Sometimes people say that water io.h pzg+*:og n e ji:)fq;s.kjx7(fsns removed from clothes in a spin dryer by centrifugal force throwing theofqs es .oj+.:7iz;nh()*k gpf xjgn: water outward. What is wrong with this statement?

Will the acceleration of a ezt+aa5spf1ga. ,x+q car be the same when the car travels around a sharp cx +ps,e qtf.ag+aaz15 urve at a constant 60 km/h as when it travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a hiho3c lwra 0ud9e -s(v-lly road. Where does the car exert the greatest and least forces on tsdl ovw-(u 0-a9 c3rhehe 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 child riding a ho s*v6 2gs..2en6e fuzsc7kd tmrse on a merry-go-round. Which of these forces provides th6 2v cs zsmtd7. e*skfg6.eun2e centripetal acceleration of the child?

A bucket of water can be whirled in a vertical cirwj zsx17i(2 hrdu 4c5hcle without the water spilling out, even at 21ih75(drs4zhc w uxj the top of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? Therve9pz jii5f 9v(otjid8 ) (6bue are at least three controls in the car which can be used to cauvej6zp8(du9 bjii tf i)9(ov5se 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 iqg9 -hx)lun) cn Fig. 5–31. His sled does 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 Newhl) -u9qc )xgnton’s second law.

Why do bicycle ridersuc)jbbd ih15d//7j0e ywln s1 lean inward when rounding a curve at high speed?

Why do airplanes bank when they tur nix9audt*b(( ger1y bn 4;f5wn0ke-t n? How would you compute the banking angle given its sp nx* f4y1n5-bgw90dr(kt(e;itbu n e aeed and radius of the turn?

A girl is whirling a ball on a string around her head in a horizontal p k*p.tfrre9e ,lane. She wants to let go eft* 9p, rr.ekat 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 gravitatiokp vqk77 mhvr0g31 ie8+ xme2+afnnne5m/3bonal force on the Earth? If so, how large a force? Consider an apple hmvqe/7 mp+ 250g8me1v3i f7nx anbokenr k+3
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways woul3,cz w 7 j;b6xnbzh0rud the Moon’s orbit be difhr ;zu7xjcnbz3wb06,ferent?

Which pulls harder gravitationally, the Earth on(s/k 8c5tl y mx8s)nf i/fe:eh the Moon, or the Moon on the Earth? Which acc x8tsl8s/e) k:y/fme hif5 n(celerates more?

The Sun's gravitationald0)iyydb4ng3+ aw0t n pull on the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Congt0 i yw)nyd a0+4nb3dsider 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 porba n 35b3tkt(.uj6l zles? What two effects are at work? Do they oppose3 z 6 b(3bln5t.ajkrtu each other?

The gravitational force on the Moon due k x. x,jwwe.;2n8 ( dn+klwlddto the Earth is only about half the force on the Moon due t.kwk (xllwd, 8ndn;+x.dj2ewo the Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit arounrpnv(ew,s 3j hu k5j,;d the Sun larger or smaller than the vp 3,ue,kwn ;rhjj5( scentripetal acceleration of the Earth?

Would it require less speed to launch a satellite (a) toward thetyr0,pm6,n8vup v cs 4 east or (b) toward the west? p t8v0c,y,n6msvpr4 uConsider the Earth’s rotation direction.

When will your apparent weight be the greatest, as measured by a scale in a fs biij*j31hb5t, hy4moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in fb,5fib*h yij13 ht 4sjree 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 ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long period xz6js0 l8-euss 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 j-s8 6zlusxe0astronauts 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 “free fall” or “apparent weigf5wo by-n 9phh; t:zc6htlessness” between stepc 6;h p zf5wothn:yb9-s.

The Earth moves faster r dgmc*fq,b.t*g +b al6q-z:d in its orbit around the Sun in 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 main factor is th-.b,glz fr amdb* *td6qc+qg: e tilt of the Earth’s axis relative to the plane of its orbit.]

The mass of Pluto was not known until it was discovered to have a moon. Explain kn9a.gux(b;a7m3w n p28 i6eea8 byo vhow this disuey8 amp 3o 6 abxakgenn(wv9728ib.;covery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger than the Moow4hv py8+fhe56 h9gj tn'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 8he4fg ht5jw9yvp+ h6 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 19-ie(d(eqtef 680 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 Earth in its orbit ar + v o10veqd 0c27fm9qnbvep-wound the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth'1en 2ove+q9w-bdfmv0 cp0v7 qs 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 is exerted on a 2.0-kg e f;r:(t ylpvvub5r(8discus as it rotates uniformly in a horizontal circle (at arm’s length) of radius 5vyu:t (l rv(pf r;8be0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km fafnp4vhxu ,j/r r. 0h7rom the Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of 0nxr4hu7.hr,a/ vpj f 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 acceleraxx4) t3e zc-,pjg+ibstion of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the jiz b3xtp-,) ecxs4g+wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform rat5-v8d0 dl *rix b0 jpef,mudsuyz6xm +c q*a63e in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its zpud5rc0m e3mvidjyalx*,66- x s *u+q8fdb0 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 whicq5+ 2vfuz mf9d9pxzx*a 6url xi3 d62gh 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 urx 3fi26m*zx96zpfq l+ 5a9 xgduvd2 0.80? Is this result independent of the mass of the car?   

  How large must the coe,u 1ehdn.q og)fficient of static friction be between the tires and the road if a cag nd1qu,).oeh r is to round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jetc y+dv xnkg3:*rp:0qh 6ehlnyyl b97u9k,q0 u 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 hn6 kn7ey ,ryc:qqh09 ypgu 0+:b u3vd klx9*lis 7.85 times 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 froutylo i,0yr1( m the axis of a rotating turntable of variable speed. When the speed of the turntable is sy,l0ru o(i1yt lowly 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 speed must a roller coaster be 0p g5vr9 x :le2i;bq8;lzn.uc0zh abwtraveling when upside down at the top of qp0:0lglvecb2.a zz ;9h x5i;unrbw 8 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 (including the driver) crosses the rounded8.k;hgxudov y9n:; :z,5ahrsk9ts y p top of a hill (radius =9ks; sz9:.y:todak5h 8; gxup,rnvh y95$ \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 vngn2 tt0 wzy 5e4uo* tc8*e9eFerris wheel need to make for the passengers t 2w9*uen0eot cgtt4y 8n*5evzo feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a verticad/l8 5 dbi :r)i vw4,vxi)saffl circle of radius 1.10 m. At the lowest point of iw i) b)i/dal vi8dvrff4:,s x5ts 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 cel v ewympc8yq2 e-j3b,j+yo8n7 r+q1nntrifuge rotate if a particle 9.00 cm from the axis of rotation is to experience an acceleratil3+ jy82+oen8qym qwy v, 7- rnjbpec1on of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated in a cylindricavr t6jk6p,t *acx3 7vklly walled "room." (Sxrvk jac6,vtp7kt63 *ee 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 frictionless airc*u kj:q; ua-r-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a centukaq*:jr -u;cral 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, +dkr6a,he,uwgbx o59s68o i*,txt hjby not ignoring the weight of the ball which revolves on a string 0.600 569h w8 ue b,hkt,,6xajorgi+o* txds 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 banked for a car traveling 75 17rgnrlw-4c v$\mathrm{~km} / \mathrm{h}$ , 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 massesz ps0tyx-.3 ln ,gwrt0 $ 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 maneuvne79 f6gz9lymer 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 centripetal components of the net force exerted wgxn( 0;dbw: vtw5: ap eral94er+ g.won the car .5wea;er4lpw+rg0 vwnbt: ( axg9 w:d(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 ac/riv ob;/l lz5w k+af-+o e*wxcelerates uniformly 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 ilex;/-++fwao bikz* / 5vlorwt 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 slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90so93u5axro6:i w j 8je.qm1tv m . At a particular in v9j w.musi o5r: a186xt3ojqestant, 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 Earth’s8sx0k4rl,pq 87sbby5x 3wn7rr-cuf b gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its bf0bx4r xcn8 7ls 3wp ,rsu-qkb857yr mass is 1350 kg.   

  At the surface of a certain planet, the gravitational acceleration g has aq kj/ 1/pwgf2w magnitudgk/ fwqwjp 2/1e 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. The Mbq2t 1f 1k7xs(9mdo( w p/ghstoon's radius is 1.7721gqsd m1stk (bx/fwoh t (p94 $\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 tha9:qac,k0oe * opiv1v xt of Earth, but has the same mass. What is the acceleration due to gravity near its surfavvkco *e,oi0 x9qp:a1 ce?   

  A hypothetical planet has/qc2hshrecr(8bu 7 8 l a mass 1.66 times that of Earth, but the same radius. W( 7sblqce 8rr2h/ c8huhat is g near its surface?   

  Two objects attract each other gravitational ec::gh(cnfxty89y:mjs- es61w - enply 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 accelerasve5kxv 4* : /thmwlsv 8,sav8tion of gravity, at (a) 3200 mawtem*h/45sv 8,:8s lsvk vvx    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth where tm ti2-j)v 0zy0g-xu mugb w19whegravitational acceleration due wuw1mvj0g 0im)2but- 9 xz-y gto 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 /mh f2;ydppk 0packed into a sphere about 10 km in radius. Estimp/ f0p2;dmyk hate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an average star like our Sun 11s62w .kqzmx af0xegf 41k8jsuq8cvbut is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is t2q8za1kff0gsxkxm1s4jw8 q e.ucv 16he surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless while orbiting inm7a/r 9 irijy:d 6s2gb6(mj xn the space shuttle. Your 9ar6 :yx/j idjrbim2 76nmg(s 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 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 jd l.ntl1 5ms7g/) idu0.60 m. Calculate the magnitude and direction of the total gravitational nd gl)t1li5mj/d .u7 sforce exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planets l8f pm+x622ci9cmi os aine 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). m c f8io6cma s29p2xi+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 u9 92 rd.mq9a y5ppojgsurface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the mass of Mduo q rm ay999pjg.2p5ars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the period of the Eart --35v3nvnhb hxp)rob h and its distance from the Sun. [Note: Compare your answer to that obtained usinbh-nbrnpx ho)5v -v33g Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite ug fv, ao49zxny: q0z(p c7r/zmoving in a stable circular orbit about the Earth at a height of 3600 km. n( a z vc7zg/f4x9 0pour:q,zy   $\times 10^3$

  The space shuttle releases a satellite into x9zwr.y3 :ongt6dz7x a circular orbit 650 km above the Earth. How fast must the shuttle be: zgxwn76.3dytorxz9 moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants are to experience2huc ouews f0t ogt* fzo7683( simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the c o e*fwf(zothgt osu72386u0 time needed for one revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit the Earth in a circular “nsp,jrh5 v1h3p1e1 qxdm qhmy 64oo ;c)ear-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is very small compared to the radiushsvx)y 4p , cqo1p 1mrmeqjo31;65dhh of the Earth. Does your result depend on the mass of the satellite?    s ~    min

  At what horizontal velocity would a sate3opft gt1ivukva -9k,dr7 + p+llite have to be launched from the top of Mt. Everest to be placed in a circular 3-vkdp7, roia vkuf1t+g+9tp orbit around the Earth?   

  During an Apollo lunar landing m 6j hqrkm5s -es, u9qg*k;grfw/m+3 hnission, the command module continued to orbit the Moon at an altitude of about 100 km. How long did it take to go as qjhf,qg*rkm k95;h 6 gs3 -uewnmr+/round the Moon once?    s

  The rings of Saturn are c xrqj9.puul0a g(-)o 6g)3ttp9payc tomposed of chunks of ice that orbit the planet. The inner radigla p tu-xy.0p3aot6q9u c t)rj(9p) gus 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. 5–9). q4k(gw7gb ;lsWhat is the ratio ogb(4; wg7lsk qf 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 km from the center of th-xu zzo8 k1h/qe Earth's Moon in a zhu x q/8zk1-ospace 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 starn7yocl 5g;jgr 0)9be ds of equal mass. They are observed to be separated by 360 million km ag0enj)rlbgd ;5y oc79nd 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 for hr3 zny,e3 4alr9 +yifthe weight of a 55-kg woman in an elevator that moves iza,y ey4rl3+h3rn 9f
(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 anbi9r9ck 9kac 8 elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevak bikca999c8 rtor’s minimum acceleration (magnitude and direction)? a =   

Show that if a satelc/nz0md(j 7 lkvu,a v+q, hng(lite orbits very near the surface of a planet with period T , the density (mass/volume) of theqz , 0vgcknd+,lmua(/n jh7v( 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 (27.4 d) to determine the perir 7h*rkqi)nfq a9xw qs3;iu6:od of an artificial satellite orbiting very near6fq9i w3)qk;q *:aix u7hrsrn the Earth’s surface.    s

  The asteroid Icarus, though only a y7 b. 8,fqn;c;h.gedbnvcfh-e3 kwp 1few hundred meters across, orbits the Sun like the planets. Its period is 410 d. q-hw v fk7n ,;c. c.n1de3b8pb hf;eygWhat is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distance of uggc)+ ls,0 ik 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 ev s8x 7kr7xnmaq0qf ;)38a k*lyvj9nhvery 76 years. It comes very close to the surface of the Sun on its closest approach ryx m8 a7xn)j*snqh q7vk9a0vkl8 3;f(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 Sun.]    $\times 10^6$

  Our Sun rotates about the centerg,8 ykk)u*)i1gzr ;fmm x4x eh of the 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 for the four largest moon+ )9h h7ltkhbhs of Jupiter (those discovered by ) +7bhhhhltk9Galileo 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 p 6 spcdh+mf 3z2nf1l2jeriod and distance of the Moon. jm d+s6fzcf 13np22 hl   $\times 10^{24}$

  Determine the mean distance from Jupiter for oeekaz z l2.6-4 u2qnmq cyah6t bs**3each of Jupiter’s moons, using Kepler’s2k eya3 t.6 h4l* eqq6acms2oun*-zb z third law. Use the distance 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 many fragments (which sow0 zpmb1 e fmu/elp:;8me space scif8/zme1;m : 0pel wbpuentists 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 aertxaqma l1u -t +g5- ki-g):nrtificial "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 th -+ ggt :-lkqrei)a5nx1uma -te 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 fmu zj3:gg2x a r504kk3rd2b hprom a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N k 5d2rxg:rg32bpa3kmh j 4zu0on 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 the acceleration of gravity beci b(;2wntsrv/a72j b half what it is on the surfa7j2inbbvr wt 2a ;s/c(ce?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands,3.rax g: nk ;uh jpouy27tc6w and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and their indig2wuxh ynro763, u:ajc.k ;tp vidual 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 acceleration of gravntqr v cmr6 nkbj0n(-/n;ly,/ity at the equator is slightly less than it would be if tctnk0 q/n 6vbjmy,nrl;n( r-/he 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 will a spacecraft traveling directly from the mmtzvr/n8 8e,fxzeblf;22 t 7Earth to the Moon experiencrzm ev8m2;7tx/ z8nef lb,t 2fe zero net force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathroom scale in an elevaz h w1zkaj:x qzb+4 9kf5lw25ator, it says your mass l 5afz+k2z:4h9wk1xza5jb qwis 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space statnv e 2;ry-;)oop y.gelion 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 di2.;lpe)g r;o-eo yvy nameter 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 gx8od ;rnti8rc3qo l r g2anw52y3 w:4in 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 mass of a planet in terms5/ 3 3n 2gonnmcdqck0sxfs6:e of its radius r, the acceleration due to gravity at its surface and the 5gc3nf o:nsxqc6/dsm2e 30k n gravitational constant G.

A plumb bob (a mass m hanging on a string) is def rxd3z//xik,klected from the vertical by an xkk, zi r/3/dxangle $\theta$ 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 9+ ckyumui+9la design speed of If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle the cur9ukic+m u9 ly+ve?

  How long would a day be if the Earth were rotating so fast that obje(zag.uiqct ;uq *t,w,/ 8ehno cts at the equator t ;wu*/c(gqna8uoie.,q t,hz were apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant dwpj :r6(yuqo2:, ct /ye.ojcnistance 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 constiq)7e3g 8hu aq3yov w2ant speed rounds a curve of radius 235 m. A lamp suspeiu37 8 ow)g 3vq2yheaqnded from 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 beelmlcg.ng *g/j 97t4t yn claimed that a person would be crushed by the force of gravity on a planet the size of Jupi9.jgcgm /n*ll t74ty gter 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 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 deduced the 2oi fz( wh*3t,ndvu6 ad(sc:rpresence of an extremely massive core in the distant galaxy M87, so dense that it co*h6oz(nur,wa vdtf (: dic 3s2uld 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 constant speed as it traverses the hill 6)wcf6vt0x zks3)) j( 3rl srauu1kipand 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 go withourps1us))3xjrku)f c6w0azikvt 3 6(lt losing contact with the road at A?

  The Navstar Global Positioning System (GPS+x(mih 9cwz+br 7f x :wt)rdb1) 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 Eaf(:t br9+z1w7 irchmxw x)+bdrth, 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 Asteroid Rendej/zx ,xvvez. i1 c 89l9sajj/s-pwta8zvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km . Eros is rou 9c,it 8-se v91v x8x/.a/pzaj swjljzghly 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 space shuttle pursuing a satellite in neeoyh- 2u-tg)6/zeq) ozfzx6f zumf9z+d of repair. You are in a circular orbit of the same radiy6 fm o6zfzge2 qhtz)z)z --u9/ou +fxus 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 is its mean distancz5z4nc ow kor2rc 5/d-e from thec-or5 /r kd2ncz54woz 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 "feel" you+fwi k m7z+jcv-t p12vrself gravitationally attracted to someonemwj2k f+ vvc1tz7 ip+- near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the centa 5o*ljt8h)ewer of the Milky Way Galaxy (Fig. 5-46) at a distance of about 30,000 light-years frome*ajth )l58 wo 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 y -gget0+p/pcml/b4* ou,lz)9vn rj qthe corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg mass placed at the midp+ *nrzgt-,/glov 04pcmjp/blu e)q9yoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbpqsp s-ms(2. zits 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 exxq+iw/* x. l 6ycqd5b:tlsu4*63 q r7tvjohzkperienced by the tip of the 1.5-cm-long sweep second hand on yourw7:o6z*jyq vlx t/+r bhlcd 6 k*t.u3 4siqq5x wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight around in ajh,a/ /qo58.ol:pvch9 dbj uv circle below you on a 0.2/h5jvj,vad.p:o 8uc/bol h9 q5-m piece of fishing line. The weight makes a complete circle 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 rta f3 ogf +pfvd68j0.sjht 35fadius R in a new highway is designed so that a car traveling at sptp5f f.30j jsfov3+d8ft6ahg eed $ 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(dcy n w1uu+l( 9nsh 6zk0tba- 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 ). (a) Calculate the friction force needed on a train passenger of mass 75 kg if the track is not b u(ud9a n+(c1kl0s byh6t-nw zanked 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.$\approx$    $\times 10^2$