Sometimes people say that water is removed from clothes i-1 ofs ll. oezi22iwzon xa-(,n a spin dryer by centrifugal force throwing ts e i lwzo-al.2x1(o2zio-n,fhe water outward. What is wrong with this statement?

Will the acceleration of a car be the same whenbr w+r0 j0rl4n the car travels around a sharp curve rrj b 0+wlrn04at 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 hilly road. Where do:26q bsraiv 9ges 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 level stretch neb: s6rig2vq 9aar the bottom of a hill?

Describe all the forcak3o,8xf f 5s wvxwax -uwu)65es acting on a child riding a horse on a merry-go-round. Which of these forces p3xws- u5au ,)kx8 fv5xwo6wafrovides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle without the water spillu+byzpub nh*i(;fn5*say s y 33b6n: fing out,b:us35zy buy(3*fihn b+snp* yn f6;a even at the top of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? There are at lea8.1l n jx*cj 2qf)qk3r0q idltst three controls in the car ilrd8x2l1c*0tqq)nj 3 fkjq.which can be used to cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest of a smalay7npu96j62 utplek7) i gly; l hill, as shown in 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 dlauy ekpy; 2t ijgl7p)n96u 76ecrease using Newton’s second law.

Why do bicycle riders lean inward when rounding a curve atophbi8x9+zkuaq.7a 30x.mtar9 v l z9 high speed?

Why do airplanes bank wb0ib 8lp7at36 m/er )tz3phvohen they turn? How would you compute the banking angle given its speed and radius of theb 76v83tbe)o izm a/30 thrplp turn?

A girl is whirling a ball on a string around her head in a horizontal plane. Ss8d:hibl x j5/he wants to let go at precisely the right time so that the ball will hit a target on the h8l j:5sxbid/other side of the yard. When should she let go of the string?

Does an apple exert a gravitational force on the Earth?ff4 w2:otlv-j0k w.eu If so, how large a force? Ck2e-j.uf:ft4 v 0owwlonsider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double wha z06y/zz7dr 8aogk2i jt it is, in what ways would the Moon’s orbit be 68jgk /dry zoz0 2i7azdifferent?

Which pulls harder gravitationally, the Earth on the Moon, or the Mtzg7 ;treaa2 ,ah/ka;lcg)s y0yz 0c(oon on the Earth? gazse7 yt r;aahl/k2y0 ca;z)t( ,g0cWhich accelerates more?

The Sun's gravitational pull on the Earth is much larger than the Moiz s3*sdovk-a mo8*0i;zmq y8s7 9oxyon's. Yet the Moon's is mainsz9s xsio-a*qk; y y*odmz83i0o 7mv 8ly 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 moo 2e/ldc96k cbre at the equator or at the poles? What two effects are at work? Do they oppose eaccd c62ok /elb9h other?

The gravitational force on the Moon due to the Ee9yik,:ca0 or arth is only about half the force on the Moon due to thk0 :e,yiorc9a e Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in itsjdv 8mu rbg/3l0* q--h wat:mq orbit around the Sun larger or smaller than the centripetal accelerath/80mdu qa rqjg*lbv3:tw--mion of the Earth?

Would it require less speed to launch a satellite (axhc; /ui0:*mzw dtr7 nyu*xj- ) toward the east or (b) toward the west? Consider the Earth’s cr:/d;iw*xzh 0yj7-u txu*nm rotation direction.

When will your apparent r3bjxmn8 7vk** efq6f weight be the 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 the least? When would it be fe jm*n*3x86rqv7kb fthe 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 space could be adversely affect2)h3avzun44g2a9z gs u2 cueged by weightlessness. Onz 9guh4ns2 3guce 2zua a4)gv2e way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how this simulates gravity. Consider (a) how objects fall, (b) the force we feel on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “free fall” or “)m yfy3c o;v)vaabf d0yt u)wb03oa8)apparent weightlessness” betweet;0 aw)o3yyf 0fcamv) vub)d3)8ao ybn steps.

The Earth moves faster in its orbit around the:ubzz1tr 6hp3zqgz 897 rg0ox 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 the tilt of the Earth’s axis relative to the plane of i8rzx79:q1z6grz g p ut3zohb0 ts orbit.]

The mass of Pluto was not known until f:cyd 2-mal l9it was discovered to have a moon. Explain how:am y dl-l29fc this discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger than the Moon's. Yet tv( ;e3)u8j gjemy0vf ihe Moon's is mainl jyv;migf(eu0)ve j38y responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]A child sitting 1.10 m from the center of a merry-go-round moves 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 1980f)qzddy),pb edvj.94wfz n0yj - xl7a1r 8h.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 Earth in its orbit aroux*cpr yv; :5oznd the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orbit is a circler;ozvc*xp y 5: 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 discus as it rotathwhk2-n) ax h0.lb.e7pafq)aes uniformly in a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the spelhn .2qh-kbfa)7 ax0wh ).ep aed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth's surfacero; +s*d rec49l2chesn/sg o8, and circles the Earth about once every 90 minutes. Find th2o c l*ogs/s ;cs d+rer4en89he centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of a 0 qozct.zib)q9:rr f +speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diameter iqrq)fi0.:zot+rz cb9s 32 cm?   

  A ball on the end of a string is revolved at a un0 u8f e*xemr+ziform rate in a vertical circle of radius 72.0 cm , as shown 0ufexmr z8 *+ein Fig. 5-33. If its speed is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

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

  What is the maximum speed with which a 1050-kg car can round a turn of 5kn g99l:jvw.a7w p0i zhlzz3radius 77 m on a flat road if the coefficient of static friction 9lzlh :i 3jn97 wwp.05vkzgazbetween tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static friction be be+lce9+z i c9xetween the tires and the roadzi+ 9cecexl+ 9 if a car is to round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots is designed to rotate a :qgp81v6 k tz7uzid0ucf2 px1 trainee in a horizontal circle of radius 12.0 m . If the for1qgdxfpt16uz0c7ki2up 8v z :ce felt by the trainee on her back is 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 from the axis of a rotating turntable4/fob )/fl ukkhf xy7/ of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coeffi / x7y4fkh lfkfb/u)/ocient of static friction between the coin and the turntable?   

  At what minimum speed must a roller coaster be traveling when upsid*yl.x5,gp,l wh xvbd*e down at the top of a circle* wbpx, gvh, ll.5y*xd
(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 drivo.)b tkio+rpg2i l,bqt dyk7yyq(-97er) crosses the rounded top of a hill (ra og )p7.y92tkotq ,ky(ibdqr-+y ilb 7dius =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)cputqk izc0p/cx ..+/;ul fa a 15-m-diameter Ferris wheel need to make f; zc.a)cqxcitp+u. fk0 lu/ /por the passengers to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whiibhuv5weif81b5as.f9tc ;55qb 0 l berled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting thebqaf9ih.505scwb i ulv5eb1 ;bfe58t 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 asl.n-k,.i qccer,yeby j1i -: centrifuge rotate if a particle 9.00 cm from the axis of rotation is to exc,ei,yj.c -.r-1:ibeksqy n lperience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a cay,lyt1x/1st3bu 66g oe n-whwrnival, people are rotated in a cylindrically walled "room." (See Fig. 5-3ew gn- yl63x,/otyt ubw6s11h 5.)
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 rota (ipji94mkp,vp marfjhn80+ er/z*f4ted in a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord conne far0( zpmjnph*ifvr4+p8k ,9i/mj 4ected 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 ignorin e u,frrie/)h 81lu.yeg the weight of the ball which revolves on a string 0.600 m long. In paryie flreh8 1/,uue.r )ticular, 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 banxp2)(x* 9rtbtnf*di r/ k nnm1mujv9:ked 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 massesw5af+m ig/de z50 hca5 $ 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 vertically at . h+gqw xfu6gmw;, ,mo 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 forci5 fqvjpzqyy4hw4g86nyz2ed6;(sh* e exerted on the chef48 pg(* wzq6v y 6s idj54;yzqyh2nar (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 acceeygc 90hf ,v)klerates 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 it is halfway through the turn, assuming constant tangential acceleration. If the curve were 9he,y fc)gk0v 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 hykp80z.o5 qggr2-oadorizontal circle of radius 2.90 m . At a particular instant, its acceleration ok-pz8ygr0.2 5gad qois 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 gravity on a spacecraft 12,800 km (2 Earth r1jrr q6 o(butj7gks6.adii) above the Earth’s surface if its mas (6bj.qjs1rg 7kro6uts is 1350 kg.   

  At the surface of a certain pnp+ta83sl9f zq.o 2 :ae5wiszlanet, the gravitational acceleration g has a magnitude esntp3z+w9.ai 28 qas5z: olfof 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 tyw4ovpp udg+zx (;2b:o gravity on the Moon. The Moon's radius is 1.74 z pu ;wg:(yd p42+vobx$\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1ylq6m;jdq0z rga,go :(f gu8).5 times that of Earth, but has the same mass. What is the acceleratio86o0du gg,q; : rqy)mafj(gzln due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, but the sab/stm-n -ku,f6x ew+wme radius. Whaw--metk , nxfwb+6 u/st is g near its surface?   

  Two objects attract each other gravitationally with a force of 2.t6cbjphts 2p9+calb5w43u z1z uj54g 5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of gravizzh56v0mvh(nl yp /0t ty, at (a) 3200 m0 z0 m65tyh(vz nphl/v    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Ea cjy9(9ynh 4oqrth’s center to a point outside the Earth where thegravitational acceleration due to thq y4jyh( 99cnoe 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 )md(r fr8+bftep*ft ()0wt sbSun packed into a sphere about 10 km in radius. Estimate the surface gravity on this mo()8ft * dbr)wf+ sp(0et mtfrbnster.    $\times10^12$

  A typical white-dwarf star, which once was an average stah4;zg+z) p n-7lbpxrjfp l)n5r like our Sun but is now in the last stage of its evolution,jl7+x; f -grln )5)pzz4pnhb p is the 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 weightless while orbz,4nu r(zhq-q df,xfp 52 3cznte5k:xiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that n(q5fzrkfc, :utnh45qzx -3z2xp, ed 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 corn93woy n thf))4sjq4 rters of a square of side 0.60 m. Calculate the magni ors nh9qfwt4 y)4t3j)tude 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 line up o 41dzk,kk u6yrn the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, an 6rk4ykd kz1u,d 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 acceloks4r0*yqn jd kp7 /+peration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 km, determine the mass of roq40jnpy pk/+7d ks*Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the period os/oz sl;icj 5)f the Earth and its distans cs)j5oz/ il;ce 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 satellite moving in a stable circular orbiwxmbpct/77s8 3 .yrc3or5 kj 1sp+na rt about the Eay 1n77tc5ar+pbrk r .ms/8 cw xo3p3jsrth at a height of 3600 km.    $\times 10^3$

  The space shuttle relee-qg ixqx h.31ases a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs? x -hi.g13 exqq    $\times 10^3$

  At what rate must a cylindrical spaceshipqxxvy ygfwt(7;)q( a* rotate if occupants are to experience simulated gravity of 0.60 g? As *(gxa (xy7y;q)fqvw tsume 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 the Earth i( *bd z vetg6b4 qi)i8ysk/2bhn a circular “near-Earth” orbit. A “near-Earth” orbit is one at 6 tksh/bb b8yi2 g) zd(4ev*iqa 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 satellite?    s ~    min

  At what horizontal velocy;w-ili7 g,vvity would a satellite have to be launched from the top of M vwly i,7i;-gvt. Everest to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landinuml ,;a1n.+(k4 8uqtbbwmt8fc/k i r zg mission, the command module continued to orbit the Moon at an altitude of about 100 km. How long did it z8t4b(81c;btqwmm/+r.,f ak lkun iu take to go around the Moon once?    s

  The rings of Saturn are composed of chunks oe n5x j+ sne1n)+yamy ,qt5b/an();*tnc svadf ice that orbit the planet. The inner radius of the r+n)(n* s)et;ny/xneq dtc5vamab1y5j a sn +,ings 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).gz ufh1a6dfy9s9p y 2* What is the ratio of a person’s apparent weight to her real weight (a) at the top, and (b) at the bottom 2y1spfg9*dh 9yfuz6a ?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 k9ye 9c) c.ahttm from the center of the c.y)9ettha c9 Earth's Moon in a space vehicle (a) moving at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star system consists of two stars of equal mass. They arn)aslaqunpg bj/p 0a 4598b -rv l:p.pe obserrv qb8gp5p/0lj.l 9a nns4-pa)up: abved to be 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 for the weight of a 55-kg woman in an elevatg 9j8 4xd8vgmhor thatg889x 4gvh jdm moves
(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 th l vyffr(hz.a/j/r 1f5e ceiling of an elevator. The cord can wi1.l 5r/(ff j yzvfhra/thstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orb/t,stm+8eo6bre mo4e its very near the surface of a planet with period T , the density (mass/volume) of,+/oosm t8 r6beet4me 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 x, ;zh8n:htnq of the Moon (27.4 d) to determine the period of an artifhnzq :t ,x8nh;icial satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, orbitszqtmf,(jqe 0s-/e5mk -cy 5pg the Sun like the planets. qsmgfcz5-p,m5 j /0yeeq t-(kIts period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average distan, 2vnd yrqsw(* e5:uwnce 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 vnm2pw+ 1)cwqroughly once every 76 years. It comes very close to the surface of the Sun on its closest approach (Fig. 5–39). Esti21+ pwcqwnmv )mate 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 center 8iviiy, i do,4 c u12o/hj-prfof 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 mt.t0ttst* ad5z i.1vkoons of Jupiter (those disc *.t.ttavt tks0z 1d5iovered 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 the Mobz d)n2meb fkpv8 b340on. 3bkb v0bdm42) en fpz8   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, using Kepl9lk ,; gq3(gu 7k..2 dam7ybkeazzppv er’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the values in thq.;k7pbpaez29d( myl,3 zv g kau .g7ke 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 fyl+j, tkgr)e9ragments (which some space scientists think came from jg9tr)+le yk, 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 "planetxc-bfh e* oacme6::7 i" 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 that the ring rotates quickly enoug axo*:e7f6 :icecbh-mh 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 bytjc ff i*lb*29 swinging in an 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 maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5f9lt2 c*fijb * m long.   

  How far above the Ear-2cs.om .ozkrth’s surface will the acceleration of gravity be half what r zmsk.ooc.2-it is on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin i/4*asj wcgch:r k7ln.e 9l fe1n a mutual circle once every 2.5 s. If we assume gh*1 cllce4ka.9 /7fs: jrewntheir 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 w pqn mqr+op;x9yvy0 l: 4f;i :r8.vqaapparent acceleration of gravity at the equator i :8p90vay;lofm iv:nyq+ qx;w p.q rr4s slightly less than it would be 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 will a spacecraft h)ln:2ws* h axlo0f 1.dw*jxmtraveling directly from the Earth to the Moon experience zero 1f 0xdxnh.:2)hols mwj*awl*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 i96rdc u ofd4*stand on a bathroom scale in an elevator, it says your6orciu*4fd d9 mass is 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a ciylmc 3.jhfra4cs+)+tva;ta - c )fqdau n)d.;rcular 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 aboh )ac; +-aq.damrtcffv s3auy.c)4) djnt+; lut 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 iagqkp wb 4v gg*ra cm3 f3sjp;9)*r72jn 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 terms of its radius r, the x.rjoy 7c.r;w,7eqsz cdiy5- acceleration due to gravity at its surface and the gravitational constant .wxcdzyqcresyri ,j75. ;7o- G.

A plumb bob (a mass m hanging on a string) is deflected from tgno4k ,b/mu/.sc;m qc d t hj(-j)kvy1he vertical by av- co1k s(n;)y,jdhtmgu4jqm ./c/ bkn angle $\theta$ 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 coefficient ofs5 ir/l;9em.sl jd8sg static friction is 0.30 (wet pavement), at what range of speeds can ;ell/s.g ir8m5s9d s ja car safely handle the curve?

  How long would a day be if the Earth were rotating so fast that objects at the exp s u(9e+fn*cs *1wakquator were as f+ k x*9*wnusa1pec(pparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant distanccuifa;xeh p a-;24q1zuw9ep4 e apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant speed rounds a curve of radius 235 m. A lamp zc0xbxk45tt 2t7e;d psuspended from the ceiling swings out to an kb2t ce 0dz;xtxp7t54angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive +y(oa7d4 + k ( kd*o+n yp1mppuc7fbmeas the Earth. Thus, it 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 fewo+d( 7+kekdy nm4p bf1o*mpu (+ pyc7a 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 presencewx mmbug,6 ly6+f.rl/ 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 bymgml,./b 6xfrulyw6+ 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 j.1+ac +/oxtgaaq0tlt 7p)b whill 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 larges7.tcpot1awg ba)jx+/atq 0+lt? 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 Syst-gtxa vu3s+ kf w+iawlc i/t6. w-5m+fem (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 approxiik uwm c5 gww+3f/lt 6avift.- a-xs++mately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 jl.(t f6yfh4-(iyiysbillion km, is meant to orbit the asteroid Eros f(t yi (sl4hyf y-6.jiat 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 space shuttle pursuing a satellite in ne/9mxie,z ,;oz 46)hzp vcys tted of repair. You are in a circular orbits9 z,thpz;/ vx4i)mc,o6 zyt e 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) W( yksjg cd*64ghat is its mean distance from the Sun? y g*kcj6s4gd(   
(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 valuv o*6fq p3f7ttee*kd2;fegy +e of G would need to be if you could actually "feel" yourself gravitationally attracted to someone near you. Make reasonable assumptions, like 3t 6fq o*f2 g7+dy*eeet;kvpfF $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the ces;gkvnv+ vuv-c*t31jm.j c;f nter of the Milky Way Galaxy (Fig. 5-46) at a distance of about 30,000 light-years from ; k; .sn3 vmjjfvc1-tv+cvu* gthe 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 v 4f zmm7e -ag4g0up)rthe 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 midpoint of the bottom sidm )ggm-vz0r7fp4u4 e ae of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg or/o svlw k(9 ukl*tt:y-bits the Earth $ \left(\operatorname{mass}=6.0 \times 10^{24} \mathrm{~kg}\right)$ and has a period of 6200 s . Find (a) the magnitude of the Earth's gravitational force on the satellite, $\approx$    $\times 10^4$
(b) the altitude of the satellite.    $\times 10^5$

  What is the acceleration experienced by the tis/5q4 g 3;wh+ ltty:vxf8ff inp of the 1.5-cm-long sweep second hand on your wristf f/gs;3 y+nwvth4x fl:8iq5 t watch?    $\times 10^{-4}$

  While fishing, you get bored anffy*jl gpe*)8 d start to swing 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 the angle that the fishing line makes with the vertical? [Hi)y*pg flj* 8fent: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highway is des ygsrgzd8if7ep,h. 8lxnaix 5z m63 6.igned so that a car traveling a5fgnrxl368xhi.i7zpe8 d.,6 ygasz mt speed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed tr)wkiq/ lta3.bxe b q:tbg (;c/ain, the Acela, utilizes 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 l:)/g3; bewktqi tqab(b/ cx. 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.$\approx$    $\times 10^2$