Sometimes people say that water is removed fromyi:a ))ln+na np, qfrz e(r2s/ clothes in a spin dryer by centrifugal force throwing the water outward. What is wrynf za+2:)nr p s ,rla/e()qniong with this statement?

Will the acceleration of a car be the same when them+ba.gu:qm:j -o0* rexapgz , car travels around a sharp curve at a constant 60 km/h as when it travels around a gentle curve at the sam:e*uxqp0, mbam j+z g. ro-:gae speed? Explain.

Suppose a car moves at constant speed along a hilly road. Where doesn,4sx hhk:mggaq6qv*3 6t8 kd78jvgx the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip betwee dk4s8 hk7hmgg 6g8 vva3*nq:,t6xjqxn two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a child riding a horse on ed0d3il9iq: ho5gn / va merry-go-round. Which of these fori 9/edo 5q0nl :3dghvices provides the centripetal acceleration of the child?

A bucket of water can be whirl ip7.jjsi76eit7 m4peij/ c9wed in a vertical circle without the water spilling out, even at the top of the circle wpijmitecj/ewps 46 .7ji9i77 hen the bucket is upside down. Explain.

How many “accelerators” do you have in your car? Theres7 fhb/ fo+ /z-rv.jav are at least three controls in the car which can be used to cause +vfszavf/j7o r.-hb /the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest of a small hill, as shown in ivw4ehkt) 4,k Fig. 5–31. His sled does not leave the ground (he does not achieve “air”), but he feels tk ,)w4h4keivthe 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 when rounding a curp56f vpbkv i kw1:sf/0ve at high speed?

Why do airplanes bank when they turn? How wouldpzwflbj/e9,b)dy2ia6 q1 k b8 you compute the banking angle given i,)a e/f1lj qz9byb68 dkwipb2 ts speed and radius of the turn?

A girl is whirling a ball on a string around her u n4ey 2j;4-gsnbwi b.head in a horizontal plane. She wants to let go at precisely the right t b.b g2uyjn-;4s4enwiime 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 gravitational force on the Ea8 5ouq 0hwd 4h+ma -h5k)q1dqfb,rnhgrth? If so, how large a force? Consider an apbhrdfqkn54aqu+8om ,05 -w1hdg h )hq ple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were doubl-y rjlqo4qnej:;,) e3rqp5npkz 5p;n e what it is, in what ways would the Moon’s orbit be diffeq3p 5n-:eqopy rnj54kq)l jez,rp n;;rent?

Which pulls harder gravitationally, the Earth on the Moon, or the Mmpxfsv(fqx1-n:, g ap j;*k- hoon on the Earth? Which accelerate fkvxp-,ps(:ng a -qjx 1f;*mhs more?

The Sun's gravitational pull on the Earth is much larger tha(dur m:t8mr6 8aft ii)n 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.]ft:m r)adri m86t(u8i

Will an object weigh more at the equator or at the poles? Whav 1 hrc. rbzwe5692r9jym5*t(ufaii qt two effects are at wor9fz5(ec.uitb w mv*9 2ahj 1rirr 5q6yk? Do they oppose each other?

The gravitational force on the Moon due to the Earth is only abou n k:yv k-c2ow892tmg4rmk6es *g /mvzt half the force on the Moon due to the Sun. Why isn’t the Moon pullesz8y:m4 mv9*6 ckmrkgn ev/w-t2gk2o d away from the Earth?

Is the centripetal acceleration of q3fjo qzx s3h1s9yz*- Mars in its orbit around the Sun larger or smaller than the centripetal acceleration of*33qzszysho1-j q9 xf the Earth?

Would it require less speed to launch a satellite (a) toward the 7gl5 wh cul.0aeast or (b) toward the west? Consider7ul g0l.ha5wc the Earth’s rotation direction.

When will your apparent weight be the d l8 :)j2raiqws(s-dg greatest, as measured by a scale 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 ir:jg)d2ssaqw8-d (lthe least? When would it be the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer sypk*r ;5 0(vjo 3xzpjbpace could be adversely affected by weightlessness. One 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 fob(o35jpyjr 0;pzx k*v rce we feel on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “fr4sqbi)eej .fy;n:n+s ee fall” or “apparent weightlessness” between snf qinesj.se+ b)4;: yteps.

The Earth moves faster in its orbit around the Sun dnyc4 nx 3:+u kcof*s1in January 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 producing the seasons — the m 1:y*s 3o4n +uccndxfkain 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 wasgx:y)y 5.tciu6tqtb1pnx z *r6ix:+d7h ejs7 discovered to have a moon. Explain howu 1rq *:6z. dt bixj77h65iyc:+et )tyxsxgpn this discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational puua66tg y1 cbt8ll on the Earth is 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 1t u gat6c8by61.10 m from the center of a merry-go-round moves with 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 o9ih/v0b 4 7qf urwhz- 1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 centripetal avyj5vtwwg-qz4 i +mf8il: 54tcceleration 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 m v-z5t4vw 8fy+q4g:wi5il j tthe Earth's orbit is a circle of radius 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 i8z3m7y9 2 l ohc(cfd z.vm1jyhs exerted on a 2.0-kg discus as it rotates uniformly in a horihl hm( m7cy dcj.y928 1z3ozvfzontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttlemjpf 478 dm nb9nvu9t4lf *lt* is in orbit 400 km from the Earth's surface, and circles 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 Eartt7bdm* *l mup4ntff 9vl98 n4jh's surface. $\approx$    g

  What is the magnitude of the acyu;eewli zcr4/1 d; .p t+g*wrceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if g. c;y*ti;l / wzdu re+4w1perthe wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a unifo8xjqwvt(06m sfyyv-p g);u5prprv : 1rm rate in a vertical circle of radius 72.0 cm , auxs6tj-qp;f g)p0 vpm8(r5 1:yvrwyv s 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 rob.v ;u+ /ds c)lnb*arkund a turn of radius 77 m on a flat road if the coefc d;sa.* /b+vbu )nlrkficient of static friction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficieyl/ep2zbp)xy dg y-9)ig,;wnnt of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of9n)y ;-b geydzw )iyp ,2glp/x 95km/h ?   

  A device for training astronauts and; *xpgpjpr2(sj s:4fx jet 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 times her own weight, how fast ispgxjprs( *;x2:sj4f p she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0z9,q, 5nhnvwv:vh e 1i cm from the axis of a rotating turntable of variable speed.w vh h nvnz1q:,,5iv9e 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 turntable?   

  At what minimum speeqk2o7ogq-vyz2q-v9knfp)m 0 4ohz2od must a roller coaster be traveling when upside down at the toppho -o7q9vo0n2 v2zf-yqmg o4zqk)k 2 of 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 950 kg (is4z(z(;wskc(fn+s uy ncluding the driver) crosses the rounded top of a hill (radiusu;fskz (sz (y4cn+s(w =95$ \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-diameterbs) hvd)roqt l)v s.c(t .2n*k Ferris wheel need to make for the passengers to feel “weightless” at the topmhvnrb* ( t. )dq scosk)tv)l.2ost point?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radi be: 2f)*cljynus 1.10 m. At the lowest point l:jnc) e2ybf *of its motion the tension in 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 particle 9.00 cm fr c acji9nv+*ko+dpfba/z.9o yelm7: * om the axis of rotatizp+* amyjlvio7f*edn :9 k9+/ca .c boon is to experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnivals5+o,/ :ebahio3ic vd, people are rotated in a cylindrically walled "room." (vd e5/ci oo3s,h:b ai+See Fig. 5-35.)
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 circle on a fritd9b rj(5a)tfuvdm7 3 ctionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a central hole as sho 9dmfdtr3bu t(7j5a)vwn 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,kdp50dpr3r /g/ 2mcho by not ignoring the weight of the ball which rep0c 3gr5hrd2d/kmop/ volves on a string 0.600 m long. In particular, find the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , 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 bankevh2vo7.a 3t an9ll +e(cmj* aad for a 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 masses yromuwo0baa p4; ucd40 e6k2($ 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 diving verti)id6cz8 g js+scally 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 :m7v+/ wensc2bn j+sacomponents of the net force exerted on the car (by the ground) in Exame+s7n avn+sw:j2/b c mple 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 uniformly from the (0t:fz9i wf/ nj wftf4pit 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 coefficiet9/fwif(jfn w0 4ft z:nt of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90 m . At a pij 6mhi3v(l8k . ,ogoparticular instant, its acceleration is 8l,op(i. ko6im vg3hj 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 Earth’s gravi-i98pg ,jftok8em3 hf ty on a spacecraft 12,800 km (2 Earth radii) above the Earth’s 3koemh p -jg9f, ti8f8surface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravitational acceleration g has a mxk x(xnh:9a. u3u 4 nw)nx(xrmagnitru:n(x 93 .xaxm(xw4hkn n) xuude 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 gravity on the Moon. Th- i5 77czkusv/mr yw/l8co7 rie Moon's radius is 1.74 5m7u/lc y/rzi 8-cri7 kvsow7$\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, butl/fk9jb2 w.h ga49chg has the same mass. What is the acceleration due 2flgh9j9cgk wh . /a4bto gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, zk /7wng i8rys -y68xmq8ovly e0bg- 5but the same radius. What is g nrnby8k7y e6wm05i-y-q8 z sgxvl8 go/ear its surface?   

  Two objects attract each other gravitationally with a force of rx fga3tny/ */ 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 , themwhncyix va(h2 /1 -4 iy:viw6 acceleration of gravity, at (a) 3200 m ycaw-hmn2 ii:vxwh1vy(6i 4 /   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earg4qecd 8u(g;ouy2wp0t( se1 wth where thegravitatio 1w 4gey(g(ou0u ps;d8wet2q cnal acceleration due to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has0-lfvzzs4s*j 6e )sq s five times the mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on thil vss*zs6 f 4-ez)qjs0s monster.    $\times10^12$

  A typical white-dwarf star, which once was an average v::g+ s8ba noistar like our Sun but is now in the last stage of its evolution, is the+ o8isa:vg:n b size of our Moon but has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightlesnf3*i( pj 2qsc *.femdc3k3durz5sb 9 s while orbiting in the space shuttle. Your friends respond that they thought gravity was just a lonb m3 erc*.d29d 3i j35(kc fsus*fzqpt weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are located at the corners of a square 8y4 j y5jqeuojuw,z am( +n8q4of side 0.60 m. Calculate the magni (ya4 w5 8umu +jjnjeq,4z8qyotude and direction of the total gravitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planets lid5ko 1luf ;1lcv tby-zs+z(;h)tr xm2 ne up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5-38). The masses are +lsmvckf (;1udzl;) tbz-hy1x2tr 5 o$M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , 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 surflj wvxwz8 szxm(s 6q./e z.5/yace of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the mass of8q.m yz(jwev/w65s lxs.z zx/ Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value a2memodezt8tv: :l9 ( for the period of the Earth and its distanze(mdm2::oa el t89t vce from the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellil6t3d d 6j td*ab8jb0tb4yq5dte moving in a stable circular orbit about the Earth at a height q068db3tt 54ddl *ja b6dtjby of 3600 km.    $\times 10^3$

  The space shuttle releasezsv 97wq ua6*is a satellite into a circular orbit 650 km above the Earth. How fast must the shuv7z uqi*swa 96ttle be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceshiu/kfv6(sk 36hyynzso l* 5g9 np rotate if occupants are to experience simulauv( fny/lyhgs5 63k*snz 9ko6ted gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit theasqh)0 7ne8sxj(z.dk 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 the satellk)z.hq7js(asxen8 0d ite?    s ~    min

  At what horizontal velocity would a satellite have to be launched ea804cqz*o x7xev4b* yruqrff4m3 g0from the top of Mt. Everest t c7e4* bxryqafzg8me 0*vf3ur40oq x4o be placed in a circular orbit around the Earth?   

  During an Apollo lunar land y; 2vzw,ln;miing mission, the command module continued to orbit the Moon at an altitude of aboutv n;wmlz 2y;i, 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of iceaf3bazozdiw(w+n7 f h3b- 8vw wym0/; that orbit the planet. The inner ra w i3hazaw8dvyw7z fw;-0 /b(3 obnm+fdius of the rings is 73,000 $\mathrm{~km}$ , 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. 5h3vcx/:iay9 i –9). What is the ratio of a person’s apparent weight to her real weight (a) at the top, and (b) at the bottomv9/: i ih3cxya?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from thj .j*oaofnqo411xfhxkma 3 4+e center of the Earth's Moon in a space veh4hx3m+*ofa1ano. k 1 j4qx jfoicle (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 systeu1 ( gw. risf(9e(fppnxo+7zcd1 fry.m consists of two stars of equal mass. They are observed to be 1(7ffp .e .icur(s n gfpd 1wo+r9yzx(separated by 360 million 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 de*0 v:5ne *taaz7sq ffor the weight of a 55-kg woman in an elevator that f*z5e* svd07et:anq amoves
(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,7jh b tgl :ilz8o,8spo(oz tsa-2b 6u of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What wa,zt-bjp 6t7oo2:ls s8, bh8i l gza(ous the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very nearm)t+jfxqifid- 5.fg: the surface of a planet with period T , the density (mass/voldm. jiqtfg:+f5x) f-i ume) 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 and the period of the Moon (27.47pfg,lra:*vfc n: d/ z d) to determine the period of an artificial sld, nvpcaf:zf *7:/ gratellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, orbits t wzno.1z1fc0n9 jt:l khe Sun like the planets. Its period is 410 d. Whafo9zcjw0t: nk1z .1lnt is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distance of +v3ptgu00kgwwpau+ 10ecv4fk ,enu9 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 ymt3l lt1z4kkzax u.1 y;8ia;nears. It comes very 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 ;zal1knxt1.mt4 u83 ;zikyla 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 Sun.]    $\times 10^6$

  Our Sun rotates about the center of the Galax/5bi.sv:xay j y $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ 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 (utkwlu;b+4 w6 2oopcmoons of Jupiterco;to+ kw42(wb6p lu u (those discovered by Galileo 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 the known period and distance of th3+c2ea+jf9yvgp n fz -e Moon. f +ave2cy93pf jg+n -z   $\times 10^{24}$

  Determine the mean distance from Jupiter for each o fpvsv; arl8x99+qfos g28of .f Jupiter’s moons, using Kepler’s third law. Use the distanceox 2vfp8+ gos;f9.rv f9qa8ls 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 Jupiter consists of 2 n0i:frnm m;t.d+cgwmany fragments (which some space scmn2d;0. mrt:cnw+ gfiientists think came from a planet that once orbited the Sun but was destroyed).
(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 "planet" in the for2pm:,kt 2eegv 0 qun5am of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). 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), ak vena,5emu:0q2t gp2nd 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 nwv nlm2ksf v5z56+s/arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximumk +wsv5n /2lnsm56zf v speed he can 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 gravity be half whatj1ye wo,i+k vf6p+ez; it is on the suy; +w,6koi 1vz+epefjrface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a muo*hw 6.p h9yo0(o*makezkdo7tual circle once every 2.5 s. If we assume theirpzk wao0 m97hoo*y (eh*.k 6od arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once p ua*;4 ilhf3zve:nv6 zer day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earthalnf*6zu 4v:z;hei v 3 didn’t rotate. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Ezaba5yzj33q, mv x,x5arth will a spacecraft traveling directly from the Earth to the Moon experience zero net force becausev5y3bj, x5 za3, mxzqa the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, b 2gie(;wt6k at 1.hxsmut when you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, and in which d2;ht.m 6a siktw1e( gxirection?     ----待选择----upwartddown 

  A projected space statiov+/amg6 yz 9+ ytjr7i6oym+ ern consists of a circular tube that will rotate about its center (like a tubular bicycle tire) (Fig. 5–42). 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 y+6i7vr+m6ar tzymg9e/+ yojto be felt?$\approx$    $\times 10^3$

  A jet pilot takes his aircraft in a vertical lo8 s7x;w1enxvv-ic: ycop (Fig. 5–43).
(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 o ;hrf0/57 qh5vnu5 rtyi:im oof its radius r, the acceleration due to gravity at its s5m0vr;5roi:y n7/ hou5 qhtifurface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from the vertical x-qjk6 fw ,/1lrtw (dtby an a l/ttkd16q, wx j(w-frngle $\theta$ 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 m is banked for a design speed of If the coe hzg5l- z9:wpvfficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handllzvpw-5 gh9:z e the curve?

  How long would a day be if the Earth w*u xf9amfq;**f1t 5+ 6.kquefnanwaoere rotating so fast that objects at the equator were apparently weightlesuuaaw1 n;q.*fa nffemx+ q*f6 *9to5k s?    $\times 10^3$s

  Two equal-mass stars maintain a constant distance apart of 8vlieu7lrcmpli(nbf *69s :k2e. d *nkr y)a( 3.0 $\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 speed rounds a curve of radius 235 m. A lam3 cxve;v.nho* p suspended from the ceiling swings out to an angle of 17n vco.h* ;3evx.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 beew e0ky5*sl(s 57dqu 6trn)xt* 1odvybn claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people can't survives(toxlrwqt 0deuyv1 )* ks yn*65bd57 more than a few g 's. Calculate the number of g 's a person would experience at the equator of such 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 Hubblez )qad8r6u m4o Space Telescope deduced 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 measurinauoqz6m 8)4rdg the speed of gas clouds orbiting the core to be 780 $\mathrm{~km} / \mathrm{s}$ 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 9rur+uh;c6cj as it traverses the hill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting 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 g ;rhu9+r cuc6jo without losing contact with the road at A?

  The Navstar Global Positioning jjudb+ no*sv z* 6q2*eSystem (GPS) utilizes a group of 24 satellites 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 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 ejdu6*z *nj 2vsoe+qb*ach satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous )wteexm( t*9q(NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km . Erm(w)e 9xttq *eos 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 c e;drb6*g 3sl,8ta tv4i9mh7 idjes2space shuttle pursuing a satellite in need of repair. You are in a chrism9g*3tc bet dj4v ;78ldie s ,26aircular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it.
(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. (a) What isrkff5l :d)nh6. /4 lo 2efoih/,ci 0ytom r)kl its mean distance from them6l4rfy/rd. n02 k : fh)o) 5tlio/,choifekl Sun?   
(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 if7ulcjjrrsx,39j- 8 gi you could actually "feel" yourself gravitationally attracted to someone near you. Make reasonable assumptions, like Fs,9l -ijcr3jjx8 rgu 7 $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center 0 jn-jtvwm/,43vl8m x*q- ipxlp) pvzof the Milky Way Galaxy (Fig. 5-46) at a distannmq0*lv/j4 lipx ,pw t)-8pjzxm v3-vce of about 30,000 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 cofry3y7t7 aw26) jk ze,imlar;uc7 a 2frners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg ma6a2f2wkymrr); u 7 az7e3fajy,7itc lss placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbitssbqb3gn c 6s+r( n*n)00ouawy 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 exw )/g7v)9 qbts v.w0i (:njaqa b5awnkperienced by the tip of the 1.5-cm-long sweep second hand on your wrist ww g0.9vkn bvbjw ()asqaqwn5i:/7)atatch?    $\times 10^{-4}$

  While fishing, you get bored and start to sd(3kj:julhc/4 zj q1dwing a sinker weight around in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is thelhuj(jd:/ dj 31z4kq c angle that the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a .ifzk-gf7xqf099 -h y+s: ihfpe9 aeonew highway is designed so that a car traveling at speqgy0 7o hk9sp99 z-:h- fiefeff.xa i+ed $ 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, * edoa0b5 wd(ltq4 ;ez l9mj5mthe Acela, utilizes tilt of the cars when negotiating curves. The angle of tilt is adjusted so that the main for(l 0t5oj;ea*wlmdzbm 5dqe9 4ce 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) Calculate 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.$\approx$    $\times 10^2$