A bicycle odometer (which measures fp/lk7r9jd 8udistance traveled) is attached near the wheel jr7p8 dul9/k fhub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on t 9/.5 rdhh (teewpgb,rhe rim have radial and/or bg ph,(terre w/d h.95tangential acceleration? If the disk’s angular velocity 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 desi rm 2b7x 9l9ivzwbla,r;o53c4 psk-a cribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaig38h)8x -ddj9yx0 hfoetcm w8n.

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 behiy77527kk 0,sytp fvyg dy:nmsnd your head than when your arms are stretched out in front of you? A dian7 k5vy7 mpy0yk:ty, sdf7g2sgram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whevfjh 6qrv*ar* n-nm1:el 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 gear is it harder to pedaln*v61*qhm nrj a r:-fv, a small front sprocket or a large front sprocket? Why?

Mammals that depend ozv p3wo.xmu/k,-/v xw n 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, explain why this distribution w. muv -/xp,/zk3wvoxof mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bq2r:l37p sbp mly mc(5f t83gg 28aavweam?

If the net force on a system is zero, is the net torque also zerot gbygc n;f3/1? If the net torque on a system is zero, is the net force z tcf1;b3 /ggnyero?

Two inclines have the same height buci c94l*5ho8ka9b3pvky k yu )t make different angles with the horizontal. The s l3kby9)c45y9 u8cihk* vapk oame steel 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 rolp1 h n+.kpk(kaling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. a.knh 1pk(+pkWhich 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 tlh ;ykhn 6k,m.he 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 energy at the bo ky,.lknh 6m;httom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mosm*a9x ztxip-z9 5lyps2 pc,7tt moving or rotating objects eventually slow down and st p7lt5*ptsp9czx x29amz- ,iyop. Explain.

If there were a great migration of peopj)uavr y-u10jh5,ss u le toward the Earth’s equator, how would this affect the)uas ,1v-0js ruh5yju length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rota)qz 6ow uk/2nction when she leavkcoz6)nu2 w/ qes the board?

The moment of inertia of a rotarp x2nky ch-c7zp438iweg g)dm60*zc ting solid disk about an axis through its center of mass ggx0 4 dcc)r-*y28 3phczezik7n6wpm 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 sittingv7au j5m f.*lc on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the mass*7uvf lmac j.5es, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass.47dy 4h1 mift9xk yd9:qzmm+nfl9.u z However, one is hollow and the other is solid. nt+k: mfz f.h7 xq9zyi ymd1d949 ml4uDescribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a ho qzh4y8ld78 vech7 sv:rizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleratio8h4l 7s:e qcdh8vzyv7n points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely r5gx8* uc7e2wi ts3do im qfj8(otating turntable. What happens if you walk towardg 2ef qsjid8*x t7cw 3m58iu(o the center?

A shortstop may leap into the air to cat:o- f2t.wzccy b4ewoc6s(1n cch a ball and throw it quickly. As he throws the ball, the upper part of bc-4 wwt (cso: c.y 62en1ocfzhis body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation ocu+ lv mf/ys((f angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can ouc lm(s v(/+yfperate to keep the helicopter stable.

  Express the following w0 :-mky pjy3uangles 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, p*pxn0u7*ffz2nd8cg using the information inside the Front Cover, the angular diameters (in radians) of the Sunf cz*2pf*8pnxudn g70 and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges a x4vjv 5edn a*(5a/shav,w ky7+ oyg5at ans dhnv57yvy ex/ ovj+5(a*a 5a ga4,wk angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a ratef4wk)l/ sum0i of 6500 rpm. When the motor is turned off during operatlk4m)0fiu sw/ ion, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to90olitp ;kp a9f;ibxk20 /dro another child. If the ball makes ro i0la2k0p f99pxi;;dk b/ to15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How ma3kmv sr54a* oyl.5xlirp ( ms8ny revolutions do the wheels make? 8 5vk mras(.5 ir spm4xol*yl3   $rev$

  (a) A grinding wheel 0.3a0a(d6t ps4e*djeqa *5 m in diameter rotates at 2500 rpm. Calculate its angular velocid p(6 ajeeada*q t0*s4ty 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 (Fig. 8–38). ;7 zabuxzk0 m)(a) What is the lin)b0zkuaxm 7 z;ear 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) inm,+vh y96sm hp its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ,7hu21 2smxeyc05 g 6rnewz fqa 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 centrifuge rotate if a particle 7.0 csq47;jle a;air4d d7tm from the axis of rotatide l47;s4ta7rd aq;jion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centev8;vqt.j q 4n; b4-zq*vzh ctkr from 130 rpm to 280 rpm in 4.0 4;;k4zj8 vt q*.tvzbvqhq c-ns. 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 ec+f6or4 j *q adv3p2i$R_1$ 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, astronaune,c-yjb 4 rz:)jn+xits 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 nx ez,n4yj:+)c jinrb -o 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 151;lbd61y7q 6od iw ibw,000 rpm in 220 s. Through how many revolutions did it turn yldd;76bq wibwi6o 11 in this time?    $rev$

  An automobile engine slows down from 4500 rwfbo. p;t1 f2 2y bqd.3zkaxh;pm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested fo, 1/od2jwgz ww+r / qqnbp;yv(r the stresses of flying highspeed jets in a whirling “human centrifuge,” which tqw1wv(z+jy,/b 2nq/p;rd gwo akes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformlpl r07 :uillphiytx ;*)our;+y from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have ilr )t7pp;l* rxo+u iu :0lhy;traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Iacupxmt. 5( 6wt turns 1500 revolutions before x. mcw5p6u( tait 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 revolutionsh l 5 n.(22dbyc3vfwow as the car reduces its speed uniformly from 95km/h to 45km/h The tirelydwc vof.2 nb(w52 h3s 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 reducesq3.oe4 3 1 d6zxeodnkfeji/0ppxb-k) its speed uniformly from 95km/h to 45km/h The tires have b0ekxd)e1 ojpep-/ioz3 6 q4.nx 3fkda 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 e72d glw4ml .df8;oz7ix6lt hkach pedal when climbing a hill. The pedalsz7ltd8x. f2lw; 7i4 6ohl mkdg 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 of a door 748oc33+uzxb;cut1p5/9atzppz r , es l cm wide. What is the magnitude of the torque if the force is exertedz/oc et5p a+ulstu9,pzx zrb8c3;p3 1
(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 aj agwg+f37j y1xle of the wheel shown in Fig. 8–39. Assume agj+w7 jygf 13that 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 ends of a massless rod whic h my.+ahp/w*:no6i m2fxgsv 9h pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net :/vmp x*2+i h6wy9fon a s.hmgtorque on this system.

  The bolts on the cylinder heada0fave o5my5zq6v99 r of an engine require tightening to a torque of 38 marov6 zq50 5 yvf9e9a$m \cdot N$ 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 sphernqrn 0u2k+ui uo-1v. ye of radius 0.648 m when the axis oq+k2nu1nyuir-0vo . uf rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia y70ieet2zm , jof a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. Tje,i7yzmt 2e0he mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end ofbm3-- duocja+h(6pv7 f w 0cck a thin, light rod is rotated in a horizontal circle of radiuscm + 67j acopvh--(bw 3u0kcdf 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’sc2p l)/pv5ier wheel rotating at constant angular speed (Fig. 8–42). T)/ rlc pev5pi2he 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 object d9w r8yw:rw/4 vth2ihs shown in Fig. 8–43 aboi4:wv89w rhyt2rdwh / ut
(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 cons ;f,h zyr(x1cm 9ryrg*ists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spituzyeje a/9me;k6ix .ovh. 5uw/5tn3nning at the correct rate, engineers fire ft6 e /oy95xmj it.5wu3/z;ak.veuh neour tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each 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 d5 1vhsygzin w/h5/i- and a mass of 0.580 kg. Calculath/1i5nd 5hzwv /g ys-ie
(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 bzcgot:sp( o+;s n+, xqat, 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 ohl5n9ni-) lk4k- fxae(eg , jdn 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. Assun,if)lkjekaxe g-9n-(hl d4 5 me 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 i,ss xt0hracc53ra ,e 4w n90fk62kmivs shut off and is eventually brought uniformly to rest by a2scnm xk03,i4 r9ve s0a cw5kf,r t6ha 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 accelerates a 3.6-kg ball at 71ws4cz q 0-zo hvr/(bk $m/s^2$ 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 t eh9h-r;s i hny(m:l6fhrown solely by the action of the forearm, which rotates about n sl:f (9;h6himhr-yethe 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 long thin rod, as shown in o7)uu7 xj( ec8tai97 qh ylat/Fig. 8–467 )thljq/uox7(ceyta7 i 89au.
(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 cons8 4m su:wpu70bdrya9 l0p.fadists 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 the hammer from rkiy2c rwu* riqjuacsr . , h0zw-235:6cc;jaaest within four full turns (revolrw j a65cu., rh: zq0u3cica;* 2jaciykr 2-swutions) 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 inerq0m;z5;ycnu y0y s3 jhgk- w8vtia 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 a1ze b ewy1,b5fgd/ye/f6r 0ve torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kgawo x mfz06- ,ow;lvda,5qu u9iok( )c and radius 9.0 cm rolls without slipping down a lanodkz ,x o9(ou a;afqvl)5u cimw-06,w e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Smpbx k0)f pz:rx-vc49 uwl,.pun as the sum of twk-fvrl xpb.m9 4up: w)z,0 pxco terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg an( u(rvpjmjz4s8m, ny0 d a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotauy(j8nrz,m v0j( m4 sption rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and s7l1hr s6pojft2d, 222*j5pwtn u p y:kohm;e rolls without slipping dowm no226 d r*w2;jyj1fo5h:7 lpeupkp ss,t2htn 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 vet3z s he:y/q1ertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of hq1y t:s 3/ezethe pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin s::u*ba rcriqdh;3 1v btring in a circle; *1 v:bdaquhcbi3: rr of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-km 2s;iluw8 uxim/ ka+k; fz3nn3t m1u9g uniform cylindrical grinding wheel of radius 18 cm tnm skaf;+3n/wuzm8 i9ku13uli ;2xmwhen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands t)im:n7z s.poat his side, on a platform that is rotating at a rate of 1.3rev/s If he ra7mz o:n)tspi. ises 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 he byy*p9c67 cxo*kvgbn l m )b-9n40dwjr moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position 7wn9oyvy* l dk0nmb x6bcj)4cpgb-9*. 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 initial rate of 1.0*,1j-a2k8s,i l kddi-vd( f cryu el:r rev every 2.0 s to a final ratjc i- ,rdrddlia kv*1-u 2sefk(8:,lye 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 t( p.. fxt4afor hga;oi iz(w9th:r8b* hrough its center at a frequency of 1.5rev/s The wheel can be considered hpft(o4.birh 9ix z . r8w:;aotg(a*f a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater0lmo- iz+,hg z spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular mofpvj.f)e: sl6r0a wa -mentum 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 of moment of inertia I is sn, qrum 48yqv (hw ao6l p0v,qk*(vu+dropped onto an identical disk rotating at angular s hsrn v*qyp +k qva0846q wl(,omuvu(,peed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns gbg.u;*qjl xw qc20:h at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rota73x 5 dbm5vjlxbf)p t7ting merry-go-round platform of radius 3.0 m and m55df) b3v plxtj7xm7boment 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 / +-k lfthx15w +bzxxkfreely with an angular velocity of 0.85fxxkx+ h+lk- b/ zwt1 $rad/s$ 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 eventu;-nh8:w s xm8b:rer ygally collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(rougrxn -: b s:yr8mwh8;end 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 12wjno7-z;0 sw r0 $km/h$ 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 upix.lpc7 nshb87 ik8,lq* h)$ 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, initihei ,z :(w9iaqj f)xk,ally at rest, that can rotate freely without friction. The moment of inert)ej hf:ak9w, xzii,(q ia 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-m diameter merry-go-r kl..t .-warut o,sl*cound turntable that is mounted on frictionless bearings and h.s okw-t*,ll.rc au.tas 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 th(,gyz k53 njep eyh83/qr ,kale ground with the end of the rope lying on the top edge of the spool. A person grabs the en5ygpnzk 3(r qyh, 3kj/e8,aled 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 unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always fai 1ht owf hu u4vynb1xhd*/7/n-o*j7a ces 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 orbiting4o 1x/hayob nndtuh/ vw* i7-17f*h uj the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 ttx49)w t-cv-t6q ikrm/$s^2$ 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 acquj fv.y7q-c+wu ires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcul .uq-vjf+yc7 wate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made h yp0dwo1 ;5py c.vs;jgut*2 eof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of masstg p;yw15vocp. *; 2udseyh0j 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed ogs4w;ls04wtv3n )qx rf the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as+zdv0 3 vwodth e(c8r8l 99ulg 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 l 9 eglr89u tw(cvh3dzd8ov0+ 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 anz 2+ue.lxf +kutomobile is for it to use energy stored in a heavy rotating flywhe.fn+xe2l zuk+ el. 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 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 ifm(hbb:-q*g $H_2O$ 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) isx 9 ;sm f;gx(4gp;qq1uwrrru8 rolling on a horizontal 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 and length L can pix-yqad*9ur uhdv02j1n0 s73w sf* qzr vot 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. [**9x1n0r 2ru- vqh0jyqsfwds7adz3 u Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on tdh c6oai53kn zrr 88r zb28d)**rwkmihe floor, and we want to exert a horizontal force F at its axle so that it will climb a step ak w 288ni)ord* rzr365*ah bkczidm r8gainst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makinglizju1y. cp. . a turn with a radius The fozj.ilp.y.1u crces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling i2)ay frf0l*lof 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 2if0*r)fa lyl is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, maki9z.ea/ff;r ui ih,jo wh/+3iyng reasonable estimates for thh/+f fe,z;w3hioi/ 9juyar i.e dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whu :n/f8xnz6wk ich consists of two cylindrical plates, ofwz8/k un f:nx6 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 radiummm0ex 1 ysf je1 0 erbl-:mgt8mo;3y6s r rolls along the looped rough track of Fig. 8–58. What is e:m -romlmtf;m sm8 b0y 061egj 3ye1xthe 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 track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu mzx++c aeu:4xq5 h (pli0bi1vt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the k tr58mq./ 4 ;mop / jvrr+bp1frfilx0car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eac im./0jo/qr k xrl 5fp+rv;mb8f1p4rth 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