.m vti z+ss11 m,kfm 90km/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