Sometimes people say that k 3a eax3:yge(01skf0sret.x water is removed from clothes in a spin dryer by centrifugal 0fr eks 1eg:(axx.sa33 etyk0force throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the caotfs h3q8i +(cr travels around a sharp curve at a constant 60 km/h as when it travels arouoc8h sitqf+(3nd a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a hilly road. Where does the car e1j6f wr k75 xzdm6tx omgon io62h8a6z g;:c5txert 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 stodjt1:i7m 56x r8gz5m6 o6czfwhtn;k a6g 2o xretch near the bottom of a hill?

Describe all the forces acting on a child riding a horse zy7 zp; eh0xl2wnu1 j1yb aomx-;8m5fon a merry-go-round. Which of thu 5e7yxap12 nbjy l0hw;mm ;z1xof 8-zese forces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circl* pf(+bjx. yr*v1lgrwe without the water spilling out, even at the top of the circle when the bucket is upside down.w+.1ry * jg*f(bxrvlp Explain.

How many “accelerators” do you have in your car? There are at least three contku6-bzpol:l 9/dio6 vrols in the car6b6lov 9pi/-z ld:ouk 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 crestrly7bw-sd3goe+ w( 7v of a small 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 thid 7l+7w3 sobryg(ewv -s decrease using Newton’s second law.

Why do bicycle riders lea2e t,j 8sm 0qufrfx3a,n inward when rounding a curve at high speed?

Why do airplanes bank when they turn? How would you compute the bank4 .10 9ff-ltampbitv ling angle given its s0 4 mtplf -f.alvb9ti1peed and radius of the turn?

A girl is whirling a ball on a strho( rpfwvc63b4q n93d ing around her head in a horizontal plane. She wants to let go at precisely the r9rhn3pfw(6dv4bqoc 3ight 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 gravitational foux2y, -q-cr st *t imoiicu-+7rce on the Earth? If so, how large a force? Consider u m, itxi+ryu-*ti-7soc -q2 can apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass werep49 7l4ouaq8ots ; xzs double what it is, in what ways would the Moon’s orbit be diffx 8o7zltsops4qua ;94 erent?

Which pulls harder gravitationally, the Earth on the Moon, or the Mk i38k*hoc 14lyy mdb1;)v hscoon on the Earth? W*;dhmh8lyc1 1k)vo b43 kc syihich accelerates more?

The Sun's gravitational pull oneihm1i2ikz6s) v 7g nbi3/7 xp the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from onebesx6 /3)iiz2g17mk ipvn hi7 side of the Earth to the other.]

Will an object weigh more za97(71mj v 4h a0 tvy*jjdkkjat the equator or at the poles? What two effects are a(j 7k0 1jm*dkj 9h z7y4vjtaavt work? Do they oppose each other?

The gravitational forcefm a )p9/t-np *ldorwyz5g;f; on the Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn’t the Moon pulled away from the Earth;- f zlay5*ngpm/ wdf9)t opr;?

Is the centripetal acceleration of Mars in its orbit around the Sun larger or smt/(fve -/ppnv n:jp0 bal0/:p-ppfj (n b/vetvnler than the centripetal acceleration of the Earth?

Would it require lesb2po b; icv2(js speed to launch a satellite (a) toward the east or (b) toward the west? Consider the Eo vpbi2cj(; 2barth’s rotation direction.

When will your apparent weight be the greatest, as m9 i9)g,mv2)cw drsybyeasured by a scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fa9v)yr) m,bdc 2w9s ygill, (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 longr wybk;-6 l0qf +jpyq) -wn)us periods 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 fow )y;-r)-byq un0qpwl k6+fs jrce 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 weightlessness*z4gf)( tqizsn6ye,xzy .w e+” between s(ee )+s.tx z*qzygzwi46nyf,teps.

The Earth moves faster in its orbit around the Sl -( yi0ge*)fi j6lwfaun in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of*ij0 y-gel fa wifl)6( 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 discovered to have a moon. jrul980d7g l:xs f.uzExplain how this discovery ed09s:.ux8 lfj7zlg ur nabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much 6utxip4(7l-d lyhx .hlarger 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 o4puhd lxy(.x -il6th 7f 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 1980 vpxdvy7oqb ,z hq25-1 lfp0-b $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 Eayb5+*6uwlb q( o.,rzusk6xxnz 3 :sbfrth in its orbit around the Sun, and the net force exerted on the Earth. Wn6x5b u l( *zysb+zr.bxkq6o:wfu,s3 hat exerts this 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 21rojs0 )bc1 i2xoy* j4d+8bgkq0 N is exerted on a 2.0-kg discus as it rotates uniformly cs4 d rb2b kio0)o1jx+q*8gyj in a horizontal circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth's suqg,yejt, )+b fnck5:e rface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Ea: g y)tf,+kb5cejne, qrth's surface. $\approx$    g

  What is the magnitude of the acdfqe sda ,+sutz.n .h07 q+r)wceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolu d sa+hf.qq0stz,.w)ur +dn 7etions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform/qzq86 .h5 hqa ,dnnhb rate in a vertical circle of radius 72.0 cm bdq .hqq/ n azh56n,8h, as shown in Fig. 5-33. If its speed is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

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

  What is the maximum speed with which a 1050-kg car can round a turn of radius 77/: p1babba- s mra0lb, m on a flat road if the coefficient of static fb b/l-a pbmr10,a: sbariction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static 1jzx,n/re54j1sgx x0omzg 4mm 4-jbn friction be between the tires and the road if a car is to round a level curve of radzmjme5snbjg 4j x4gx1m4-x0,zn or/1ius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pilots is desias(, ll6gtoh (wo(,sogned 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 at ,osgs,wh((o(l l6 ois she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the axis of a rotating turntable of variabl y 8,szflb;x(lgq(j2te 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. Whajs2bt (,(;ly fxzq g8lt is the coefficient of static friction between the coin and the turntable?   

  At what minimum speed must a roller coaster be tr) y8xez.b cw7 ggef(d,km* y7eaveling when upside down at the top of eye(.gdxmgk)f yw, c7 7be8*za 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) rhr( ts 3.) se27hrfjtcrosses the rounded top of a frsh3t7re. jr)t(h2 s hill (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-mgbnwtc;jzg/:2-6 3 az ryzeqn 8fx g19-diameter Ferris wheel need to make for the -x8e g31agw;jntgfzz yz9c/qr62nb : passengers to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a vere q(e2sl:pwwhq z4.4htical circle of radius 1.10 m. At the lowest point of its motion the tension in (le sq :wqp4w.z2hh 4ethe rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuge rotate if a particle 9.00 cm from the axig /fug,c*:8 a ddrzl8 .rmmu/ybvfz79s of rotation is to experience an acceleration ofmf rud z :gagcb*yr/.d/m 8z97v 8ufl, 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carniva odw+lhdz n*h 3r(zi39jro9) wl, people are rotated in a cylindrically walled "roo3( o rjd *zrh+l dwnz)3w99hoim." (See Fig. 5-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rotated in a circle on a frictionless air-hockey ta(yev xko)8,xvw0vx -p8e l9gg bletop, and is held in this orbit by a light cord connectedgyew ovle)x-g vp(088k x,9v x 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 igs7 yf57wbi9k)e7 eu wqp-;tdwnoring the weight of the ball which revolves on a string 0.600 m long.7ebwu ey;57 q k)w7tfdps9iw- 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 trvds 7i 3jcy90yaveling 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 masse kpc+c 4, sq2wvo; sc)ju igp76hl4pk.s $ 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 evasi8i v 4p6u/nbjo+ aw ylcim/a;c.xe+g 6ve maneuver 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 centripetam hzq /n1g8v2t /xtms b,,mvd(l components of the net force exerted on the car ,8 x2mvbs hvm/mt1tg ,/q(dzn(by the ground) in Example 5-8 when its speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerates uniformly from the pit /-yrt 7l.kpaaarea, going from rest to 320km/h in a semicira/-yal r.tpk7 cular 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 slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.903yvy.x8yv hgb 1)9,u6c;qu ec:g myvf m . At a particular instant, its acceleration is 1.05vybg uvcyv 3y)8 : fme9.hcq1x ug6y,; $\mathrm{~m} / \mathrm{s}^{2}$ , 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 spacec8x-g:uu6qe zf0 b;sji raft 12,800 km (2 Earth radii) above the Earth’s surface if its mqz :xbi f ujs-;80g6ueass is 1350 kg.   

  At the surface of a certain planet, t qi-prrpu r)*.he gravitational acceleration g has a magnitu p*urr.) rpq-ide 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 gravjg, it5mh2;nn) w f, l.l1lncfity on the Moon. The Moon's radius is 1.7lnnl;wc2mf5g. l , jfi1nt, )h4 $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 timej7 o(*r*ymn8zy,qv v vs that of Earth, but has the same mass. What is the anm(8zqy,r7v *o* vjvy cceleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that ou*68bw5p5cfeg :g iw gf Earth, but the same radius. W6u85 gcef 5wwigpg :b*hat is g near its surface?   

  Two objects attract eacu(-p.)93ih2 lkbz xz bevj 2tch other gravitationally with a force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of gravity, at (a) p b8k;i g2ac*h: d-dld3200 mh;lb cd k* :p-dagd8i2    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’lu1 +3s tgx5l(s 2gmpks center to a point outside the Earth where th(+ls gt sul2 1xk5mgp3egravitational 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 tcmlot,nl s:zr - y61k.he mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on this monster.ncm ,s-ll:trky .zo61    $\times10^12$

  A typical white-dwarf star, which once was an average star like our Sun b h -h7dmz7,vv.to go-tut is now in the last stage of its evolution, is the size of our Moon but has the ma vgt,7 - .-ovodmhzt7hss of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts lk7w*am wx9 3yzf; k/vfeel weightless while orbiting in the space shuttle. Your friends respond that they thought gravity was j mv *yfax/w;l937zwkk ust 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;;lcfe-u f+e8 -cgjdhy;qg-k corners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravi8c;d;qg- jyf ck f e-;uhel-+gtational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planexi vlebx9lo/3n0- 2 uc5 i 0(ts9ofmlpts line up on the same side of the Sun. Calculatoxi 2ifc tl30oe(u9px/-sllv n5b0 9m e the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5-38). The masses are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , 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 surfopz-4j g.s+;+hlejrvsu ;a7j*hz-v i ace of Mars is 0.38 of what it is on Earth, aneajv;r h gzh-;suzilj*7v4+o+ ps-j.d that Mars’ radius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using thew4oq yra8;c wh+9cr )p known value for the period of the Earth and its distance from+rw8) oy;pac4 wh cr9q the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellitevm5jrakd -d . 22 bon67te+imk moving in a stable circular orbit about the Earth at a height of 3600 km. b-n2i6mtom25jddev a. k kr7+   $\times 10^3$

  The space shuttle releases a satellite into a circular orbit 659ova*x9 vbwtloj)y/. 0 km above the Earth. How fast must the shuttle be moving (relative wv o/tl9v)*jxb.ya9oto Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants2 azc34yjurx8g5y fw . are to experience 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, Fig 5x 2uaf j853ygr.ywzc4–32.)   

  Determine the time it takes for a satelliteepij.afnm 2dh0s u 1-x3 s-t7l to orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surface of the Earth which is very small compared to the radius of the Earth. Does your result depend on m2l phu13fd-ns.x0tsji- 7 ae the mass of the satellite?    s ~    min

  At what horizontal velocity would a satellite have to be2svq uu 99ju9z launched from the top of Mt. Everest to u9z vs9qj9uu2be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continued to ok)mu + bkidkj6a f8.0pgjvu y )3p,:kirbit the Moon at an altitude of about 100 km. How long did it take to g kpam,.0v k8 dikg36jufjiky pb)u:)+o around the Moon once?    s

  The rings of Saturn are comb:,+w8 be trskposed of chunks of ice that orbit the planet. The inne+w:bs,br8 tekr radius 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.oousykz+,x (s43q, u n5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to he4ux(s,y zk,noo3su+qr 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 th,fj)p zp.w-q pjfcb )t:i9v9de center of the Earth's Moon in a space vehicle (a) moving pz . d:9tjq, -f)vjpci9)bwfpat 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 are bnkzuh**db/1 observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between thekbb/h*1zu*n dm. What is the mass of each?    $\times 10^{29}$

  What will a spring scale reasgo.,,lm j;rod for the weight of a 55-kg woman in an elevator that molr .;,o ms,gjoves
(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 suspendwhw s*ogf*w0 2ed from the ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direfw2*ohw*s w0gction)? a =   

Show that if a satellite orbits very near the surface-toz(7td5y g bkh*vt 1 of a planet with period T , the density (mass/volume) of the planet isbko tvtdg 5t-h*yz7(1 $\rho=m / V=3 \pi / G T^{2}$ .
(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 Mzn n: g+tn*jg/oon (27.4 d) to determine the period of an artificiagntz/*nj ng+: l satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundwy:*)/h e girnred meters across, orbits the Sun like the planets. Irnweg i)/hy*: ts period is 410 d. What is its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average dis/emy*bbar . 9+xkwm5itance 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 *w1*cs-kv jq 4i rkb:donce every 76 years. It comes very close to the surface of the Sun on its closest approach (Fig. 5–39). Estimate the greatest distance of the comet from the Sun. Is it still “in” the Solar System? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance d1brkw-s *ic jk*:v4 qfrom the Sun.]    $\times 10^6$

  Our Sun rotates about the center of t 0*vvx ws24owocn+r(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 moons oapaex/:la97/h lzf 4 qf Jupiter (thol lea fpq x:9az/ha7/4se discovered by Galileo in 1609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the Earth from the known period5;nminn)-quhrijx 2ps b .:0 c and distance of the Moon. )nnbh:;0q mcrpji.-25 si uxn   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, using Keplm 5xt;ga5 gmn0er’s third law. Use the distance of Io and the periods given in Table 5–3. Co ganm;5g0 tm5xmpare 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 consif-1u t(8i5 ult3rhfsp 1e sf+ests of many fragments (which some speu- f1 f+5 3rtti(1 sehsfpul8ace scientists think came from a planet that once orbited the Sun but was destroyed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale describes an artificial "planet" in the form of a band tvzef5 a, k3;ocompletely 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 makv o ,kz3fae;t5e the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by swinging in an arc from pk6w*v u(g;t )dxt-kx a hanging vine (Fig. 5–41). If his kg;(uw xv6-t )dtpk*x 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 5.5 m long.   

  How far above the Earth’s surface w xx0m4z:ig(f,a xh 0ucill the acceleration of gravity be half what itahi 0x(u, x0m:xcz 4gf is on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in ( hl* abt:5xzva mutual circlel zhva5x:(* bt once every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates oqj c,ki0x7 2vgdplb 6;f:h(c jnce per day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this e:bp cg2 l0jkhx qj6f d;i7(c,vffect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraf;fz3:0k5k5+ e khczm9nwppyh ) wpgp8 t traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with e zpp5;n zwp0f8pmywk35hh) k gk9+c:e qual and opposite forces?    $\times 10^8$

  You know your mass is 65 kg.5 oy u95 /zh6;kqlt+duk,th/ e qriox, 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, ao 6hey;dur5 9+i, uzqk/tqxot5lk. /hnd in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tube that will rotate a82 oacjf tc30fr*fh 71kh uji:bout its center (like a tubular bicycle tire) (Fig. 5–42). The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt?cf uhcf a30ht2 *of:k 8rj1ij7$\approx$    $\times 10^3$

  A jet pilot takes his aircraft in a verti uri;jj1 y7:xvcal 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 masg9,wd gb4py wsn0b c;/s of a planet in terms of its radius r, the acceleration due to gravity at its surface and the gravitational constant n4 w;gpw/0b b9,gy cdsG.

A plumb bob (a mass m hanging on a string) is de(12ec wdmbw3 ue-ip(yflected from the vertical by an wicm2b(-p(u d3w1 eyeangle $\theta$ 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 oc( v4 3g3npczhf static friction is 0.30 (wet pavement), at what range of speeds can a car saf(z3 vchn4g3cp ely handle the curve?

  How long would a day be if the Earth were rotating so fast that objdum*c. e sp2,zects at the equator were apparently weighzs,*eu2.d mp ctless?    $\times 10^3$s

  Two equal-mass stars maintain a cons-:88wwoei( g l d p n5atn6s*k,lkfmw,tant distance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a co2 y7*2,hl ghiq:yyguu,e1bv tnstant speed rounds a curve of radius 235 m. A lamp suspended from the ceiling swin,,l1g btyh2 y ih7e:yqu g2vu*gs out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as massive as the Earth. Thus, it has been clgp*zo59 :xc ikaimed 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. U:oi 95cpz* kxgse 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 Telesc;kvmwdg; /:y6aa qn-p) d8 ciqope deduced the presence of an extremely massive core in the distant galaxy M87, ;;qvddaacg k-)qwi:8n6 p/ ym so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be 780 $\mathrm{~km} / \mathrm{s}$ 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 as0r4p ve *tqs-4m9lfen it traverses the hill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature4e*r- nsqflp 0m 49vet 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 without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24u v2 eu.gzm 1t+ jqw6g8yu5ev) 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 Earz 5v1g28 uwgy)ueq ju.6mve +tth, 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 Rendezvous (NEAR), after traveling 2.1 billio7di+hfa jp h+f+-1wnrn km, is mefj+pf7 +wa nrh-+ihd 1ant to orbit 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 space shuttle pursuing a sat vqjid5i,h+q3ellite in need of repair. You are in a circular orbit i5,qijq +v3 dhof 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 periodhf 290+yweo ao of 3000 years. (a) What is its mean distance from the Sun? 90w2o o+a yfhe  
(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 "z7-mfsby-* g kfeel" yourself gravitationally attracted to someone near you. bm-fgzs*ky7- Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46)am5a-; dp yt+a 3+uwho5j* ere at a distance of about 30,000 light-yohyt+ u epj5+3aa;* d mr-5eawears from the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the cornersc /4;6xb. g1 tibqb,wo xuqoh, of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a f w16tcgohxb /,qbobq .; x4,iuifth 1.0-kg mass placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbits th*lm3hs5c gu : o/hpkz6clu1)6zlh2eze 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 thb2 bl8krg 2+7vtil5pefrn *e, e tip of the 1.5-cm-long sweep second hand on your wrist wi58 ln flbbk7*t2 ,rpveer g+2atch?    $\times 10^{-4}$

  While fishing, you get bored and sta wrlzf 2wr t 3v:zmn5u.0r4hl-rt 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 cirrwf -v2l3r5w 0r:zhtlu4 z m.ncle 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 lg jmx tdo3c4;zq+ .mw24cj1eradius R in a new highway is designed so that a car travelin4z2 ;lj1x.wcq d j 4+cmeot3gmg at speed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizes6seqmv w4hib( g3v9)j 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 ). (a) Calculate the qiev(b4j3g6ms v h 9)wfriction 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.$\approx$    $\times 10^2$