Sometimes people say that water is removed from clothes in a spin dry a6 (x2k20m h( lwsxcg5vkz/tzer by centrifugal fwv zxhmcs (g(kztx0/22k6 5l aorce throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the car travels around alq:2)0wabz 4r *nnul x sharp curve at a constant 60 km/bn0r xw2z4a) lq*u:l nh as when it travels around a gentle curve at the same speed? Explain.

Suppose a car moves at consta 68,cb/onk7+walqoml4jf 3 g. m*mcyi nt speed along a hilly road. Where does the car exert the greates +kio m 3a4mgmcc,8yo.l/bn76 *j qflwt and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a chicwey4 7wuf:/j o-nt5v ld riding a horse on a merry-go-round. Which of these forces provides the centripetal accelera ucn7/ w-wv5 otef4j:ytion of the child?

A bucket of water can be whirled in a vertical circle without the water spilling4kwbb c.-6;h xuwb n+t out, even at the top of the circle +.w wbu ;khnxt- 6cbb4when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? Therez:9.kbtg- 3j tu1zy zd are at least three controls in the car which can be used to cause the car to accezkgzu13tt j bdz: y.-9lerate. What are they? What accelerations do they produce?

A child on a sled comes fly7x-br -jhlzm/ing over the crest of a small hill, as shown in Fig. 5–31. His sled does not leave the ground (he/zjbxm-h r7-l does not achieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward whencxlxdss; qt/+18 aamyv,0 mo p4m29na rounding a curve at high speed?

Why do airplanes bank when dkj+9s 9 r .hc4og0cojthey turn? How would you compute the banking angle given its speed and radius of thkgs4o0och. crd9+j9 j e turn?

A girl is whirling a ball on a string around her head in a9l, l5l3l(jyrow:j m 8yqu+ la horizontal plane. She wants to l :y+wu3 raqj,ll59ymll (8jolet go at precisely the right time so that the ball will hit a target on the other side of the yard. When should she let go of the string?

Does an apple exert a 3ap663zkanavi dgv5* gravitational force on the Earth? If so, how large a force? Conpazdvg3a 65*kvi63a n sider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it iskpp)o b):wk hw -lt8h;, in what ways would the Moon’s orbit be dif:hw bw;k8 kh-) l)pptoferent?

Which pulls harder gravitationally, the Earth24upww7x bl)w v8;yhfr l+ ot z80p5kh on the Moon, or the Moon on the Earth? Which accelerate)ykw 2x; zwhp8f4v0 u 85t+wlolr7bphs more?

The Sun's gravitational pull on the Earth is much larger than the Moon's. Yet th qbbeple(abrlwyh( . n0 .9id 657j*lqe Moon's is mainly responsible for the th7nd(l(5wl bq.ir ely b eb06 pa.j9q*ides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at thf,ej0tn nq9 u.e equator or at the poles? What two effects are at work? Douq0nej9. f ,nt they oppose each other?

The gravitational force on teh::+b.k b 0uisdn :ithe Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn’t ht0:b : d:ekiubin+ .sthe Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in itt5ruo 5zsoglx)c2: k0 s orbit around the Sun larger or smaller than the centrio csu2z5)otx:r gk0 5lpetal acceleration of the Earth?

Would it require less fq2d bay8rynbdy-s12;g 2i*wspeed to launch a satellite (a) toward the east or (b) toward the west? Consider the Ea -2yaw y21 q8fr*b;dbygn 2sdirth’s rotation direction.

When will your apparent weight be the greatest, as measured by a 3tr mn2)*cf+ fdwmcg 1scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at constant speed? In which case would your weight be the least? When would it be the same as when you are on the gromc cm 3wd1)*f ftng+2rund?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer space could be adversely affn c*z+ lg0,x jwa qhh+b6 pmz-80ndhr2ected by weightlessness. O bhap8c 20 qxjh+znzh*drnw- g60+ ,lmne way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how this simulates gravity. Consider (a) how objects fall, (b) the force we feel on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “:7dsun 76 oauoicxfy9 coj7u/jp;4e : free fall” or “apparent weightlessness” between ui:jo eno7x d4uaou96cc7 y7/p s;fj:steps.

The Earth moves faster in its orbit around the Sun in m s/exq (uz*ed g;h4z( w2/gamJanuary than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in pm*;qaezuw/s egd4 /hx( mg2(zroducing the seasons — the main factor is the tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not known until it whxm */agof/n3as discovered to have a moon. Explain how mx 3ofa/gn h/*this discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth ije)(xva3; b *u kpya;4t pd6ghs much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]A child sitting 1.10 m from the center of a merry-go-round moves ux6 kjat;pv( 3pby) hd;a 4g*ewith a speed of 1.25 $\mathrm{~m} / \mathrm{s}$ . Calculate (a) the centripetal acceleration of the child,   
(b) the net horizontal force exerted on the child ( mass =25.0 $\mathrm{~kg}$)   

  A jet plane traveling 1980 6wu c:a8v,)sku75md re +g ojj$\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ pulls out of a dive by moving in an arc of radius 6.00 km . What is the plane's acceleration in $g^{\prime}$ 's?    g's

  Calculate the centripw (hszptxyst.) 8u 25(wzkg6setal acceleration of the Earth in its orbit around the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orbit is a circle of radiusy.)ktp ( w z z(tshguwsx8562s 1.50$ \times 10^{11} \mathrm{~m}$ . [Hint: see the Tables inside the front cover of this book.]    $\times10^{22}$

  A horizontal force of 210 N is exerted on+(e ufga ;a7w;4e elbw a 2.0-kg discus as it rotates uniformly in a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the sfw +47gba ;e(aw;eelu peed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth'sx30zlgevq /v rfy 97j gvba7)0 surface, and circlesfg0zgj7 vvax9rv q70/bl y)e3 the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clay on thm)j s04x / y2rlj m)vocpr :pq2l3ib5ze edge of a potter’s wheel turning at 45 rpm (revolutions /c:23 o4rpxlr5plz))j0j 2b q vymmisper minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is reay tntrh(esq q;.)6e vlb3n+0volved at a uniform rate in a vertical circle of radius 72. nne ).batt vlqryh0es3q(6 ;+0 cm , as shown in Fig. 5-33. If its speed is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

  A 0.45$-\mathrm{kg}$ ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3 m on a frictionless horizontal surface. If the cord will break when the tension in it exceeds 75 N , what is the maximum speed the ball can have?   

  What is the maximum speed with which a 1050-kg car can round a turn of 1a0p8 8fs2 ac)6upg neybzee*h4cg (aradius 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is this result independent of6 syap8)pgn ua0 f eca1eeh4c(b2gz8* the mass of the car?   

  How large must the coefficient of static frictionq(o2w s8 9ezg)b )nnse be between the tires and the road if a car is t( eg)zbn o8s)ws2 9enqo round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jvo-kp)b:owqg9 *n *tq et fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back is 7.85 wo9* o-:np*t kqqb v)gtimes her own weight, how fast is she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the axis of a rotating turntable of variace0oln)* m1pd ble speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between the coin and the tur0l pnde m1o*)cntable?   

  At what minimum speed must a roller coaster be traveling when upside dowxcla v2ql;2* o 3 0(5gbljciljn at the top of2 0g5l3ibc;c( j x avlqo*ll2j a circle
(Fig. 5-34) so that the passengers will not fall out? Assume a radius of curvature of 7.4 m .   

  (II) A sports car of mass)azt6wri18d h 950 kg (including the driver) crosses the rounded top of a hill (radius =95dr zh186t a)wi$ \mathrm{~m}$ ) at 22 $\mathrm{~m} / \mathrm{s}$ . Determine
(a) the normal force exerted by the road on the car,   
(b) the normal force exerted by the car on the 72-kg driver   
(c) the car speed at which the normal force on the driver equals zero.   

  How many revolutions per minute would a 15-m-diameter Ferris wheel ami6 ucw9-.q,zi 2jhpneed to make for the passei2waj- .qiuz ,m96 hpcngers to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg8 uuiju :k.k.r 6hcj2e is whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension k2ujk .86eih r .j:cuuin the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuge rotate if a pj4 udt5ng;ha9ua vqg;q -/k-marticle 9.00 cm from the axis of rotation is to experience an acceleratij/; agtu q-4q -9n vagkh5mdu;on of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, peoplefn nl.ya*mtjm3 /6g2+ khx2irf:uf.v are rotated in a cylindrically walled "room." (See Fig. 5-35 t+623n:.f. /uyfxafj2h r kimnlv*mg.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rotated in a ctm3 huv rp19j dp-03hcdg1 h)nircle on a frictionless air-hockey tabletop, and is held 3gc31dvu-h9h jrpp01hd t)nm in this orbit by a light cord connected to a dangling block (mass m ) through a central hole as shown in Fig. 5-
36. Show that the speed of the puck is given by

$v=\sqrt{\frac{m g R}{M}}$

  Redo Example 5-3 , precisely this time, bwskcx(n6 470u8p yk skcf5 u1i4bmzq.y not ignoring the weight of the ball which revolves on a string 0.600 m long. In particular, find the magnitude oksicz5upuk 60ns7(4c4bw fx . mkyq81f $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of 88 m is perfectly banked for ay2- tn5z( 3j+cndprzd car traveling 75 $\mathrm{~km} / \mathrm{h}$ , what must be the coefficient of static friction for a car not to skid when traveling at 95 $\mathrm{~km} / \mathrm{h}$ ?   

  A 1200$-\mathrm{kg}$ car rounds a curve of radius 67 m banked at an angle of 12$^{\circ}$ . If the car is traveling at 95$ \mathrm{~km} / \mathrm{h}$ , will a friction force be required? If so, how much and in what direction? $\approx$    down the plane

Two blocks, of masse5u3/ yd klrgw0s $ m_{1}$ and $m_{2}$ , are connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency f (revolutions per second) on a frictionless horizontal surface at distances $r_{1}$ and $r_{2}$ from the post. Derive an algebraic expression for the tension in each segment of the cord.

  A pilot performs an evasive maneuver by di abil(u7)axd-bm+2f xving vertically at 310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripetal com72dyqhd7pt gd0 i07betj6f i.;ga 2scponents of the net force exerted on the car e0hdjbpt2qdt;ig6cgid.s7ya70f 7 2 (by the ground) in Example 5-8 when its speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerates uni-srdim+ iz 2l1h i/ ebk*mub,w2v327hrcl3 gvformly from the pit area, going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the r3 le/rsi22d2i vmc7kbgwu,rb lm -zh1* hi+3 voad to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90 m wjc5 a6h8w-k hrb8//udsaee. . At a particular inst w/ da8 esu8ekbha/wh5jrc. -6ant, its acceleration is 1.05 $\mathrm{~m} / \mathrm{s}^{2}$ , in a direction that makes an angle of 32.0$^{\circ}$ to its direction of motion. Determine its speed (a) at this moment   
(b) 2.00 s later, assuming constant tangential acceleration.   

  Calculate the force of Eazl,dtb d:t3jnt8xy w2:z8+sbrth’s gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 kg. tn,+t y3x2ztl :d:wd 8 jbsz8b  

  At the surface of a certain planet, the gravitational acceleration rqck8mhrj42w5u2 i q 7g has a magniwrmr2uk qq j24i 8h7c5tude of 12 m/s$^2$ A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet   
(b) the weight of the brass ball on the Earth and on the planet? $W_{Earth}$   $W_{planet}$  

  Calculate the acceleration due to gravn0yi c6an: fe;ity on the Moon. The Moon's radius is 1.74;0 n:6fney iac $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 times that of Earth, but has the same masw4m hmv7-uhxw+u. 1,xfeqd l/ s. What is the acceleration due to gravity nea +lwhmd/u uh-.4v7me1xxw fq ,r its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, but the same radoref*,kcg-sn+lp 0 .xius. What is g near its sus l f*p0ongr+,.eck -xrface?   

  Two objects attract each other gravi7 9qxvkfhg- 0utationally with a force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of grgx sbnp6x78xfhkw 4hx: ; if)1avity, at (a) 3200 m 1: )nwi8pk7h; xxs xhbxf64fg    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s mwr 60w- atg ftfl4-)qcenter to a point outside the Earth where thegravitational accelerw-f qg t-)lrm60waf4 tation due to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has five times the ma5cboo 45 3tybxss of our Sun packed into a sphere about 10 km in radius. Esyt5o3 o4bxb5ctimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an i9jn+x 0lmfmfnxd1 :0 average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but +x lfj0i0 x19:nf mdnmhas the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why ast( rx *dz6a1)ar :klovcronauts feel weightless while orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravioar)vc:*1z rk(lx a6dty 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are locas9r7*1*fb gb0k ds8eq8mji euted at the corners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force8qsu1g** dfbs ki98m7jr 0 eeb exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred yearslt)h)bxa3vb ) most of the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planb) )3bvxa)ht lets are in a line (Fig. 5-38). The masses are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , and their mean distances from the Sun are 108,150,778 , and 1430 million km , respectively. What fraction of the Sun's force on the Earth is this? $F_{Earth-planets}$ =   $\times 10^{17}$ ${F_{Earth-plantes}}$ \ ${F_{Earth-sun}}$=    $\times 10^{-5}$

  Given that the acceleration of gravity at the bxz5c3q w:tt b ,vk2k5surface of Mars is 0.38 of what it is on Earth, and that5 vckzt3 b qkb2x:5,wt Mars’ radius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the perio2lv-,f pqogh ylbf+ 6,d of the Earth and its distance from the Sun. [Note: Compare your answb+y v,fgh,p -lfoq62 ler to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of 1ilwe iv m;md/9:v6qjg:, id wa satellite moving in a stable circular orbit about the Earth at a height of dmmvv:i 6/g1wew;: jl9i qd,i3600 km.    $\times 10^3$

  The space shuttle releases a sateaag9o xqrc 5h)ujbw. mv*m3 9.w8y h3ullite into a circular orbit 650 km above the Earth. How fast must the sh3 38hmga.qa99u ucv r)wymjhb*w. o x5uttle be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if ocwi6/qllc 0.ykcupants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as t 6l/.yqwki l0che time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellit .ita ffa (wz1)qb6jx1e to orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is very small compared to the radius of the Earth. Does your result depend on the mass of thz 16i()x a1.tf bqwajfe satellite?    s ~    min

  At what horizontal ve63,,7baeoacyzj eh24 .ugio ulocity would a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbioy,obz,c6.3eha a u7guej i42 t around the Earth?   

  During an Apollo lunar landing mission, the command mos6 iy +d9lu,d.;,t ev +jnk0nfur1yyqdule continued to orbit the Moon at an altitude of about ;nq,+ ,1 u ukntd0erj9fli ysdy6y.+ v100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunkslgj4rqg 3bk /, of ice that orbit the planet. The inner radius of the rings is 73,000 bqlr /j3g,k g4$\mathrm{~km}$ , while the outer radius is 170,000$ \mathrm{~km}$ . Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn's mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 $\times 10^{26} \mathrm{~kg}$ .


$T_{\text {irner }} $ =   
$T_{\text {outer }} $ =   

  A Ferris wheel 24.0 m in diameter rotates once every 15.5 s (see Fig. 5–9). Whafd,( x)c i8n5bv0inu gt is the raiivun8(ng, 5) 0bxd fctio of a person’s apparent weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 s/vv)9,mk k /lvj2sc 58qzu;vw)gxk wkm from the center of the Earth's Moon in a ,;ks8wj/) xlqmw/ck vvvuzgs) k 92 5vspace vehicle (a) moving at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star system consists of two stars of equal mass. Thg idq lrvz/3xv2f) ,wq3eq0-yzjhs ;-ey are observed to be separated by 360 millionjzqq/ qy,i wd 3v -;ez3vlx2h0 -f)rgs km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read forl vtp.wzg 57; pgog*b8:xv. te the weight of a 55-kg woman in an elevator that moves * zlp5v7gg:gtbt x;.p 8vewo.
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspended from the ceiling of(mb+te9x8:ddeih hq3 phm: / xz -*m+fcgpgd0 an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevatgxmdi9z(-m xd/ edmc*q:t3ph:8++bpg e h0 h for’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a plane-8kn- x6g :dwna)j ,l d k0pqbgom4p9-koie6nt with period T , thed-kp-lkp4 gk) dn ni -mqeb68oxgj a9 0,o6n:w density (mass/volume) of the planet is $\rho=m / V=3 \pi / G T^{2}$ .
(b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min .

  Use Kepler’s laws anhq+3 pn2 )m*5i zzd5gzs 5t.qeroaibz9hr 7t9 d the period of the Moon (27.4 d) to determine the period of an artificial satellite orbiting versr z9r 5 ahnp d b)i735q o*ez+iz.q59gtm2htzy near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, oz6 qb0 ki4w7rwu44bdz tt 0ww:rbits the Sun like the planet4 b zwtkt0zq 4brw6 0w7d:wi4us. Its period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average dista8u7azzp x1z:1ddje t iso9u41nce of 4.5 $\times 10^{9} \mathrm{~km}$ from the Sun. Estimate the length of the Neptunian
year given that the Earth is 1.50 $\times 10^{8} \mathrm{~km}$ from the Sun on the average.    $\times 10^2$

  Halley’s comet orbits the Sun roughly once every 76 years. It comes ve* zn/ncj ;iik*(zns ,jry close to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the greatest distance of the comet from the Sun. Is it still “in” the Solar System? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Su ;kinzj j/*innsc z,*(n.]    $\times 10^6$

  Our Sun rotates about the center of the Galaxy fr:iw3f.m2,- aldyjp1x y y t8 $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ at a distance of about 3 $\times 10^{4}$ light-years $\left(1 \mathrm{ly}=3 \times 10^{8} \mathrm{~m} / \mathrm{s} \times 3.16 \times 10^{7} \mathrm{~s} / \mathrm{y} \times 1 y\right)$ . What is the period of our orbital motion about the center of the Galaxy? $\approx$   $\times 10^8$

  Table 5–3 gives the mass, period, and mean distance for the four largest moonsec o8zxg8npq2( l*.fzgdq6 t 3 of Jupiter (those discovered by Galileo lz * xc23n(8g6t zpoqdqgf8e .in 1609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the Earth from td gr5u a5f6p.jhe known period and distance of the Moon. jfrd6au5.g p5   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jqx.t e -o:q fef9v0u;ecgo1z 2upiter’s moons, using Kepler’s third law. Use the distuoq zf.1ex:g evoct -;f92q0e ance of Io and the periods given in Table 5–3. Compare to the values in the Table.
$r_{\text {Farip }}^{r}$ =    $\times 10^3$
$r_{\text {Guarymale }}$ =    $\times 10^3$
$r_{\text {Culimio }}$ =    $\times 10^3$

  The asteroid belt between Mars and Jut+bi: ,sakubms78a e0p8 dl. /g)tahvpiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destad7b0itmat+v g8uassh, ekb8 )/ lp.:royed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale describes an artificial hhe * )rb19 (dz/hii0zorm l+1wbvj;az 2tc0x "planet" in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noom+ letc9aj; w)zh hbdzr0r oh2i(1bvz/i*x10n). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by swinging in an arc from a hanging vind:k0dz tw0 gy5myr npsm+(;,ze (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he ms5 ;t (p:0gdd mwyy0k+r,zn zcan tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration of gravitydpr4w.) h3rh( cgv4ad be half what it is on tp4(w)g4 vc 3adh .drhrhe surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mutual circy 1zkbxwwn))c) y;2 mxle once every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pullibw)ynkxxw) zyc; 1m)2ng on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent ac f *p-)3n eai3.ivbabjceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this effect. What fraction of g is thiv3n-3.i*bpfe jb aia)s?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft traveling directly from. q9 u4ywcm9u0aw5l bs the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forc0 9.59sbcuauww ml4qyes?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathroom scale in;ezj- -sqc5zqhary 2+ an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, and in which di;-j-r5 zze q+ah2 csyqrection?     ----待选择----upwartddown 

  A projected space station consists of a circular tubeo-r5r2 lam -zzxc;1qw.k kvv/d+a mc- that will rotate about its center (like a tubular bicycle tire) (Fig. 5–42)5-1zzw c.;dc vm-kaxm/kvq-+arro2l. The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt?$\approx$    $\times 10^3$

  A jet pilot takes his aircraft in a vertical loop (Fig. 5n0ta)e oc -,vqbm mx9m2w eagj8g v.3p ;r:rx1–43).
(a) If the jet is moving at a speed of at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s.    $\times 10^3$
(b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle    $\times 10^3$
(c) at the top of the circle (assume the same speed).
   $\times 10^3$

Derive a formula for the mass of a planet in terms of its radiusg ;2k t76az+qcxczy ,c r, the acceleration due to gravity at its sqyg c kzx7+; cc,zt6a2urface and the gravitational constant G.

A plumb bob (a mass m hanging on a st v.mbfi//w8sq ring) is deflected from the vertical by an a8fsm iwb/v/. qngle $\theta$ due to a massive mountain nearby (Fig. 5-44). (a) Find an approximate formula for $\theta$ in terms of the mass of the mountain, $m_{\mathrm{M}}$ , the distance to its center, $D_{\mathrm{M}}$ , and the radius and mass of the Earth. (b) Make a rough estimate of the mass of Mt. Everest, assuming it has the shape of a cone 4000 m high and base of diameter 4000 m . Assume its mass per unit volume is 3000 kg per $ \mathrm{m}^{3}$ . (c) Estimate the angle $\theta$ of the plumb bob if it is 5 km from the center of Mt . Everest.

A curve of radius 67, c y;3r8oo t4,meymfc m is banked for a design speed of If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can 4ye y8m;, o3cfc, ormta car safely handle the curve?

  How long would a day be if the Earth w v5+fz iq tg5w oaa3q5f3a;7tyere rotating so fast that objects at the equator were apparegf53f3toa5 a wqv; +iy7qtaz5 ntly weightless?    $\times 10^3$s

  Two equal-mass stars maints 4kejc vd*6(xain a constant distance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant spm k- bioe:j-uzm, z2w4eed rounds a curve of radius 235 m. A lamp suspended from4om- -2embjzuiz ,:kw the ceiling swings out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, it has been c pk20v 4tjcxf+ y onrsn+n1*-glaimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of sury 2g j1 4so*+np-0 nfvxc+nktch a planet. Use the following data for Jupiter: mass =1.9 $\times 10^{27} \mathrm{~kg}$ , equatorial radius =7.1$ \times 10^{4} \mathrm{~km}$ , rotation period =9 $\mathrm{hr} 55 \mathrm{~min}$ . Take the centripetal acceleration into account.    g's

  Astronomers using the Hubble Space Telescope dedu / o r8jjdu),4zmy9cmv6kts k6ced the presence of an extremely massive core in the distant galaxy M87, so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be 780 8mo9 ,6d mzsrk4 6uyvjc)tjk/$\mathrm{~km} / \mathrm{s}$ at a distance of 60 lightyears $ \left(5.7 \times 10^{17} \mathrm{~m}\right)$ from the core. Deduce the mass of the core, and compare it to the mass of our Sun. v =    $\times 10^{39}$solar masses =    $\times 10^9$

A car maintains a constant speed as it traverses the hill and va5x ndu h8,uq;.ccifn *lley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting qduh58x* i,nu ;ccfn. on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites *dnue2zwi-q *orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites 2weqniu*- *zd in each of six orbits, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (NEAR), aftew- q/.6 axcxybr traveling 2.1 billion km, is meant to or./x-x qy6 bacwbit the asteroid Eros at a height of about 15 km . Eros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the s)qj azk fb9r(1pace shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it. a(bz1fq)9rk j
(a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km?$\approx$    h
(b) By how much must you reduce your orbital radius to catch up in 7.0 hours?$\approx$    $\times 10^3$

  The comet Hale-Bopp has a period of 3000 years. (a21icpp -:70w 8rkff vr zg3hih) What is its mean distance from the 031cirgi8wfk:2zh pfv h7-rpSun?   
(b) At its closest approach, the comet is about 1 A.U. from the Sun ( from Earth to the Sun). What is the farthest distance?   
(c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5–29, but smaller distance travelled; see also Hint for Problem 59).]   

  Estimate what the value of G would need to be if you jbbkst 8i u)l(r h+(n+could actually "feel" yourself gravitationally attrac)(s8(j h b+kl +britunted to someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46) at a dista.)d fzmlb0vf. +sjqb (nce of about 30,0jm bzlbfdv(. s.qf)0+00 light-years from the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the corners of a svi,j2 xj5xf; dquare 0.50 m on each side. Find jdj xi;v f2x5,the magnitude and direction of the gravitational force on a fifth 1.0-kg mass placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg o: 6t a-yrsmojf 5blify0+1qy:rbits the Earth $ \left(\operatorname{mass}=6.0 \times 10^{24} \mathrm{~kg}\right)$ and has a period of 6200 s . Find (a) the magnitude of the Earth's gravitational force on the satellite, $\approx$    $\times 10^4$
(b) the altitude of the satellite.    $\times 10^5$

  What is the acceleration experienced by the tip of the 1.5-cm-l, fasd*+hw.un8qu ab-ong sweep second hauw*q,s+- hn .u8 aadbfnd on your wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight around d p5umfw8y *b urxs1zu/ 4;3elin a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circlewu m4z1 d8e5lusrypx*fbu; /3 every 0.50 s. What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highway is designed so that a car trave:io tf86n(abeling a(t nao86e:fb it speed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizes tilt of the cars whe6())jfpcac i5nzt ;ybk. -lqh n negotiating curves. The angle of tilt is adjusted so that the main force exerted on the passengers, to provide the centripetal acceleration, is the normal force. The passengers experience less friction force against the seat, thus feeling more comfortable. Consider an Acela train that rounds a curve with a radius of 620 m at a speed of (approximately ). (a) C(yiabpclh-qc6 .) zj) ;n5kftalculate the friction force needed on a train passenger of mass 75 kg if the track is not banked and the train does not tilt. $\approx$    $\times 10^2$
(b) Calculate the friction force on the passenger if the train tilts to its maximum tilt of 8.0º toward the center of the curve.$\approx$    $\times 10^2$