A bicycle odometer (which measures dista(rsb)q (tc)w/g d olr9r.g*vonce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you uset*ogs(w 9q r)) /o d(grbrc.lv it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocit 9.gh*7r*lfj .rhot tcy. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point9fhl.* t r h.rt7ocg*j 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 ofpmolt(le af,9 ,- eso0 2irf,e the angular velocity $\omega$ Explain.

Can a small force ever exert a gys79 dch 1oas5reater 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)r ;cm+wqllv 3a5mtgq o*fda*uu2 ;cswf.a4; your head than when your arms are stretched out in frc) m 2fgav* .ao*wdr;q; c+uu 3sl5law tq4mf;ont of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and th2eex s m)2tdv,ree at the pedal cranks. In which gear is it harder to ped m, et2vse2d)xal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on beingchqg( t ;.ykt(y .7ncm 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 hytyn7cc(m.gqk( t; ., explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) cnefxwd;z sbxes -7f .z*l751;qub rt )arry a long, narrow beam?

If the net force on a system is zero, is the net torque alukdpl0g )yn::qck( 17 b g0bihso zero? If the net torque on a system is zero, is the net forc(0b:q cugknpiy b:7l d0gh1k)e zero?

Two inclines have the same height but make different angles with the horiza8jcw fh3w ex44d6h )jontal. The same steel ball is rolled down each incline. On whi4x)ehw6jh 8jd3 caw4fch incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling 1 a4i rd6twr7r(from rest) down an incline. One sphere has twice the radius and twice the mass or ai4tw7r61rdf the 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 cylinder have the same radius and 7i/wuzw:m w *wthe 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 hasuw:iwzw7 /wm* the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum anr- lkzpz m)hd7wr9 9s.d angular momentum are conserved. Yet most moving or rotating objects ev pszkr hmz- 799wrl).dentually slow down and stop. Explain.

If there were a great miucta /1a- mnr+gration of people toward the Earth’s equator, how would this affect the length of the t/- cu 1amr+naday?

Can the diver of Fig. 8–29 do a somersault witho *v;scte:r8 m2 4pltybut having any initial rotation when she leaves the board?4lpbmt 28:yt ;e vsr*c

The moment of inertia of a rotating solid ;(8sj0- s izeb7znq *7lkq pmidisk about an axis through its center of mass ;e7qz -qzb( lk7ii0snp8 *jsmis $\frac{1}{2}WR^2$ (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 hold 7hjbxea+;p( ming a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease7pxa(hjb+me ; , or stay the same? Explain.

Two spheres look identical and have the same mass. However, on7eyn4 0oi7r(una3+tq e kk4tje is hollow and th 7kurt07ino4 (nke+3 4j eaqtye other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatix/2tdn(dhuw4 k+ spux1i5 50n)yoqwj t0 8lygng on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular /yx0kw tsquh5y 1un4wljd0 gpt ixd25n) o8+(speed increasing or decreasing?

Suppose you are standing on the edgvhd+ cndm-deaqh2j2:iv 4tf-m -;8xv e of a large freely rotating turntable. What happens +m2vj-ce4qd mh ; xnv82i-d v-dtfh:aif you walk toward the center?

A shortstop may leap into the air to catch a ball and thro bw2 g+a t2)pwjert550qvi ld;w it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will j;0) 2rp qwtbd5gve2tliaw5+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 momeia02a( hjza2h ntum, discuss why a helicopter must have more than one rotor (or z0a i(ahja2h 2propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rag id(lni0y5 * j37epqvdians: (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 th5:usadq)y8 hj)u ghems:/rinb 6t-l6e informatis ben8:uqus)ty)/rmh-agl d6i 6:5hjon inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 ki*p qbmx)*yk .m from Earth. The beam diverges at an ang*q* pk).ib mxyle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender roe.ccg e dv+k:8zg1v4j tate 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 angular acceleration as the blades slo1z.ecgkgd8:v4 j+ evcw down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m tow(xx)rxbz4ltuygm a8 +:,7 wh another child. If the ball makes 15.0 revolutiox,b)xua+ml h( 48t:gwrw zy7xns, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolu54;7cxvu+a qo 6 j7ithumprh(tions do the wheels make?7pm ia r4q;u6hv hc(to7ux +5j    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates axsobl mh48;9:3kf xs lp- +v8rhw wwv+t 2500 rpm. Calculate its angular velocitxwr+v-k3h s:lfp sv9 h;m8 owb48+lxw y 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-ro xko+j6: xe0uhund makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed6u0kj +xh:eox of a child seated 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 v e) x f/ea*qosqyb6*a2elocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pj;vjcq nm+ay9r /:yh1xpr syg )gf+4*oint
(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 centrifu t1v:2p ub3n*ruymck8 ae+z4 cge rotate if a particle 7.0 cm from the axis of rotation is to exper v1mc ckybu er *a32tz+un8p4:ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 r2ruphqmnet(r+b9 f*0 pm to 280 rpm in 4.0 s.bf *mh 0e(qpr92turn+ Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius ,qjde1 0y o /jn5uwlj/i/pbw4$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft pu 8nqdw j*x8wobbq*5h,nrocj: *(l h(lt themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no n5wx*h:b dqorbjlh j8o ,*q 8 ((lnc*wrotation 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 un) kd;+saeaw r8 2w,w2b1ra ybkiformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? ;aabsedr )w 8y+2,2wk wbka1 r   $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5) hb8)eenx,mt 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 flying highspeed jets in a wpl1b 6/s ywd)yhirling “human centrifuge,” which takes 1.0 6sl1y /wdbyp) min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s,+ e7fgvzl0w2kvff *r. How far will a point on the edge of the wheel haveff2ez v+0,g7fwkl *vr traveled in this time?    m

  A cooling fan is turned off when it is 8keys k6ti l8gta6 e cn/+t3d4running at 850rev/min It turns 1500 revolutions bef6el n8/4tdt3skea8 yk+t6c g iore 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 65cm do700 mvf3 w1m0dbv revolutions as the car reduces its speed uniformly from 95km/h b70d0mw1mcdo3v 0 mfvto 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 speed un ;-7qz3eburb ciformly from 95km/h to 45km/h The tirebrbz-c7 u; qe3s 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 pedal when climbing a he.fzl-c,v)y:si +unz 8q t/rf ill. The p:tn.f ys cu+ l-q)f,zie8r/ zvedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end.5(t ( ycd1d/a wngxco of a door 74 cm wide. What is the magnitude of the torque if the force is (go5x/aw . 1td y(cdcnexerted
(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 s .gq:,ru4uo3g-ye 9clxv p4 tfnwod0,hown in Fig. 8–39. Assume that a fx u u4fp:d noco0,r3 v9 tl4eg,.yqg-wriction torque 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 thphngbk0s4 *s-) xboj4jh3v,o ur.e 7le ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the4hk.4r bn*j7l,x)hej p0 ogvbo 3ss-u horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightea:pwzklz: mrcq21,b ,ning to a torque of 38w2qrm,z1pc::a,zl b k $m \cdot N$ 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 10.8ykkr t c9cg .er*,a,4n+ jh/mh-kg sphere of radius 0.648 m when the axnhhe4ym c* jr/gkrkc+.9, t,a is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of qhh,)2pxi0 s t fa9nlj+k9vdz5 /jvl. inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire havh+vjvf/ nzd)sqx,a.52l099i tljpkh e 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 end of a thin, light rod is rotated i8jl+: kvquyw*n a horizontal circle of radiulqv8 jk:*y+w us 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angujp*d4hca.4ga x yl,k *y: f9tmlar speed (Fig. 8–42). The friction force between her hands and the cllf pagx4* k,jt4d*chym:9.ayay 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 faou92 a8 vitce235wjdc /d,sthe array of point objects shown in Fig. 8–43 about t23o c58cu d2j sfvi,ed9aa/w
(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 ox w y+i2 za lz/nf:z*0lp*evb1qygen 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 spinning at the correct rate, engi/t1 rv hlksi*;/l1ycnneers fire four tanglk*ivt / c ;sr1h/ln1yential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the 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 cm and a bhldas3) 5r 1xefn i6c;- qcj,mass of 0.580 kg. Calculateces5)c,6lx n -r3b;1j da hfqi
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fromnyrt6 o)v si6*az 6nm8 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 tangentiv jwm ,nm sn:d1((h6boz,re v9ally on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque.nsev(b:h o (zm,j ,rmv1ndw69 $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventwh(ko mnzxa (ya;,9k-q+f. r( kmrj.lually brought uniformly to rest by a frictionakjr ,rx;k+w(.((. n qlmmk-fyoa9 hzal 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. 8–45wb mmpfw*)5;cpe. 5hd 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 action of the forearmqzjx)c);i xoi-5y 1 vc, which rotates about the elbow joi ;v j1i)czqx xc5yio)-nt 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 consid8 )jzh * 06qptefz rn7w*o;ltjered a long thin rod, as shown in Fig. 8–46.z7lw0p*qtf ;6hnoj*8t )jez r
(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 twoxe6 ,.r/ fvfyb 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 thrhw (bub:a /1mm1ht (fe hammer from rest within four full turns (revolutions) and releases it at a speed hw: m u(r/1(t abmh1fbof 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 o-dxdq *i n:a-ef $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque o1e6lkn,2asjkvafc2 0 f 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7. *m:b djf u,z6,yqqn,n3 kg and radius 9.0 cm rolls without slipping down a lane at ,,:z j*6,ymfbudqqnn3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun s u0 j94qp9iaf1v-iw vas the sum of two termsv9f0iw jui4 qv1p as9-,
(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 k7e6gp- slx;sf5s yp01640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotat-kl7 pp;gf0se6 5yssx ion rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm an :5iq;m,s rw vk9ktqr,d mass 1.80 kg starts from rest and rolls without slipping down a 30i;kw qr5rsk9:tq,,v m.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 tipq/7 y*( tvauf bjq6m;z/4oozc. 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 it hits the ground? [Hint: Use conservation of energy.]obu4o jyq/a*zv t6;qf7/m c(z    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating +ssueh xaq)k8w7;ji 6 on the end of a thin string in a circle of radius 1.106;x+jeswu7k q )8sih a m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical un dm4p( cw6)d v8ggs:grinding wheel of radius 18 cm when rotating at 1500 rp:sv)(4g mw cun g6d8pdm?    $kg \cdot m^2$
(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 mwns4y 2ik2m .rate of 1 22ik .sy4mmnw.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia vuwc;3ck*3a;1+rz kolho , diby 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 in the tuck position, what iidw1ckho*+;3;k vroua c3 z,ls her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial r uspa4i*i4p-hate of 1.0 rev every 2.0 s to a*4 u -ihpp4sai final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a verti*tmm cx.fn )h42;qhrsf3n5vgcal axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto;3.) v 2h5m*4cffqxgnrmt nsh 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 s;vizbn2qsz962cpb 6+p2hz umomentum 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 angulac w,0eyu9q hui)8wkh6lz7q z 4r momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk tc/w5b(-sui lr)wo x9 of moment of inertia I is dropped onto an identical disk rotating at angular spi xwwr bs- o(9t/cul5)eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4y -*+wmljzonnreim65 4 b x75b j.:moy $rev/s$ 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 rvmez: ,6c(ocz2 b7txadb z+n*otating merry-go-round platform of radius 3.0 m and momet(bnv xd zz,:2ob+ezc* am7c 6nt 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 rotating freely with an angutejqeop1k2)-vjc0 8 f lar velocity of 0.8 8 1t-pjok2vf eceqj)0$rad/s$ 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 iljxcvzh4o f:d-xzbkp r) 5fw54a/v60ts mass in the process, and winding upzj4acx4l :bf50)hv 5 kd6o/xzwpf- rv 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;j 73otmnk5wrjor 2w7 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 vpaco;j i+7*v$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially azz+ b)l,/ j*llyg2pii.nn: jqt rest, that can rotate freely without friction. The moment of inertia of the person plus the platform iq/)j,: pz +.n nljz 2yb*iilgls $I_P$ 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.l hoa- u1e0w7j5-m diameter merry-go-round turntable that is mounted on frictionl-oj ea0w7l u1hess bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the ropxge- 4g8 qcrbh5(qx abg (wm:ud )m(y:e 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 slippigebrx (qd qa m-:4uxgg8cwb( h:()5ymng. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such t0n4l ty:j-b2 ;mesbodhat 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 particle orbiting the Eaesm2: b t0ob 4;nj-yldrth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/rs+fqjch 7(g/m0 -nf k$s^2$ 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 unx 3 ej+z0e te+e*,wfpxiform cylinder of radius 0.20 m acquires a rotational rate of from rest oveeee, xtx 03*jfpw ++zer 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 cylindrical disks,z q 5c8zqnj6qpnnl b)r76 9m;c each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches themncbl ;q8nr5cqz6q7) z6j9np end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular spee +-;zlo0twt:a2s8+u 9y,vfa) pxojbvvk6kw wd 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/l10 ,kx qi1l,t3iq 9pnklwqr 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uni 0p1x,i3q /l9w rlln ,1ktqqkiform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollun/) jhtuams3y jn36x8 tion automobile is for 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 diameter 1.50 m and mass 240 kg, and should be able to traj nh /t3smna) 3uxj6y8vel 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 hyb dr 8jl3*c+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 horugy,y6xl v wf:v/0) ngizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M pemi/: idu4 evyu(;z9fmz8m.and length 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 releasim(4:meyz vm/zeuif 9 u.d 8p;ed. 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. It is standing vertically ong.:sog5xvmfi 0h por w(p*:w2k, 6a sp the floor, and we want to exert a horizontal force F at its axle so that it will clim6 s:xmp:o * wpo(wfka hv2,g0rs.g p5ib a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m8 (3mgvmu8jnkeyvn fx*88 hx8: :ihvu /s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the nh 8uy vnj:m(8:*x h83 xg8vk8emvuinformal 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 andis9er p 5x(fo4yd 348et8mktl begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm afdeo4y3mtx 5fes(pt8lki98 r4ter 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 hadpouk, jvyxz vb3lzc./) 7 5njek 859er 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 with arms held tightly against ,c9jex3klo vj8ukvan.z7bp5 /y d z)5 her body.   

  You are designing a clutch assembly which consists of two cylindrical plates, of.dw9r 5 g8uqv.fvtvwk t;fc yr2, :9oy mav9..:5vy 8to,;cft g ukfrydr9wv w2qss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rolepe5if 3. gl6u82 h0o1wbahg wls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the tra e6e f.ab3hhg5i0lww8 1 uo2gpck? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumes8ns;k w((ey.ekc (cshw vp,2 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uya5jafp2 vj: u;-px .fniformly from 90km/h to 60km/h The tires have a diame. ujav:5 - aj;pffp2yxter 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