Sometimes people say that water is removed from clothes in 0zq-dgrvnl 62 a spin dryer by centrifugal force throwing the water outward. What-2nv6rzd 0q lg is wrong with this statement?

Will the acceleration of a c,6wsg ;kxqh-m ar be the same when the car travels around a sharp curve at a constant 60 km/h as when it travels around a gentle curve atxmkw6 hqs,-; g the same speed? Explain.

Suppose a car moves at constant speed along a hi j5u5+0e +d-flrkon pylly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a levellfo5ur kd+ny-pj05 + e stretch near the bottom of a hill?

Describe all the forces acting on a chx4h w, cbp5( fgqg04vsild riding a horse on a merry-go-round. Which of these forceg,0w 4bgcxf pq(54hv ss provides the centripetal acceleration of the child?

A bucket of water can be whir cfjlzh5/g(ytd.*zg.y0 3wh aled in a vertical circle without the water spilling out3*a5 wzh c.hg(0yty gljf.dz/, even at the top of the circle when the bucket is upside down. Explain.

How many “accelerators” do x +3vtw+,+ oxxhpp ryo;2kx+ nyou have in your car? There are at least three controls in the car which can be used to cause the car to accelerate. What are they? What ac3wn,o+xk 2xp+vx ;+hty pxo+rcelerations do they produce?

A child on a sled comes i .2i,v 3yj ,byax2jzdz y)mx3flying over the crest of a small hill, as shown in Fig. 5–31. His sled djz2y, ,3a2zii yb.)d3mvyx jxoes not leave the ground (he 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 hhr)j9ab;v+alean inward when rounding a curve at high speed?

Why do airplanes bank when they turn? How would you compute the banking anglzs p0 qliy,sp)p)xa 07l7 dbl:e given its speed and radius of pl7lp,x dq)00i7s) bsypla:zthe turn?

A girl is whirling a ball on a string around her head in agbqzx: d8,rj lue+r/, horizontal plane. She wants to let go at precisely the right time so that the ball willz,:du8+/g bx jlrqr,e 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 Earth? If p*/etam*.jux2paka7a/r6h b so, how large a force? Con tjrp.a e/amu6h2px a/*ab*7ksider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways woulom.muq/uca, 0d the Moon’s orbit umqa.co /,0 umbe different?

Which pulls harder gravitationally, the Earth on the Moon, or the exy 73bo fif( lph (m212sj:eus;r (ttMoon on the Earth? Which accelerates br(y fpm;ehs(j 2o13:l e7f(2suxtit more?

The Sun's gravitational pull on the Earth is mucyfkqal:stt:ei/*,h5th8 9 tf s sy /l6h larger than the Moon's. Yet th: f la68sit*ytfhs9se5 kt /ql,/hy t:e 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.]

Will an object weigh more at the equator or at the j *g ;9ezi+z r(ugl:wmxgkn71poles? What two effects are at work? Do they oppose eek r7mw zjg(gu1*:z +ig;9xln ach other?

The gravitational forkt+dtxg 5:* eam6ib8l0)uw d uce on the Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn’taek5tg6w*80 dul ti :x)+bd um the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larger or/c)yq +c5z7+ kj3emlf c-e dvx smalley cc/7)+ 5 mzdc3q+ke- lxfevjr than the centripetal acceleration of the Earth?

Would it require less b/m +kzzcv9b/,n ,eitspeed to launch a satellite (a) toward the east or (b) toward the west? Conside/zki9ze+bbcv ,tm n,/ r the Earth’s rotation direction.

When will your apparent weight be the greatest, as measured by a scale in a m5yx jlfg f4p( qi3) vw8l:jez6oving 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 x 8gj(5e y4i)zf ll: w3jv6qfpbe the 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 perqyf*iyi1 y y d/2ys57niods in outer space 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 force qyy5dy1n/y i7i* y 2sfwe feel on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “free fall”p4g1 el wfv x1.k1b l.gl62gte3sxh0a or “apparent weightlessness” between s11.kg0 e26l ht l 4xb.lewgafp3vsx1g teps.

The Earth moves faster in its orbit around the Sun in January than in Juqq:6.t fo.bup ly. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is.ou.bfqpq6:t not much of a factor in producing 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 was discovebx xyr /1 3l*tc-qklk5red to have a moon. Explain how rcxy 5tllqk1*kb -3x/ this discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is mu;hg4nsyo75wa9 yzh/m ch 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 th 4 hgymzawy5o97/hs;n e other.]A child sitting 1.10 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 1980 m0yn23s ik+4kpshh .o $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 acceleration of the Earth1 q yu4n9w wo8;+psaz md5/mcu in its orbit around the Sun, and the net force exerted on the Earth. What exerts thmw d1zo; cnsq85/4 pyuam+9 wuis force on the Earth? Assume that the 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 2w;x (- pcw 3+*uxmqgjo10 N is exerted on a 2.0-kg discus as it rotates uniformly i-op c( g+wmuw;x*qj3 xn a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shcc-tlw7sj6e r)bidt)1,l 5 h guttle 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 your76tclecrg shi b, wdj) -5t1)l answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the ac;;:z:gzy w2nokasc97za.j jiceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) ifyg;;iza jj sa2:nw:oz kz 9c.7 the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform rate in a vertical circ r0gbpm9 ry6bpy d(d68u7ave8ktb 7+ale of radius 72.0 cm , ad+(8k7u a6 d9m7pbavpbgr8yerybt0 6 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 wit u/l s,-nim0meh which a 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is this result inl/0-imn,e s mudependent of the mass of the car?   

  How large must the coefficient of stabc p 7qb+9pcuvs*7/mdtic friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95km/h ? +cbvs*d7mb7pc 9/ upq  

  A device for training astrona/k 7zdzei8wk0bvl lf9 eur236 uts and 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 is she2 0wk6uild z78fvbe9ek/l r3z 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 rotatingktdhp3; x:w,o sc: h/b turntable of variable speed. When thkx::t3h/ocsw p;hd ,be 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 speedow9*bj2lixne7zy;g y*w4 /y l must a roller coaster be traveling when upside down at the top of a cznwye2 *ljowbi*x7 9lyg/; y 4ircle
(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 (including th13m 4ictr yye3e driver) crosses the rounded top of a h34ci3 yte1myr ill (radius =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-diameter Ferris wheel need to maxpb2i* vs63 ov a3d veq8qisu:y1tf9l1y t 4k)ke for the passengers to feel “i62klatu1y3vpsbq3q 1o)4y8 tvd *v:e9f x is weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a v tjt 5 dqvkuo6rsano/d+3 78xms2(w1wertical circle of radius 1.10 m. At the lowest point of its motisuq 2asm1j+o n 7ovdk6 3twt(rdx/8w5on 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 rot+stjxq5e ,hgd:cg1, late if a particle 9.00 cm from the axis of rotation is to experience an acceleratioh +jt ,g1d c:l,5qxsegn of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotal1b xlmp -rqq. *ulu)j0 7qdcg6*o:u lted in a cylindrically walled "room." (See Fig. r- bqmul*opl0) * 6dgxcujll 7qu:q1. 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 frictionless air-hockey tabzou; c05k*og stmc0 f2letop, and is held in this orbit bco05fgs2um0*ct; kozy 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, by not ign/m9s1eqj rj* u1gs kv-oring the weight of the ball which revolves on a string 0.600 m long. In particular, find the magnitu gsm11uqj/k-re* 9jsv de 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 c b*momn5/ k i3e5pri h;l4gh6m is perfectly banked 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 9o+.z*fmhnt u $ 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 maneuvi:is,s*krld4ezl vd4xr 8 ;qn9z ik4m( pd9v) er by diving 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 centripetala ar.)j4iml;zamh)0l components of the net force exerted on the car (by the ground) in Example 5-8 when itsjmi) lalr0;h a4 .mza) 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 pit area, going fro ulwg/+.)zl hafo q80im 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 road to provide this acceleration with no slippingz/ .8g ouih)w0ql fl+a or skidding? $\approx$   

  A particle revolves in a horizontal circrsde36 a vly 95,9xp:uig rpc6le of radius 2.90 m . At a particular instant, its acceleration is v 3xyrgc:, p i9dp e69rslu5a61.05 $\mathrm{~m} / \mathrm{s}^{2}$ , 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 gravity on a spacecraft4n90 e*z0v: se6 qtq7dokytl k 12,800 km (2 Earth radii) above the Earth’s surface if its mass ilty7kq0nkv t *9z:0o4ed sqe6 s 1350 kg.   

  At the surface of a certain planet, the gravi3 9k p z4y3vxx:w .vvcslcabsx:hv 1-4tational acceleration g has a magnitude of 12 m/scvh-x:.x9zl 314ba yvvk ss3v4:x cpw$^2$ 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. The Moon's radiau ak(rw (9knzal2.o 8us is 1.74 ka9lowa. 8anu zk2((r $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 timhm05qt osq e +kgem08/es that of Earth, but has the same mass. What is the acceleration due to gravity near io8h keet5qq0s0/ mm+g ts surface?   

  A hypothetical planet has a mass 1.66 times that 0lo5h+, l4umu -s,rqcxry, z*m kgf5pof Earth, but the same radius. What is g near its surr+,r0l554 hu*s,ofqz-l ymp,cgu mxkface?   

  Two objects attract each other gravitationally with a 6za ct kylt+ om*,8ya1force 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 , t 9c -lsjt41rmisj -8ethe acceleration of gravity, at (a) 3200 m94ts--t1emlcis8j rj    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth wn we.zg sa-sjvr 0(: 2 goj168rdu2jijhere thegravitatijjar2ug62:(8re dzniw10v o-sjs .gjonal acceleration 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 mass of our Sun 5ur pothu9 p.)packed into a sphere about 10 km in radius. Estimate the surface gravity on this m prh5uu9.t)po onster.    $\times10^12$

  A typical white-dwarf ;u *fslppg,9cq ug 9r0star, which once was an average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on90sgc 9lgq, rp;uup *f this star?    $\times 10^7$

  You are explaining why astronauts feel weightless while orpz2,ons 9mi 6sbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yo iozmss6, p9n2urself 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 of side 0.60 m. hmj9hvx,pyn; q:f( err.7h c -Calculate the magnitude and direction ofj,-h7 9hr(y cem.qph x;frvn: 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 line up on the sam fpj-* xc ,;ias9ps;c l*2wpmg x-p*wie side of the Sun. Calculate the total pam- wcp-2i,il x ;s *p;*g9scf*pxjw force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets 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 surface of Mars i nxkda*c(bf0:wom -z,s 0.38 of what it is on Earth, and that Mars’ radiusfd*cx:0onbwz ( ,-mak is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the period of the Earthu )nwwy91r,r 1altid0 and its distance from the Sun. 1au)trwrd1li , nwy 90[Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellitegqk jxfn34lvn.(;n.g moving in a stable circular orbit about the Ex 4nvnkg.nl ;gjq3. (farth at a height of 3600 km.    $\times 10^3$

  The space shuttle releas 1wpp os xa p(7l,5ul6(xz9ujles a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurxpl,p pw9j7u(oa lz(5x1s lu6s?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants are to eii wzy*k lz 6c4-4gt6jxjgpr4 a54 r+q+vkx6vxperience simulated 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, Fk4gv*k64 ly 4c +iv54zix6p-rgtxaj + z6qrwjig 5–32.)   

  Determine the time it takes for a satellite to orbit the Earth in a circ2y;ws( comog,ular “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 radmoswo,cyg(2; ius of the Earth. Does your result depend on the mass of the satellite?    s ~    min

  At what horizontal velocity would a satellbx8dj8 ;pf8iyhz7d z :t 5i*laitk5w,ite have to be launched from the top of Mt. Everest to be placed ind j7wyt;8 5x :8, a hidzpikifl*bzt85 a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continued to orbit thcuke*3lfp f /plaw,wt 3o/s+ 7e Moon at an altitude of about 100 km. How long did it take to go aro3l7wlwp t, fe f3 u//sockp+a*und the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit the planl4 s4)i/cmw3r .jx dozet. The inner radius ofrclm)z4siwo4x/. d 3j 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. 5(+qi gd3k5s;xfnht,ddnx66xvk :j + r–9). What is the ratio of a person’s apparegxsi;(fvktk6dx +qh5d,jn n +36rx :dnt weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent 3p.adcwr3sf (fb1fp 1weight of a 75-kg astronaut 4200 km from the center of the Earth's Moon in a space vehicle (a) moving at const3prfb3.cf (f 1sp d1awant 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 me1k69 ixo4ex qcz;h3h,6 nl cdass. They are observed to be separated by 360 million km and take 5.7 Earth years to orbitihk qxe9 nl oezch1 36;46x,cd about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read for the weight of a e hs7 y+mq1 ig-a4v8826aeddf yrrut- 55-kg woman in an elevator that m mg8aq8re7t f2+4d- sv yie6ua1 d-rhyoves
(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 ofhyg43 gn co /69gxk h1 4-sh9bec5cpjb an elevator. The cord can withstand a tension ofk 5 y4/csjp 4gehh c3b6b1nc9gh-9x og 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a plnuwsf(:d6 o0lanet with period T 06d(nusf o :wl, the 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 and tyarwnq 4fm.82ex)k l 7he period of the Moon (27.4 d) to determine the period of an artificial satelliew)l 8ka4ry. 2qfx7 nmte orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, oz1j -vtd*,x kkrbits the Sun like the planets. Its period is 410 djz -kx,t*dvk1. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an averagenwjwv xx3c )b o62( g/cu-*egz distance 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 very s,dqj 3lv em r:nq)q1+ts1q8vclose 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 farthe8j l vmn:+qd es1s13qrvqtq) ,st distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center of thepo/h 0h b,mv3l Galaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ 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 f 7. w mw+6en :d71l2.mnnzqwgjw a0dgror the four largest moons of Jupiter (those disc1 ngr6ww. .wm:dn7jm 0e an +7lzw2dgqovered 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 th-) ffjpv dn1/5wezy 2ee known period and distance of the Moon. y/f2 wep e -51ndv)zfj   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, using82(m h )hwx;1of3;wmvfak3ormecnvh :k/r)e Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the values in r8k3heon mwef;) arwx 3ko1h)h;cvf/mm( : v2the 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 many frsj b :g0. .ahb8c1gmlxagments (which some space scientists think calamsbj:0xb ..g8 chg1 me 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 artificrqquy )x6loq6ipx 2eo:5p o8*ial "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 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), and t2 6peq5u)q lxio8xpo:r oy *6qhat 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 fr;*b 4ewz6hg raom a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N onwz6 a4*bhe; gr the vine, what is the maximum 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 thew sx(jk0we /x8 acceleration of gravity be half what it is on0 wswj(8x/x ek the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mu *bb9ige v/;wq d*t6u7pf9 mtrtual circle onctu ig*wbpqe/r9fmb*6 td ;97 ve 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 pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent accei v)/gm4z ; g2s3xgyeeleration of gravity at the equator is slightly less than it would beex)g;/ey32i v sgz mg4 if the Earth didn’t rotate. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth wn k6rm9cwb.)skf5;q q6wn5y rill a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forces?;c9n rr6yq5knbf 6w5).wskmq    $\times 10^8$

  You know your mass is 65 kg,ioh6)te-kbprbb93soq (2 v6a but 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 whichr6t(ao9ekv6 2bis p)-q bb3h o direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tube that will rota ttgo/5 x ;(nu0ldyn6mte 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 r/(tn l; u6ntd05 omgxyotation 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 ai7flm w7dev*oz26f44k ylr rk;rcraft in a vertical loop (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 massok 5cmlm;p9/u of a planet in terms of its radius r, the acceleration due to gravm;p okmc l5u/9ity at its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) isde hryb/hrllv ((v(80 deflected from the vertical by an angyervh b(0 d/(vl r8hl(le $\theta$ 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 dpq)v6xyl 4ie2esign speed of If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the cy46p 2 qlxei)vurve?

  How long would a day be if the El-y+v ;wr 0teearth were rotating so fast that objects at the equator were apparently weightless?l-0yere v+w; t    $\times 10^3$s

  Two equal-mass stars maintain a constant distance apart of 8.07w / xxmtdw7) -a*7cth ftcm4m $\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 x75xss u) ,.zrpn) jaopj,e,oconstant speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swings out to an angle o7,j)o , o5xs) re. s,upzapxjnf 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, iby7rf/ (vd9 ust has been claimed 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 such a planet. Use the follo7(d uy9srfvb /wing 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 :swx jmi si3ac-o7v1e+c4ih+Space Telescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense isa47icev cx1:w3 h si- +mj+othat 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 $\mathrm{~km} / \mathrm{s}$ 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 con z)ibkrbztm 2h4(udc5b( t4ulyl;67+npx0mq stant speed 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 au nrl(+0i4tm4czhp6mbb lu)2( q bk57z ;xtdycting 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 p +98qev. mlbnbqe--b System (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 determ .b enmp-v+ql9b8- qbeined 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 each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroidpk6*;. trqn0hp2evg h4)ycfv Rendezvous (NEAR), after traveling 2.1 billion km, is meant to orp 0chp6)e2vrnq4ghf*.y tk;vbit 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 sp2y aab4ug1:hr+i u r8oace shuttle pursuing a satellite in need of repairibu:g+ ruo82yr14 a ah. You are in a circular 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 30000j splfz(.3pji3 j3mp years. (a) What is its mean distance from thz3 j(3l p.sp0pmifj j3e 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 if you could actually g*jr e0us0/*mxivntd 7 *- xya"feel" yourself gravitationally attracted to someone near you. Make reasonable assumxgy* dt unme i*vsr70-j/a* 0xptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around:)4vz z/vx haqweeh/0x;k;,frp3 ik s the center of the Milky Way Galaxy (Fig. 5-46) at a distancp , 4rzek:qzvv h;hek/; /3wf)0 xixase 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 9 x,db gab;iv,6hpgb l+ol43b at the corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational forceb;d p+ob ,9b3hg,vaxi4l bgl6 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 551.6ii8dg5po lir v)s vqr,kd,00 kg orbits 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 experzipd2c6t; wq 3we8 mjj jcw33+ienced by the tip of the 1.5-cm-long sweep second hand on your wrist watch? i +w3jm ztwj3ecp23w; qcjd86   $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight f.ag- ;-enguh around in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0 uhneg-.ga;f -.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 d9 7 u5ddzv:tezesigned so that a car traveling at d: u5zetd vz97speed $ 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, utilizeiuyc(qb0a 26 96ir t*am,tlo:9k * juxczzs-ks tilt of the cars when 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 ). * ackort 9j6u-c6 iu ,(bazlk9sq*ix2ymt:z0 (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.$\approx$    $\times 10^2$