Sometimes people say that .4qqyhzo)teo7 l o w;2water is removed from clothes in a spin dryer by centrifugal fo27e w.yt;oqlh o4q)o zrce throwing the water outward. What is wrong with this statement?

Will the acceleration of a car be the same when the car travels around a ;dau) 7vipqy +y/j ,-ig7ylnk sharp curve at a constant 60 km/h as when it travels around a gent/vqyiy+dk-u ;j77a gyl, pi)n le curve at the same speed? Explain.

Suppose a car moves at constant speed along a hilly road. Where does the c.9 wgzg,h s0r.y,geyj ar exert the greatest and l .,sh 0,yyzjwg9rg.geeast forces on the 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 forces0/p:wk (.6s u o74havgyddjqh acting on a child riding a horse on a merry-go-round. Which of these forces provides the centripetal accelevop4 s60jk 7wqh. hd/uyg :da(ration of the child?

A bucket of water can be whirled in a vertical circle without the water spil cckkr9kg-98iw ff(sv+)u, awling out, even at the top of the )-i ku,ka9gvwwff( 8s9crk+c circle when the bucket is upside down. Explain.

How many “acceleratords0hgtp.n28nx lka5xb6 xw 19s” do you have in your car? There are at least three controls in the c20 1dng6xtn8 bwxlx5k pas9h. ar 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 small hill, as shownry )mdzjsq58;kbxwn:ku82 d0 in Fig. 5–31. His sled does not leave the ground (he does not achieve “air”), dnb d05:zwx2u8k)jqs8y;r km but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle ridersp1a m -bnsgl7xg4ggg j o;69j: lean inward when rounding a curve at high speed?

Why do airplanes bank ws qt4tdnq:m:,s y 13tf6n.e uohen they turn? How would you compute the banking angle given its speed and radius t4s ty:o6 feqnt,un. 3md1:sq of the turn?

A girl is whirling a ball on a string around her head in a horizontal plane. She2k;ov dfl);g j6ec;s b wants to let go at precisely the right time so that the ball will hit a target on the other side of the yard. When sg le;2j b kf;cods;v6)hould she let go of the string?

Does an apple exert a gravitational fo8qns 2-wpf)hu rce on the Earth? If so, how large a force? Considqh- 8 nu2fpws)er an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were do+v;(d,ze :i ,c *a hnenig*avcuble what it is, in what ways would the Moon’s orbit be different:c,eicna ;in *(+avg*zvd, he?

Which pulls harder gravitationally, the Eartha 5cb8 /dzs9az on the Moon, or the Moon on the Earth? Which accel5bzz8a9s/a dcerates more?

The Sun's gravitational pull on the Eqg:o,k52+ o dy5rfoqtarth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side o 2oq 5r:+,dogofkq5t yf the Earth to the other.]

Will an object weigh more at themgmsle2.mj7 v 7x )p).rpt7 pg equator or at the poles? What two effects are at work? Dvmg p7jgpm2tr)) m xpl .77s.eo they oppose each other?

The gravitational force on the Moon due to the Earth is only about half the for0n2r.7c +k+ uhztplqpce on the Moon due to the Sun. Why isn’t the Moon pulledl+ p0kr 7q. zcnutph+2 away from the Earth?

Is the centripetal acc0ftql js ;7l)x)jen,jeleration of Mars in its orbit around the Sun larger or smaller than the e)) 7ljtxlq0j,sn;j fcentripetal acceleration of the Earth?

Would it require less speed to launch a satellite (a) toward the easa)4 wwgjfdm/ )t or (b) toward the w) dw)4m /gjawfest? Consider the Earth’s rotation direction.

When will your apparent weight be the greatest, as measured by a scale in a mo f2h wspc3 7t;dkzmu/5ving ele3 w zups ;cf2tmk/h5d7vator: 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 the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in oute. 2*frs*q+pcjhhl e dmv;v3h( r space could be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how this simulates gravity. Consider (a) how objects fall, (b) the force we feel on our femfhcr2h +jshqp(l; .dv3 v *e*et, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “w27 uq k2 qu(9ort v46thggro7free fall” or “apparent weightlessness” between step2hgw 7 ok2r roguq4u(97qttv6s.

The Earth moves faster in its orbit around the Sun in January tha z , qcv.b7mr bbtv/m1m n/gn6ed;o3p*n 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 pl /,v em .bmdbmgon31pbvn6tq* rz7c;/ane of its orbit.]

The mass of Pluto watgr*y+3e odp2/- g tzqs not known until it was discovered to have a moon. Explain how t+3 y-/d gt*qt2rg zoephis discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull1 u (0midevrg9 on the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tigdv9e ( 1r0iumdes. 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 qkyx34c ;ma2+of(hw b 1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ pulls out of a dive by moving in an arc of radius 6.00 km . What is the plane's acceleration in $g^{\prime}$ 's?    g's

  Calculate the centripetal acceleration of the Earth in its orbit around d a6x98 1/rlnd zg/l)qmju ih0the Sun, and the net for q8j 1dnzurl)d/60/mi gla 9hxce exerted on the Earth. What 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 210 N is exerted on a 2.0-kg disc +6sq*q2fya;x7mcbeh us as it rotates uniformly in a horizontal circle (at arm’s length) of radius 0.90s27+f q*my;qcb6ex ah m. Calculate the speed of the discus.   

  Suppose the space shuttle is neugk0t(y0: 3j ndq7ein orbit 400 km from the Earth's surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration at the Eaej k (:3qd07u g0entynrth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of e3yokic)midk o9) nk2k a*l378vo-ds clay on the edge of a potter’s wheel3vdd ook m2-k 7oylak3i *s9 cik))8en turning at 45 rpm (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string jmapk6y/dgzy v. +25dum) ji*is revolved at a uniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4.00+)6 z2giyy*jdu5 p avdjmk./m $\mathrm{~m} / \mathrm{s}$ 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 radin squhfq,rw.-d +yqt)b:1 yt5us 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is this result independent of the mass of th+ftqsrqhq)w. -yu:1tbd yn ,5e car?   

  How large must the coefficient of static friction be between the pr7c2 g19 e d+k)oxk q6kr.mcftires and the road if a car is to round a level curve of radius 85 m at a speed of 95km/h 2rc7x g6m.pr+kc f okek1) d9q?   

  A device for training astronauts and jet figj7skt;(lc5dbt k) 4wxhter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back is 7.85 times her own weight, how fast is 4lk 7 xd)s(c;w5kbttj 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 rozphij-7zte11 m2b7 /xrox z7btating turntable of variable speed. When the speed of the turntable is slowly increasede/7 2m1rzz p xxhtj77ob -b1zi, 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 trm /+o/ lhd.t,tg zzgd6 mc3/vyaveling when upside down at the top of a cir/t yg6 3 tzd,c//hgdm+mz. olvcle
(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 (in rj,kmnk8,8 gudbp4hy3fv s:3cluding the driver) crosses the rounded top ok,3j3fsgkyr,m8 pvb4 :u hnd 8f a 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 m-l oyuj erv1 23+kmd/cinute would a 15-m-diameter Ferris wheel need to make for the passeneuc m1 ly +/o2vk3j-drgers to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.103o3 6hbmv 9g+iaid -nd m. At the lowest point of its motion dbi +dgn3m- o3h9vi6a the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuge rotate if a particle 9.0ada0 me:rffx*1 3 0i+.kf 9ixkg505dlknu zef0 cm from the axis of rotation is to experience an accelele05aefnaxddx:35r uf fzk i0+fmkk*910ig.ration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated in a cylindrically, 7ku )-xrvpmw: 2owqe walled "roo2w - r7e)vuqmk pwox:,m." (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 aidwo9;:rqqtybpx k3t(nf *zjv,iv 681 r-hockey tabletop, and is held in this orbit by a3wv6;b9,i vt: nto(fyqd8jqz1*kxr p light cord connected to a dangling block (mass m ) through a central hole as shown in Fig. 5-
36. Show that the speed of the puck is given by

$v=\sqrt{\frac{m g R}{M}}$

  Redo Example 5-3 , precisel:tmp7xdyh p * )hlx,3sy this time, by not ignoring the weight of the ball which revo3tp hd* :mhx)p7y lsx,lves on a string 0.600 m long. In particular, find the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of -nwh; e -. nwvydyzy+*88 m is perfectly banked for a car traveling 75 $\mathrm{~km} / \mathrm{h}$ , what must be the coefficient of static friction for a car not to skid when traveling at 95 $\mathrm{~km} / \mathrm{h}$ ?   

  A 1200$-\mathrm{kg}$ car rounds a curve of radius 67 m banked at an angle of 12$^{\circ}$ . If the car is traveling at 95$ \mathrm{~km} / \mathrm{h}$ , will a friction force be required? If so, how much and in what direction? $\approx$    down the plane

Two blocks, of massesc d:y ulr.obr0 oie+j7p2+u 2x $ 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 8k:(n -h )o8o ed3 gu4pzejqrqat 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 exerte tza31zmg01g nl3 2uefd on the car (by the grou10z23geagzm un 1f3t lnd) 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 g i., .d2cuqy ts5yk5k10 subfthe 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 ckb,sg5ic 2.qu yf1.t 5kd0s uyonstant 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 circlemmc j;+os rm6/ of radius 2.90 m . At a particular instants6om;j /m +mcr, 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’s gravi4ee;xjd m8j3qnwl, ,km *lde/ty on a spacecraft 12,800 km (2 Earth radii) above tl,d/;xk,8 4mnl*mj3 jq edeewhe Earth’s surface if its mass is 1350 kg.   

  At the surface of a certa.z/vw2sx 7zkf in planet, the gravitational acceleration g has a magni.zv sf7/zw2 kxtude 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 olp yy.ehqgvkq16 /j5 +v8p c-bn the Moon. The Moon's radius is 1.7.pe-q y /k6vcp+15gyvj blh8q4 $\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet has a radius 1.5 times that k-q ,cq :c9ndd 6fkq)sof Earth, but has the same mass. What is the acceleratiodqd kcq6 :,kc)-s f9qnn due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times that of nz 5ju7*dw k6(yi5jm b.y5catEarth, but the same radius. What a7w6t 5ym*kiu djb5yn5. (j czis g near its surface?   

  Two objects attract each other gravitationally with a force (op 00ydri/t pelef-0 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 effectivepuvtf)o 2l8kmu ) 9gyu/d z06g value of g , the acceleration of gravity, at (a) 3200 m )9 /2od6g 8ymtk luzu)u0vgpf   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside th nhhx56 y(xfe5/mjyw +e Earth where thegravitational acceleration due to t6f+/e (hx5 ynj5y xhmwhe Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has five times the m,7 g:yc7voiy wass of our Sun packed into a sphere about 10 km in radius. Estic7g yowy ,7vi:mate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf d;8 px j9 v jyj0v1brn:)42;e rzwayrlstar, which once was an average star like our Sun but is now in the last stage of its evolution, is the nyy)4 2jvrrj1bx0 z9a 8rld:pve;;j wsize 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 orbii l5,:d e(uy21 vpwcvcting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince thecw2, i( vu5pydecl 1v:m 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 locasqa 33 acbf0(cted at the corners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force exerted on one sphereccba3 fq03( as by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of the planets line gkedi-z90t v8 up on the same side of the Sun. Calculate the total force on the Earth du- ie0kv89 gtzde 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 gravitn)v,y/w8cjd7 +g ryyty at the surface of Mars is 0.38 of what it is 7nvyc rjygw)8 /tyd+, on Earth, and that Mars’ radius is 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value fori2 (nkg(x np3js04axi the period of the Earth and its distann2iki a(xn0p4j 3x(s gce 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; 4+i,;apz5y - jeyqvvfaz, wg a stable circular orbit about the Earth at a height of-; vv5y;we 4jizy,fq+apagz, 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbit 650 km above the 9ta )+yg cpix7Earth. How fast must the shuttle be moving (relative to Ea xp9 tgcy7i)+arth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrjvv x dg-9w+/cl ub**oical spaceship rotate if occupants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diamex9dob vv*j*uw lg+/ c-ter 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 td7t3e1 e b:p9e4pzy1d)fk ywzj +j8l io orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above the surf1je34j9z )p +dfditzp:l7yb8e1wy keace 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 veloc/na 08f ak6ts2j,uo j na8jq:pity would a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbifq6ajaasj n02k ,88jto/nu: pt around the Earth?   

  During an Apollo lunar landing mission, the commaxjcn99odk1fg / l- 2nqnd module continued to orbit the Moon at an altitude of about 100 km. How long didngn9d1k- f9q xlocj2/ it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of ict n7kindi60,6q b nf+ke that orbit the planet. The inner radius ii nqk7dtnf k6,n0+6bof 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 k n.i(g,, *bbiq7xe ep3dutk, every 15.5 s (see Fig. 5–9). What is the ratio of a person’s apnpqg,u debeb.i3,t k*,7xk (iparent 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 astrqt)e8l:5mv : *altuqo onaut 4200 km from the center of the Earth'tlvu *8::m)aeqo5 ltq 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 cons yo ire0x4i0+rists of two stars of equal mass. They are observed to be separated by 360 million km and yr+ rxi04ie0o 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 rea1 8k8jgaq +mgf(m(5,y srsgtid for the weight of a 55-kg woman in an elevator that moves s5 qf,ig s1(gg(mkm88jya+r t
(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 cg4ysa6y/vy l8jmw raaa );)((b 4luvzord suspended from the ceiling of an elevator. The cord cayb ay vagrls4m6(( 8)yvjw 4 a/ulz);an withstand 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 orbits very near the surfaznx( +uat 3 -)(wbngca)gx i,q) ucw6lce of a planet with period T , the density (massa6 (cwq-)uwbn3+g()i)uxz naxl cgt,/volume) of the planet is $\rho=m / V=3 \pi / G T^{2}$ .
(b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min .

  Use Kepler’s laws and the period of t 3rc s ju:r.0*2c5/j cgwceossgfo 4j7he Moon (27.4 d) to determine the period of an artificial satesj*csj35ce2cw74/ gogc: fu 0r.rojsllite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, orbits hn )cqr:m(:kothe Sun like the planets. Its period is 410 d. What is its mean distance fromm(qcn::k o r)h the Sun?    $\times 10^{11}$

  Neptune is an average disb z+5q lm r+8l-emq6qbtance of 4.5 $\times 10^{9} \mathrm{~km}$ from the Sun. Estimate the length of the Neptunian
year given that the Earth is 1.50 $\times 10^{8} \mathrm{~km}$ from the Sun on the average.    $\times 10^2$

  Halley’s comet orbits the Sun roughly once every 76 years. It qr0b)* /18uo)reovlpb d yk0l 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 from the Sun.] k/ol0o)l *q0r)rev8 u1bybdp    $\times 10^6$

  Our Sun rotates about the center of,8 5eovjnskl1kgb: 7t 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 largo7ja. agau0idil8pqz* ) //ubel5us ,est moons of Jupiter (those discovered by Galileo in al0i/a8/iupu oj daz*qe g sb).u7,5l1609).
(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 peripbt- z4fcp a na+d9*6wod and distance of the Moon. -taw*cn4fabz p +69pd    $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moons, t, g3+y,jk tvzusing Kepler’s third law. Use the distance of Io and the periods given in Table 5– + kyzg3,v,jtt3. 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 a t8 m6so-xmi6,b/lik(which some space scientists think came from a planet that once orbiaomb8tkxsl, -i / 6mi6ted 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 r56n eesl +(ke1;j wu y.uaj.cform 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 enough to produce an appayc6ru +al;(w1e jne s.e5.ujkrent 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 a hanzs 7 6n0p.;lsonb6xo uging vine (Fig. 5–41). If his arms are capable of exerting a force of 14007lbx; ns 6n0upoo.zs 6 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 will the acceleration of gravity jq9ruh* cta;4be half what it ista4j9 ;rhuqc* on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin i 7lh, ylm3:*emwgv )lfmfy+a 1n a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and tha7:ewvllfmy3*y +m )hlm 1fg, eir individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates 9 -oh2d mo jtyw i7glrp/.l:8ronce per day, the apparent acceleration of gravity at the equator is slightly les2omld:t8 j9.p y/g rlohi-r7ws 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 traveling directly from thzo)oo*- 3pd q ss0sr;le Earth to the Moon experiencep) q0sdros*so3;l zo- zero net force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass isjx awi+ )h,tye:f3 :brbqw e:(kvyh+/ 65 kg, but when you stand on a bathroom scale in an elevator, it says your mass 3 ire+,hjh :xe(q: aky+wvb fb/)yw:tis 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station c rh(4zqv *vx;p miyiqro/4+w-onsists of a circular tube that will rotate about its center (like a tubular bicycle tire) (Fig. 5–42). The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed r (x z+o q4-v*vr4;h/iwypqmi(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 in a vertical loop (Fiqf:4qs5g 1 xgmg. 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 radiul 2mnrc+ y6g;c7no qy2s r, the acceleration due to gravity at its surface and the n 2c+; lr6cgq2yo 7nmygravitational constant G.

A plumb bob (a mass m hanging on a ,* q al+vmvm:ap3vg-u x8wa l(string) is deflected from the vertical by an angleqgaa l:,xw+*vauv(8 3 v-mplm $\theta$ 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 of sr dh:bl4iz* anki*r/eq: o7z f*p/l3ztatic friction is 0.30 (wet pavement), at what range oipz:d* rz ni3k*faq/ *:l4/ehr7zb olf speeds can a car safely handle the curve?

  How long would a day be if the Eaw95xk nsrby2a 5sqf;l vd ;/:lrth were rotating so fast that objects at the equator were apparently wey ls;vq w59ar slf2 ;:bkx/dn5ightless?    $\times 10^3$s

  Two equal-mass stars pv n:a0zq(n)v maintain a constant 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 constant speed rounds a curve of radius 235 m. A l5aqs p4b: u5cuamp suspended from the ceiling swings out to an an5b:4uu q 5acspgle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 tiscp5l- dhlk8)mes as massive as 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 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- k5hscl)p 8dl 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 preseq5nrc:u4 oabp y (8v/cnce of an extremely massive core in the distant galaxy M8yq45 bn: /vp ar(uc8oc7, 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 as it traverses the hill and valley showhjjm s 1c.nr3qv 04q3in 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 w1r33.0jhvsinm4c q q jould 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 (GP rn9:kgx9dj jg 5;xyp/S) utilizes a group of 24 satellites orbiting the Earth. Using “triangulati ry95gx/jxjkd 9p ;:ngon” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion ko k2yg s w)d:y1sr33jtm, is meant to orb2gs 3w):ost j3k yry1dit 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 shuttlelq)(l u bazg90e, /-q5vp;oth4ti1icvud)b w pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (4uope/h 4vwi, qlqvz1(u0)9 ga5b )ctbld-ti; 00 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 ye k;u.ey:d9:r3vi jwjg7t1 m9kos :obcars. (a) What is its mean distance from the Sun?.koei1yb : tsjdm;3kr:o9gu7v c:w 9j   
(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 act(w xkqf d/sp.l zdi.zr5 (+8qj 95nbslually "feel" yourself gravitationally attracted to someone near you. Make reasonable assumptions,bk5.jq+5n ldx (zdli .s8qzs9pw f/( r like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Fig. 5-46) at a disr9y 0gku +. onh1loo6htance of about 30,000 light-olh+ 6yu9oog h.kn0r1 years from the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the cf3b(os3yue iksa.4 i;-dnfh , orners 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 thu i kby-d ,e3i4osa3fs.;h(fne midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kg orbits th xz)p.w1q(: x2 zhdwbce 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 va4 tx;*s+lcf+vog 0iexperienced by the tip of the 1.5-cm-long sweep second ha;glvc0*v o++f as4txind on your wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weighcj u36o :cuwx urxud9);4 3niat around in a circle below you on a 0.25-ia4 d3 uxu;c u6)o3ncj9rxw:um 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 radius R in a 88ta/uidd/r e new highway is designed so that a car traveling aiedrd8/8tu/ 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 h, )hi+ )cu4nndh*yfiAcela, 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 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 frictiy hnd4)nhcuiif,*+ h)on 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.$\approx$    $\times 10^2$