A bicycle odometer (which measures distance trave)ecunkt(, jv*led) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-i*ucvn(j,k )etnch wheels?

Suppose a disk rotates at con 4a7yasmwax ghwt 7ftk d up22mr45(gp88u4/gstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular veloc xd m788ytg42a agsum7 uwgrfa(/4kw p4 tp5h2ity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a /5;c ano+q.glej) p6cox4 zn4spl,yu single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a great vkmqcv,ey6v )v )z4f1a*su:m od.l:g er 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 behind9 inn2.28 zvz;eogffa your head than when your arms are stretched out in front of you? A gnez9;a f2vifno.z8 2 diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three atxx cgv v i4:l6sup81b. 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 small front sprocket or a large front spxs.l 1g4vip6vc b x:8urocket? Why?

Mammals that depend on being able to ;, +ri*xk-7z a zgwy*,oxl rnmrun fast have slender lower legs with flesh and muscle concentrated high, clos,yl xg*x; zna+mw r,kzr7i* -oe 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)9x6ipr1hh*lrt m u*jbv0d, h. carry a long, narrow beam?

If the net force on a system is zero, is the net ygx457)hmynu saj7i+xua h:.eeds 8 4torque also zero? If the net torque me4uus+ e5a hd sx8hgina.x4: y 7j)y7on a system is zero, is the net force zero?

Two inclines have the same height butssp bj70;weij +h37tu make different angles with the horizontal. The same si t77e0 uhpwjs+3js ;bteel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start,bjsar5 s fw-8j aar:. rolling (from rest) down an incline. One sphere has twice the rada s j,: a-bjrasrf8w5.ius and twice the mass of 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 yd.pzo3 +(5)ln (iqd8bqarlt have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom fanoq (dl3qbdp )i(+ly z.rt 58irst? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum,l kr)o a sy0q9*ru ah58:s sani6m7cs are conserved. Yet most moving or rotating objects eventually slow down and , 6nl*5:samrs0 h7asu9oyik r qacs)8 stop. Explain.

If there were a great migration of people toward the 5agfqn4 y1 ,sekk. b(kfc-t6 lEarth’s equator, how would this affecty(bs - 5gc1,e.qfktlkn4 k6fa the length of the day?

Can the diver of Fig. 8–29 do a somersault without dzf2u 0jozis 54fmw4,having any initial rotation when she leaves the boa5 jz zu44fo0,wfd2msird?

The moment of inertia of a rotating solid disk a/ag:qml6ivd i 5r/3gtbout an axis through its center of mass t/5a:/3 vrm idqg6gil is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotatinaw g m:/pf0gwcx2i(p+g 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 samew0mc2(xg:wipgp f+a/ ? Explain.

Two spheres look identical and have the same mass. However, one is hollowymdzn 4ijv)tk+; al n f/:zw2a9pn1)w and the/mnz l 4)nayw;jzw2:v1f+ ip)at9n kd other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In b*kx0 e406jg5zuhv ozh .ec,vwhat 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 tc 0,v6zzh*eb .ejvx5g4k0uh ohis point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on thnt::3b7iob ) 8 kusiibsxcj. 2i;5vvie edge of a large freely rotating turntable. What happen3ns7i5bu bc;vtki:2sb8 o vj. i ):iixs if you walk toward the center?

A shortstop may leap into the air to catch a ball and thr8h 4a skbd:)l kre/ pnzj7+s:on;*rrx l2lvl3ow it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (F)+ h4:ll/rl7bss ;xkj38kn a*lzpd r rveon:2ig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a k(phv fe ;j8kn451dq+ d4cpwqhelicohn 5(d81 q;4eqdpvw fkjc 4p+kpter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesg6wv*z b.svib*,s*t u 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 ws(u-:tsc.87 j+x ern +t zqbgbecause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diamte7rtwjn-b s+ g+(c:.z u8 sxqeters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from ;zaq98 ze -b8wsu x5mooj5qx6Earth. The beam diverges at an angle a8bsxuxzw-zq9 q e8o; j5 mo65$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motu 1ykw,n 4pvt7or is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow do,kpv wy41 nu7twn?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes9cn-tnj+/eveb4g pn* 15*c/-bpe9v +g tn4nn ej.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How mz14u4d 3wet muany revolutions do the wheels mak341 zdt uwum4ee?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rp .07a5 lbzde (x1bkpbqm. Calculate its angular veloc1zkb p5edl.ba7q b(x0ity 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 comp.r 4 jx(tyrv6yif9m -rlete revolution in 4.0 s (Fig. 8–38). (a) What is the linvi 4 yt- xr(r6y9jf.mrear speed 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 velocity of the Earth (a) in itsk-j (rk(p l+ 1c7jr6bcx 3mvwa orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poiftt *1)vd0 ibbnt
(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 partic*5q:0klnxa p lmz6 ixqc(o1g5le 7.0 cm from the axis of rotation k0ix1nz q5 :(6 c5 lg*pqmaxlois to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ce3,e /rscj0apg3mbqj a) yg qc.i 58uv*nter from 130 rpm to 280 rpm in 4.0 p),*3 acr i0/u5j jg.3bgce8 mqy svqas. 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 -r6iu vxmc+9(3f4qae86v m5 vnrunat$R_1$ 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, astronauthk jfwz i:n ei:(dt1td )aan7p:i;a5:s 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 revolutiotn( i5:)adae;:fhz pt dni17ak : j:iwn 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. Throuw44 i + dnz llb*,xdxv2vzwi(.gh how many revolutions did it turn.2,wni4+ xb (*lvid4dvxzwlz in this time?    $rev$

  An automobile engine slows down from 4500 rpm to unu- q,mdsyj10y6we2tc ,4 (vlqex8p1200 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 f lxnby-:vrr(7c uf/7jor the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.bxvl /7u r(fyc-7jn: r0 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 e6 .ks v 4p:ue36tvjwps. How far will a point on the edge of the whj ve66 tkvepu4.ws: p3eel have traveled in this time?    m

  A cooling fan is turned off whe.rbu ik*nq*zu9,)s zrn it is running at 850rev/min It turns 1500 revolutions before it comerr.squu i)zb *n9 *z,ks 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 calkb .1nox3j:yv)i*y ar reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8aj.nblv*yi)1x o3: ky0 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 makeb i)rbc eftutg469-p)*w3ypm 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The titug)b*m6tcfb9i) ppe34yrw -res 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 pucje;4w :o( +zf3lssits.i n/zts all her weight on each pedal when climbing a hill. The pedals rotate in a circlen .j c( 4z+t:s3wlz/i;so sife of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 c cca6tkfom52v5 w--cp m wide. What is the magnitude of the torque if the force is exerted cof6kc-v5m52a w-pct
(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. 8–39.3oty bzxz67b h 2:c3af Assume th 7ofyhzzxb6ta3 23 :cbat a friction 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 the endmvbp2/yo pe6:s of a massless rod which pivots as shown in Fig. 8–40. Initially 6v2pbo:e p/my the rod is held in the 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 rez1c2tg r+yo v+quire tightening to a torque of 38 r + ov2cgy+1zt$m \cdot N$ 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 +u 0qos6gv,cg 6 bu:fpz5i l0gof inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through+ob 6vug,z5l fcsgup :gq0i60 its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a d0z( jnrt)*; jc :rhcb8pxee8bicycle wheel 66.7 cm in diameter. The rim and tire hav;h jrpx(jen8c0r zdcb8) :* ete 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 4pr3 zy339haogcey-z end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculat9-yy3c4g3 hrezp ao 3ze
(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 r i-h(2kxy8ej hotating at constant angular speed (Fig. 8–42)-y kx82e(j hhi. 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 the array of point objects shown in Fig.0bf8 b/ 4ztwf3gb 83fv usluv; 8–43 abs3038u;zbl4 utfw bgf/f bv8 vout
(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 oxyg:p:wle6 rfz p9en 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 co gpi 91- ,g*7grbqpn1zx k0jvprrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite h g-k jpbvg7n*piz1g 0qpx9,1ras 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 himg5)b,sk 6d6-g l:scym i.mmass of 0.580 kg. Calcdik m lyghsm.:6igc ,-)6mbs5ulate
(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 frsadg q+c)c3 c 8 8x3)zjr;vgubom 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 h; aif 8viibs58ljq 5*pand-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 oppavb588; iflijiqsp* 5 osite 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,t fk91/w ( kpyzp7/ 16-vlzr2ayzq8oo9iavs s300 rpm is shut off and is eventually brought uniformly to rest by a frictional 7 y2wppzfza/y8s9 11i-lo6k9z k av/(qvtr os 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. 50xr9l,p:nytfbbp5b 8–45 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 forearm, whic6 vclr3tfe9s(ibr e(e ;u 1,dq.jn9 pxh rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fr399f;jbs envdceur(ie1,t. 6r q(plx om rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8)t69 k +sz0e 2,eg,pw izx pu/7- ;kupajcbpzf–46.fz ptsaw7 ig/,j;,kz)b 9z2upp-u + 0cpkxe6e
(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 consistdf/0oafef (r (x.6ng bs of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates t4eov9 *7tkv9mv n g)trhe hammer from rest within four full turns (revolutions) and releases it at a speed of n9vr*7e tgm4 vk)o9vt 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 mneuw;;md9s(i6 s 9hzyoment of inertia 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 280mi t+gca;8jsgp( fh1* $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg an.x 5*e0hmb -u97xlsi/x,r /zee ayvoa rf3l1hd radius 9.0 cm rolls without slipping down a lane rxh 0xe a/z7v,-obhleexr. u3my l*/19is af5at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of tpb4u 2*oipx c(he Earth with respect to the Sun as the sum of two terms, o4*2p(ubip cx
(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 7.50 m. How mu i;(u0rlbo9lub,7kd m .m:,uh2 khsf och net work is required to accemflhk so,bi( u.uud9;bk h2 07r:lmo, lerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mas1jtm;,uwoz6r i 4a4y hs 1.80 kg starts from rest and rolls without slipping down a o1,wrmaiuz 4yt ;6jh4 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tia/d*48pii 9k9 frqhzd p. 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 rf kp9ii84d*qah d 9z/the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of aj7+fo:wab bl 7j yh)/u 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at ao:yl 7jabf)+ h/jbw u7n angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum o g0.ws-xi4 .w as5fffdko /9sxm:je p.f a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatin s:wf5f/j f xg.isexdam.k 4. so0p-9wg 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 thax(q.0)d r8 s*;czc.u ektw +b nq)tfprt is rotating at a rate of 1.3rev/s If he raicnpe *r fxq .tc+k.rqbw0; tszd8)u()ses 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 re+bkqxym+cf jv*.3z0 tduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If shev y03.bz+fx+qt*mck j 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 ra;l yc t.5+t e3yq1dqpnte from an initial rate of 1.0 rev ever;qp.tyt1 qe 5+dnl3cyy 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throug v4h+b,mi +: nek-xbfdh 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 k ef+dhbb4vin,+-x : mradius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at(bs. ua(a sy2t 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 Earth6q tp-;id**o yz6b +olxuo1ji
(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 ofgsv;ia setvw l; 3*lkf3:d7n 2 moment of inertia I is dropped onto an identical disk rotati3tgf2 vl3*s;;easvn:l iw7dkng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at n5dhqvv1 x.x-ajq698(c1egpr hqv7w2.4 $rev/s$ 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 28 :9kfncf8dr pv)sidthe center of a rotating merry-go-round platform of radius 3.0 m and moment ):s8ik pcd ndf9vrf2 8of 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 angular velocitytc7pztyf6 hl m 7;fj*1 of 07*flpj6ct;t zm 1h7 yf.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half itytun)-6 5iivt s mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angvti t-n6 uy)i5ular 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 involvej72ju me-x v82hk)oau winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass pe myoqixl+4:1d50- pos s6xv$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotat/w5f. ea uet)he freely without friction. The moment ou) htea/wf5e .f 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 gm)pk79tv;joki j2,g a 6.5-m diameter merry-go-round turntable that is mounted o;2kmk) t, j7gg9 poivjn frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the groun9z ng(af5f+.1 bhm*ezzq hzu- d 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 q5z h*unfe( az-h zb.1f9g+zm of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side aft ,.9fd2c(rjaruan ,xgn 62 elways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about itc,9 ffn,du.(a 6jgnrxar2t e2 s own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest-)**fh9ma2u 3kgerqlhi/ ce y 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 acquiree4wwf h:+m e5t3d*ktqs a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the5 4d:km+wthe*3ewqtf motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and .vkmm* dwhs ex sb0-4z ,a3)ndb9ab. xdiameter 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-lonh db,.9.me)*3wbzxdk s 4nv- msx0aa bg 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 z2 hcbuo)n -a4p:r )ay666c;ukubrfzthe rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotasrlfu3;if qm7 b 6j59mting 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 uniform spf63fj rm9qu;s 5bmli7here 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 enph*xf4y0 enc)/ t)i jpergy 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 abch)ppf0* it e)x4yjn/le 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 an grkungkdv(+7n 76tt1us:4 rdj- sye3$H_2O$ 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 sphkwa /rs 2fk-(eed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can piv(hh j:0 sh-btmot 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 fo hb0t(sh-h j:mrce of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floo1st ngua91fbmbbw2 ny/85n3ipb 1;m qr, 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 stebyfn2q8ts3n1 9unb b gap/m b51miw;1p has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wick 59jr1wyj be8a -,0tnk k*vz*pcgio s8l-j (th a radius The forces actin(bc1 9zk j*jysgipk0a8v l8cet -o5-*j kw,n rg on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sliakau qoa2d-1hmo: ar7 . uwc9+ng of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat7+wd1q9ukr :-ao2.cmohaaa u , and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylindkockh t9 (m 89(zhfte(er and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretcho89(ft hztm(9k(chek ed arms, and with arms held tightly against her body.   

  You are designing a clutch assembly w ty83 l2mxx4ebj 8(diwhich consists of two cylindrical plates, oftiw4m(jx 83ble xyd82 mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 njmg 5radp/,9 :usdm,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 leanp ,/a9:,5rmgu dmsdjving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumkeb7v t4r.b3 ke $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its 86icn f u9td5bspeed uniformly from 90km/h to 60 t8b6dcnf 5iu9km/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