A bicycle odometer (which measurroa/+u kgvp 333,; d7xtne x/6dy jxcwes distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens i +epgyou33a/w j 3;xr/ 7 dc6,xktdnvxf you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doe g* 1p:ha5ox:nf a8ebos a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases unifor8opxgf:ae*5h1bo a n:mly, 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 sin0 hc n/xya*e)zgle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explapu4,gv.ft dhswc * 6.pin.

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 eyx. ,,k cnj:rvh+ jgb.h:e:/izjc f. hands behind your head than when your arms are stretched out ig /j:nz,ei: h:rby+ h,c.vexf.cjjk.n front of you? A diagram may help you to answer this.

A 21-speed bicycle has se3bxhc zbp.80a3wf;rd ven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gafdr h wc;pxz83b3b .0ear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able4po2d1tcb0e(ukayb 9bbt ) 8yezc2 5 e 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 dynamicsy12zb kc4 ee0)92btc bbop5ea8(uy td, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry fe)*mk gvh/8 0v w8, yz.jmfmsa long, narrow beam?

If the net force on a system is zero, is the net torque x -9xlozw:h/ 8-v m :dkpdqkq1also zero? If the net to-w x/kxq oh9 d:vk-:8qzp 1mldrque on a system is zero, is the net force zero?

Two inclines have the same height but make different an/pqmk+*1/ cd n7blj tmgles with the horizontal. The same steel ball is rolled*q+ njdc kmpt 1/mlb/7 down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rescdv6 2bgq,uqizqxq5y(;f03yt 7l f q3t) down an incline. One sphere has twice the radius and twice the mass of the ot2yx0ycl3; fdqfgq (i7 b6vqq ut z5,q3her. 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 z:xp9acax.67 qow l52g, bwoy 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 great2ba6 ygalx5wq,xo79: .oc z pwer total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserve92uy+iu 0+(yd phv(u:imuzhs i g7bk3 d. Yet most moving or rotat:iumd3 h hi27gyp sz0( u+k9+uy(ubi ving objects eventually slow down and stop. Explain.

If there were a great migratioi*ev,k+vo 417g azu rgn of people toward the Earth’s equator, how would this affect the len71,*ivgkav ue + go4zrgth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatio ow kq2y-bg(0tn when sog (b2 kyqt0w-he leaves the board?

The moment of inertia of s rdjul1*z j9-a rotating solid disk about an axis through its center of masssd 1*uzljj-9r 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 rotating stool holding a 2-kg mass4ohkh zrqt5, m).5vu t in each outstretched hand. If you suddenly drop the masses,o4z5.kq ,h5m huvrtt ) will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. How*a)yyg; lsdme1--j sqever, one is hollow and the other is solid. Describe an experiment to dety*d ega1sq)-ys- j;lmermine which is which.

The angular velocity of a wheel rotating on a horizontal axle xoxm/pxdq t1m8 4y e3hf(q89j 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 spe xmf/8(4 8t9x 1 qpomd3jyqhxeed increasing or decreasing?

Suppose you are standing on the edge of a large ndokll -1ot) f+m b.dk7x(ib5freely rotating turntable. What happens if you walk toward the o.mk b-1l+xdo5b(lknti 7) dfcenter?

A shortstop may leap int t he8p 67zxsn+w8-:iton,k weo the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his bs 7ze+ ntitxe:8 -n8wp,h6w koody 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 of angular momentum, discuss why a helicmcpfb;g( k:4 zopter gk:(z; p4c bfmmust 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 angles in q2,+hr*b6 .vl btjh aeradians: (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 coincidenc5i*dj8hxm1w e)vq ilax.,kj7 e. Calculate, using the information inside the Front Cover, the angular diamejhl,e1)jxw qix.5kdviam 7* 8ters (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,d7+pz . 73xenysbs8 zo62vbgu000 km from Earth. The beam diverges at an angle gy7uzenx2pzs76b +38 v.b osd$\theta$ (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 mot s9u4dklb5+of1 myvl0or is turned off during operation, the bladm4y5u f9sl b1old+vk0es 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 to another 9 8ep9dw rvfa/wd )*p hioqj2*child. If the ball makes 15.0 revolutions, what is its diametejq v )2w9 rdi/p9aweoh8p** fdr?    m

  A bicycle with tires 68 cm in diameter travelsegzz:b :p xr+k)ow6i+ 8.0 km. How many revolutions do the bxg)orp6:z+ez+ :kiw wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate it - syow:ymh(j7o4ib 78 wniow+s angulan7y iwom -4oyj hw :i+s7bo8(wr velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one;ng.2w,k/n. x2ri dh( l 0eppbujguy ovi12x3 complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sxg;kg/u 2r3(i dw ipp boj,v 12l0u2y.nxh. eneated 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 (tepm -os g0p8;/ cj.mba) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe 4vsj5escesy9dm (d49ed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a ceng-q7x,a sx6l itrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration ofg7slq x6-,i xa 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fro-ua/2 2g-vvu cf;g.r3lbc c,jsbk-i lm 130 rpm to 280 ;julcv s-2bv2.gfac --rik,lg/bc3u rpm in 4.0 s. 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 radiusij895j ev2qq7fb. rhin ;/adq $R_1$ 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 Moonlh,3 f)7pi le0ed 9;v+ +cdjflz6cluf, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of ascz9c,ujf 0edf;l +i)+76hl l3 ev fdlp 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 unifor59w ;cdbz kjrhx(a+nd.1 * wdgmly from rest to 15,000 rpm in 220 s. Through how many revolution9 *gwxhwzkr5d a1j+dc(. bn d;s did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2ktri7.i 5(5m8cpkni 3d/ s:vb x1gdu i.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying hin9c rswu42cb:g 7d- ;n; iicv o*b3wyyghspeed jets in a whirling “human centrifu4;irccvbg;u dbcnwyo n29ws: -*y i 37ge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerapzx1kjq 0hcpv5 .gu nqyi*4:,tes uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel ha hz,1ypxq.v4q c5nu0pkig*:jve traveled in this time?    m

  A cooling fan is turned off when it is rx 6imdxwf o*48jq)gi4unning at 850rev/min It turns 1500 revolutions beforg4i8d4j* wox)6 ixqmfe it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces4.3 o x(a*lnbps)isoe its speed uniformly from 95km so3 a.p*(exns)b4loi/h to 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 revy)ffpq 7vq8s6 ijs0 a+olutions as the car reduces its speed uniformly from 95vssj fq0+ q)a7p6i8 fykm/h to 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

  A 55-kg person riding a bike puts all her weight on each pedal when c64k0* k5eqv/a mn6 w9htebzqnpu bp.8limbing a hill. The pedals rotate in ah tnq5 .qkpeu kv n0be p6*w468z/bam9 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 p,.atl cvupi9r5 ok2, door 74 cm wide. What is the magnr a,pu5lo2p9.k,c tiv itude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque abou- 0 orr4qit((ef kt/+2tfcxqt t the axle of the wheel shown in Fig. 8–39. Assume that a friction torqit r(kcto0qf(4+xqr e2-tt/f ue 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 ofw3u ca)5uq0 8fwtnx) u a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the)cn5u wfu0uw a)xtq38 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 requicgfd-z mj* du ;m5pm.,re tightening to a torque of 38. d;5mfm-c z*dmugp ,j $m \cdot N$ 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 10k +e+ 0ar5x wwfeid*;w.8-kg sphere of radius 0.648 m when the axis of rotation is throua5;w0w +f dre we*+kixgh its center.    $kg \cdot m^2$

  Calculate the moment of inerf yx.u:;b5j+fhv9rqth rg7 ka.w*jm9tia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored fx wm ; 5h jt7yubak:g.jrhv.qf+9r *9(why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizonmsd0ayou/+, 1z xvn 4dtal circle of radius 1.2 msnx d,0y14+u/oz m vad. 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 angrgg72 v s7l6rwular speed (Fig. 8–42)rgsl7 7wvg 2r6. 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 show u q8frkqw 6ird,k:ae:;8 mfz(+3 tqzln in Fig. 8–433krlr ztf6 qf8(:z,q ;:iw adeum8+ qk about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass isbaxhk9l)38wn mk .4w s $5.3 \times10^{ -26}$ 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 cylinh7*h1qdajy v; drical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and ad j1;ahvy* 7qh 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 witha(lt v l+wbsx yk)(z., a radius of 8.50 cm and a mass of 0.580 kg. Calcuszb(k.,yv+ )xl(at lwlate
(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, acceler, ql4okf. so:b5ab9hmating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and qgo; ra i6bily6d9w71xct6e1 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 has6rx7a d1y61te i liowqb;g 6c9 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 x8 w sxi5n-y*ais shut off and is eventually brought uniformly to rest by a frax-iyn x8*w s5ictional 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 u-qotar4i-4 l.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 bsei.6p;ih,f xnt;abg3z 2w j+l:7 qhq y the action of the forearm, which rotates about the elbow joix2, le :;hibn6g 7wiqz;t+pf .aqj3hs 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 considered a long thin rod, as shown7 5f x 3pcvtjf).xuas5 in Fig. 8–46. tx 5s.uffp 5c37)xvaj
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse)qy u 6muxv3,hs, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turuxvl95+4oso /f6 t azxmhf9f.d)d 5;wxw 8aqfns (revolutions) and releases it at xla9ff;) du4w89dmfztswov5axx5. q 6h +fo/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 (i0yyl 9nrts x :(y9 lb:5odfdof 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 torqubpjp6s11u5 , ityqpt3 e 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 kg and radid,hev.:m.8h0f a54mu s wx kykus 9.0 cm rolls without slipping down a lane a ksf 5d0m8.4hxey.wuk,hm va: t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum o9grrjd-h g1rs(f4z ve(q(;qsf two te(rzf1s- (s9rg(4 gr;qdj qevhrms,
(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 qaf x2eg:ou(8x(1lm6 . y*o0ikg ulxemass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylindgxeomf(io lg.806 x1ax: q2 yle*(kuu er.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolb *1:rip8klfhn9 :txq7g):e.bvl 9azom9smnls without slipping down a 3:8:rh9el. tzmp)b:1anx9m f 7nsb q9ilg *okv0.0 $^{\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 tip-taf- v 5:i4 (lcafy4:s pkpfm. 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 lpf 4 :i -tc54v(mpyff-aka s:energy.]    $m/s$

  What is the angular momentumwbk;+3 zap-vxqiedy y r *lr++5f o*/u of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed q- ;5ufox3y+b*+k+dp*/laize wrrvy of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindric dqmmo,/)fsa q;9qw d*ac0p4y al grinding wheel of radius 18 cm when rotating at 1500y ds)d49q; q/a* ofc0wapq mm, rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that 9 yxnhjw -.j2m52 vapf14o z cnf4cpy,is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreas2c2j- 1yym n5zf x,4p oj.h4f 9pcnvawes 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. 2k izn- 3cu;jy8–29) can reduce her moment of inertia by 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, jcyu ;kni32-z what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase he ;ta(rsos)8ev 4gv dj)2oq+tv9i k 1nqr spin rotation rate from an initial rate of 1.0 rev every 2.stvka2e)s +oj;)ot qn1rqvi4( v9 gd80 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 rotj4nuzrk ;z sf. 2x(:tpating around a vertical 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, approxi2xpj;f u ( 4tsnkzz:.rmately 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 fiz7q:-qlwfv4+b0gzcd gure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eart/ixlqxgin (2 cc8yq:3tsc :1sh
(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 dropped 4;ipie)1 fq ayonto an identical disk rotating at angular speedy i)fqaep1i;4 $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2c j3gewrywa kdnke,3 f(,4*,h .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 0gu3wxc5(0qne bw9uk of a rotating merry-go-round platform of radius 3.0 m an3c ewug kb5w 00(qxun9d moment 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 angular ve0cqey 5(2 gt evlfxkc .2ai54*hfuw1ylocity of 0.8 ta f1wihu2*q2c l c5ke 4.5yex0f(vy g$rad/s$ 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 z8resate:4z4) -h1b dle 3 lsp3qi0gt white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of shr4g083 ee4 e bpqzds)3-t i1atl:zlits 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 wind8jsdjb4mt .5 7gioo9 bs 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 ffog0z5ep ) 2v$ 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, initial9h ;ai0j :yiduly at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform ihii;ud aj:09 ys $I_P$ 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-rod:sy 6u1x 9qurund turntableuy1qds:x9ru 6 that is mounted on 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 ground with the end of thezevvx 9h s6pl3dv, x4- 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. h -v6zxe9vd4,s3vl xpThe 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 samf( ofdhyf5eh2dw2 5 /ve side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbitalvhf5 w2d (e2yf/hd 5of angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a pvp; uyjlr49g:w53x d 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 rad1r p((,puu, hjdqij: dius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratiijpj,rh(1q d(d ,u pu:on. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so-6 kf2 wmj i-f qbc5f7e5cdf7blid cylindrical disks, 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 calculatef ewkfq-c6jc -52 7dfmbf57ib 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 of6gy l87ale5nw.rrou)p;5eqkko ) h0o 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 o3gnhq8 0 yqwm9 )su y9if)ane.ur Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collyu0nq3)qe g.w min 89ahfy9 s)apse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a0(y*7vz ayndp low-pollution 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 flywh*yyn padzv70( eel 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 illustratesr(v w5q*7t* o ajxx4wb 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 (hoz7fy0+a u eb3xop) is 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 a)4u .zf zk2yck1 o6q2pl0 vkgpnd 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 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. [Hint: See Fig. 8–cuyk4.2lk2o)p p06 f zzk1q vg21g.]

A wheel of mass M has radius R. It is standing vert11w )b9 6wwz:kreqg+l f5zf gmically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a sgkg19 z5)w+f1mqw6fl e z :brwtep against 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 making a turn withb84g up to,ml r6hhl;*opr0i( a radius The fotproh o8l;i6hu(g0, rp4b*l m rces 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 of length 1.5 m a i7ly6oj.:rv- b8l a1y1pidz knd 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 feadz6 vlji b-rok i:y11ylpa .78t, 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 r 3xqw.v-7ea8*o3wo k 7dwkym iods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms heow3*i3mkvw ekd o8w. xa7 qy7-ld tightly against her body.   

  You are designing a clutch assembly which consists +-r8gqqzbjxsn.x92 e/ h* aviof two cylindrical plates, of mi a js8 ./gh9 n2*zqx+-xqbevrass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the pa,q+nmf m91nm+,r yy 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 track? q, ym9mf,pr+ man+yn1 Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asspmdkz75/ 2wj ;wahf1aume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces x19pxb g0, zdqits speed uniformly from 90km/ xzp dg9x0q,1bh to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s