Sometimes people say that water is remo6 akge7xjq+.sved from clothes in a spin dryer by centrifugal force throwing txe.ksq g j7+6ahe water outward. What is wrong with this statement?

Will the acceleration of a j-ja6 jwnr 4 k68sixrz:4j+ja car be the same when the car travels around a sharp curve at a constant 60 km/h as when it travels aroundxj68jw+ ar46j sr:j nz4ka-ji a gentle curve at the same speed? Explain.

Suppose a car moves at cogw9af4. a8q.ip pk lv*nstant speed along a hilly road. Where does the car exert the greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom oi g 4k.v9*pqa.fl8w paf a hill?

Describe all the forces acting on a child riding a horse on a mf58.bd:byil5 ikzgq* erry-go-round. Which of these558*.fiq:bk zbligyd forces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertica /86n7jch cnddxc8m-4n2 svxhl circle without the water spilling out, even at the top of the circle when the bucke 8hdmn-c chc7xnj/ 2v s6d4xn8t is upside down. Explain.

How many “accelerators” do you have in your car? There are at least three contrert)9v hwx 1o.i6g no9ols in the car which can be used to cause the car to ai91)9xe nvo.r tg6woh ccelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest of a small hpdv / kr rlc/bcvc:3*.ill, 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 arr*c c: pb3//vc vdk.lnd the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward when 1q: zj7vnk08xmate6i:0bj u hrounding a curve at high speed?

Why do airplanes bank when the nn2x1izl 3c9gy turn? How would you compute the banking angle given its speed and radius of z91ixn 3ng2lc the turn?

A girl is whirling a bae/w(a wf .+l zhxfs39cll on a string around her head in a horizontal plane. She wants to let go at precisely the right time so that the ball will hit a.hl9 afwxw+f e3s( cz/ target on the other side of the yard. When should she let go of the string?

Does an apple exert :u5mac(/t , sy tq9xqaa gravitational force on the Earth? If so, how large a force? Consider an appqt 5sc,m9qtxy( a:a u/le
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways would the M b-uv dm, y1,3ifmmd8jj2y( uaoon’s orbit be differ u, jfaumy2(mm1d ,yd38ibv j-ent?

Which pulls harder gravitationally, the Earth on the Moon, or th)m nd .eae6hz jrs.jglh4+1y 8e Moon on the Earth? Which accellsr1h4 nm za.d8y6+eg)e.h jjerates more?

The Sun's gravitational pull on the Earth is much larger than thws)hfhml(ly7/xjx ,5: dm w9z e Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull frzxd5 ,mw sh)flh/xy(97 :jmwlom one side of the Earth to the other.]

Will an object weigh more at the equator or at ;3cmvwm:l ;iirroi:nt e530uthe poles? What two effects are at work? ewi itv:lrci;: 5 uro 3;0m3mnDo they oppose each other?

The gravitational force on the Moon due-*ns q 7wja3(gdju v /c/a1rn(m*zgjb to the Earth is only about half the force on the Moon due to the Suj( -1vw7sqg3* m*gdr a/aj j nbn(/uczn. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larger or p5/s,ope,t em(xqik5smaller than the centripetal/ sm (pqkt,px ,5oi5ee acceleration of the Earth?

Would it require less speed to launch a satellite (a) toward the east or (bjcue,;rp ( w( 4lc37 cc,rfvux) toward the west? Consider the Ec;uclc7f(xr 4vc, wpu3je , (rarth’s rotation direction.

When will your apparent weight be the greatest, asw*zbclex ( 9s: 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 the s bz: 9(ecx*swlame as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer s aw(jch47oag+ pace 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 4 h(7a +aojgcwforce 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” betwm .*cqq*8gel feen step l8 fme*cqq.*gs.

The Earth moves faster in its orh1cp s5zu:25bv bd0hnbit around the Sun in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a facto c:1z5 0pu5bnd2bshvhr 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 m+/ 5y 6mnxa bq-bpq-t2v r9nvo5rmi +toon. Explain how this discovery enabled an estimate of Pluto’s mapaqr6b b 9q/nv2 o+ntt-mxrv+mi- 5y 5ss.

  The Sun's gravitational pull on the Earth is much larger than the Moon13b /5x.yys4wc66ew r jyzel u's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Considewxujs4z6 cel5e1 /yr bw63yy. r 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 19-w03u (hg0lya nukpq xdg:8 e,80 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 centripe(zgrzk *s tf0cpu4e 81tal acceleration of the Earth in its orbit around the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orz4 stcue8z( 1prfg*k0 bit 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 discus as it rotates uniforo7/)*01 vm:syr;sost hpssc dmly in a horizontal circle (at arm’s1sm)vs o0t ohcr /*pdy7s ss:; length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Ee8/ ,is c n;02q3qkuxsav y0e.m:w*ubrrhj8 marth'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 Earth's surfaceam hu wkinqcebj0s8*2 :8 uqres;3,.ymxv0r /. $\approx$    g

  What is the magnitude of the acceleration ai-1px *ptvd 4of a speck of clay on the edge of a potter’s wheel turnadptx4v - i*1ping at 45 rpm (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of e*cnl:cwo 02bdwdy(fj 9or*2 a string is revolved at a uniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5ewwb* lf2j9o 2(r:d ond0y* cc-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 radiuhzt.u h q.r7f+s 77 m on a flat road if the coefficient of static frihru zq t+7.h.fction between 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 between the ti pn yrbg, ;c;qt6p0d-gres and the road if a car is to round a level curgbngcr,6p; pdt;0-q yve of radius 85 m at a speed of 95km/h ?   

  A device for training ns o0w 9rc)x3/sb /brzq0hg /bastronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back is 7.85 0)wbbg s0rq h /sbo/zcxr/ n93times 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 axhr :*kts avaw(i 5bt:x*)ux -mu.f,ohzv+ pr2is of a rotating turntable of variable speed. When the speed of the turntable a*:w o) ph 5*k stu+vvmrxfzaru-b t2:(xh.i,is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static friction between the coin and the turntable?   

  At what minimum speed musut/4szor w2t- 4lbh8ht a roller coaster be traveling when upside down at the top of a cirb l4o - urw/2hzth8s4tcle
(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 m0b ysjqv8arju urq1a:b(//8l ass 950 kg (including the driver) crosses the rounded top of a hill (rvb qyjbj 18ua80/q/s :ru( alradius =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 minuonbi 1l ju: abxns 4fb;24.e4ste would a 15-m-diameter Ferris wheel need to make for the passengers 14 :bbj;nlb4nsse i4.u xaf o2to feel “weightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg isg1;ayw 7iq8o m:el0 zt whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the btlae: g8 1 zo;qiwmy07ucket 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 e 38 qjfyj.o 4+/zfp3on2d io8a2 akopaxis of ro+3 8qzp4oo n2f.o3yp8jok ea2a/ j ifdtation is to experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, peopj53)xs(w c2yo/unw 7b, ef omdle are rotated in a cylindrically walled "room." (See Fig. 5 23,w7 bxc5 yn()/jsofedwumo -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 frn38dygzmtr(3n :skj yk2*w* dictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) thr3ng2j kz*k3tys(dy: m8* dw rnough 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 igno t:l1qxoe a*6tring the weight of the ball which revolves on a string 0.600 m long. In p:l otet6q*1ax articular, 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 fotmp6*b/a*aoqo9h 4ij v2c, rk r a car traveling 75 $\mathrm{~km} / \mathrm{h}$ , what must be the coefficient of static friction for a car not to skid when traveling at 95 $\mathrm{~km} / \mathrm{h}$ ?   

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

Two blocks, of masses ny6nhdm9 i0)cof4 6/ si-xawx$ 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 3104g0:da,)(puftmqi6peq 2w, bq rr.h t $\mathrm{~m} / \mathrm{s}$ . 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 compo ic) 2,)esgeysgj r 7k-wb,pv3fhly7)nents of the net force exerted on the car (by the ground) in E2-7yp ls)cyrvhk7,j f e,ie))gg wsb3 xample 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 auyhr si. /1p0frea, going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangential a/ruhyfip.1 0s nd 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 hu1kb/;s xbfh.e 5qv 0worizontal circle of radius 2.90 m . At a particular instant, its accelx fus;.b/bq1k0 v5he weration 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 Eq*yl6y3 dcqgz4gji 9-5 gh9d jarth’s gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s su9-6cg5zgg yd yj*i3 d ql9hjq4rface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravic xvu)gb (,,o9qbc3l ntational acceleration g has a magni9 c )xvul(qgn,b3bo, ctude of 12 m/s$^2$ A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet   
(b) the weight of the brass ball on the Earth and on the planet? $W_{Earth}$   $W_{planet}$  

  Calculate the acceleration due to gravity on the Moon. The Moon's radius is f+w;o uli.+zk;b8ymk 1.74.l w ;omku;i+yf+8bkz $\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 of Earth, but has the sjjhz8h:0nfe1r+qm v 9ame mass. What is the acceleration due98fzjnv+mjqh: 1 h0er to gravity near its surface?   

  A hypothetical planeh y w4 ps98cpwi,1ckq*t has a mass 1.66 times that of Earth, but the same radius. What is g near ipi4sk9p y18*c wh wc,qts surface?   

  Two objects attract each other gravitationally with a force of y 8hj3g-5l k)7c fzlfz 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 f o5(q05jmftj.cm gg/value of g , the acceleration of gravity, at (a) 3200 m fc 0 fgj.mgm5(/5 tojq   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside thed+ 0g+kp (rfqzs:fp .t Earth where thegravitational acceleration drpgk+tzf qfp 0(+s:. due to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has five times the mass of our Sun packe efimoa39 1e tzo0qt)5d into a sphere abot 905a3eeo foq)z i1tmut 10 km in radius. Estimate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, whic-;7tmm j, gv*nbufnja77/ mbjh once was an average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sunn7b t7f m/jjm, *m7abuv;jg-n. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauw9x-iu0 5u cluts feel weightless while orbiting in the space shuttle. Your friends r0u9wuil5x c u-espond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are located at the corners of a square o qe8c 5u*:abbof side 0.60 m. Calculate the magnitude and direc*o c:8qbb5eu ation 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 o q-ywc cxygf.8juv(i(ts:3ao/us +d0f the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line .+uscf q8o is y3/j(tx0(:-gvycdwua(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 gravityulx8)oe( poi:+i9 aisf7c4 oj at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 k( si ifc9io:xo4u7ea 8p)l+o jm, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of tu m,6e/a ;krlm xrk(s-wfxs-3he Sun using the known value for the period of the Earth and its distance from the Sun. [Note: Compare your answer to that obtained using Kepler’s lfrmr asskk;(mex6,w - l/3u x-aws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stable circular orbit aboun.8 usyo (5r /nmklnn(n n:+djt the Earth at a height 8r:ny(mu+n./5snn ndn j ok(lof 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circu ,s pji,o(j6dxoeoj*j:r -*tslar orbit 650 km above the Earth. How fast must the shuttle be jt,rjpdx is(j:jooose* ,*-6moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupa3)af x i,k6taqhle0w1nts are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answet lwhei1 6f, xa3)ka0qr 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 in a circulan;ghu xf4*hg )r “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 the mass of the satellite? x4u *h)g;f hgn   s ~    min

  At what horizontal velocity would a satellite have to be launched from the to9d-r*pfsyvz o 4uh7 /1nfy:mlp of Mt. Everest to be placed in a circular orbit aroun1 7 r*l/z-u4hfpyv yfd: 9somnd the Earth?   

  During an Apollo lunar landing mission, the command module contin2adk4mq2c1wi *+zp enued to orbit the Moon at an altitude of about 100 km. How kc241pq w*2anzmd+ ielong did it take to go around the Moon once?    s

  The rings of Saturn are comk: ksf6w/0ib.+d cw5qqp- xrnposed of chunks of ice that orbit the planet. The inner r6 i0:wkf/. kpdqwcq- sbn 5+xradius 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 inktk ,hg3jw/*q s+oa+ q diameter rotates once every 15.5 s (see Fig. 5–9). What is the ratio of qkkw+,a jqt gos/h3 +*a person’s apparent weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a -qs vh(/m:to -1m 0 fnqddzu2r75-kg astronaut 4200 km from the center of the Earth's Moon in a space vehicle (a) moving at consdvrtmh( fqoq:u 0-2d-mnz1s/ tant 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. Tvgkovw;l7vmtz ) e/*t5 vp 5x1hey are observed to be separv7v5 ; / )vvoemgw*kptxtz15l ated 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 sca 43mi o.g)heqdprj7 2xle read for the weight of a 55-kg woman in an elevator that movesj ode.7m)h i2r3xqg p4
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspended from the oj 2.vmx. 0etj7bu am2ceiling of an elevator. The cor vmb tx .07j2j.umaeo2d can 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 orbit7gu::w bl cjco62n0i ss very near the surface of a planet with period T , the density (mass/volume) of the planc bns:0j2 g6lcwiu :o7et is $\rho=m / V=3 \pi / G T^{2}$ .
(b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min .

  Use Kepler’s laws and the period of the Moon (27.4 d) to determi*f jv4b/v ,qhfne the period of an artific 4fh /v*qv,bfjial satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a+dh il)(m9xy n few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance frlim x9n+( d)hyom the Sun?    $\times 10^{11}$

  Neptune is an average di55:vm- v0-xsdsnak jcstance 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 ra(8 xn,zt c sl6y.ip/tmt-5nqoughly once 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 from the Sun.] xt8/-tqz spct6l5i ,ya n.(nm    $\times 10^6$

  Our Sun rotates about the cx :h9, w-uqvdhrfpf-fa a+o1+enter of the Galaxy $\left(M_{G} \approx 4 \times 10^{41} \mathrm{~kg}\right)$ at a distance of about 3 $\times 10^{4}$ light-years $\left(1 \mathrm{ly}=3 \times 10^{8} \mathrm{~m} / \mathrm{s} \times 3.16 \times 10^{7} \mathrm{~s} / \mathrm{y} \times 1 y\right)$ . What is the period of our orbital motion about the center of the Galaxy? $\approx$   $\times 10^8$

  Table 5–3 gives the 5 xkjjq4+f )st1,;smj2 inwvomass, period, and mean distance for the four largest moons of Jupiter (those discovered bmqv2,s w;k4sj)ij1 xj t+5ofny 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 ekwhkd.)um4wt*-.7jf, vkpxof the Moon. x e kkvf.h)mtd * j7upwk.w,-4   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupite ,bg;fo7v670rxdihtx* zj0,+ieckvz r’s moons, using Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the valik,0+ ;* 0b6xzoiez 7,fh crtdj7gvxv ues 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 betwe79e-gokc3ju s en Mars and Jupiter consists of many fragments (which some spaceo3u-7kjs9gc e 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 thp sa61zfq,,tv9tgvql7zp/ zr0(7)mgvd4sz y e form of a band completely encircling a sun (Fig. 5-40). The inhabitants live onqvzrzvg t z1v7s/t z9mp)yp,ld7gfas 04(q 6, 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 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 fro;hab/bdwyu6 q.f)v* 3js6 0ya3vda zkm 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 tol/ )qwaavd;dv6bkh3zya .0fj*3s6byu erate 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 be half w b;q l e)m v+3ftgbcy/s w:z;d.b(ml)qhat it is on the surfaceb(elm;vz3 )bm.gq;lb +/tqw: ydscf)?    $\times 10^6$

  On an ice rink, two skaters of equal mass graob c4+:b ywn.fb hands and spin in a mutual circle oncw:yb+ .onb4 fce every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once perehvr n 9)ck ;b+ ikad*8bc2n(n 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 effect. What fraction of g is ndk i2 (*;rvach8bc9 nb+)ekn this?    $\times 10^{-1}$

  At what distance from the Earth will a spaf+) u- fw2(5t2b narf gp9ogydcecraft traveling directly from the Earth to the Moon experience zero netf d u9pa+)rgy5tg 2ownf-f2b ( force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathroom scale in an ele i tiq+;kp3ol:vator, it says :lt+ k;ii q3opyour mass is 82 kg. What is the acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space stat g6b-(d eaog,qqax1a:sb11udion consists of a circular tube that will rotate about its center (like a tubular bicycle tirae:116g s dbx-u (q q,adb1gaoe) (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?$\approx$    $\times 10^3$

  A jet pilot takes his aircra 2s29xi8zu.gt u (wycgrt9qq ;ft in a vertical loop (Fig. 5–43).
(a) If the jet is moving at a speed of at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s.    $\times 10^3$
(b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle    $\times 10^3$
(c) at the top of the circle (assume the same speed).
   $\times 10^3$

Derive a formula for the mass of a planet kif;g* l2yb)rin terms of its radius r, the acceleration due to gravity at its surface and the gra)l2b*; rkigyf vitational constant G.

A plumb bob (a mass m hanging on a 6xs)+fqfjl) cstring) is deflected from the vertical by a +xqc6fsf)l )jn 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 h wn+hy1: thy+ If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safth: n1yh ++yhwely handle the curve?

  How long would a day be if the Earth were rotating so fast that objects at the umw(ek(35d 1 5 y/+kyhoynbu;dj; tgm equator were apparegybtm yeu m1n;kkw; yh 3(d5j+ (do5/untly weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant diak65dl+ d85txeb4lpk,0 zehf stance 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 -5lakm1nra .qa curve of radius 235 m. A lamp suspended from the cek-5rlqa1mn a. iling swings out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times lhi-ifc3hi lq vy2/:1 b a(ms;cy 0fl*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 fyil l/yvfqci:bh -f2(li*c3mh0;a1s or 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 Telescopes o)zighu14 /v deduced the presence of an extremely massiv4zi1 vou g)/she core in the distant galaxy M87, 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 t r5zx zs/f8b,ghe hill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces 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 wit/x s5 grz8,zfbh the road at A?

  The Navstar Global Positioning4bqhh 1 hw4k0 x7gbpv2 System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a 70hx1vhpw2hqkb4 4gb 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 Rendpxse9ovrn*df;m. zp)4 .m9so ezvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km . Eros i 9rpf onoxvsd. ;zm)*e4s.p9ms 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 i5z2kce cu64)( do,oz xy )hse(pzz9fshuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it)9() u6xz c42poeeo, f (z sk5hydzciz.
(a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km?$\approx$    h
(b) By how much must you reduce your orbital radius to catch up in 7.0 hours?$\approx$    $\times 10^3$

  The comet Hale-Bopp has a period of 3000 years. (a) What is its mean distan* s09 ctd,( y/vh g r2lbv(5kqui8btrhce from thethuv 2rhl5dy(t /vrs089c ibgq* b,k( Sun?   
(b) At its closest approach, the comet is about 1 A.U. from the Sun ( from Earth to the Sun). What is the farthest distance?   
(c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5–29, but smaller distance travelled; see also Hint for Problem 59).]   

  Estimate what the value of G 8o i6f8okmaf*ht2gu uv5+v v(would need to be if you could actually "feel" yourself gravitationally attracted to someone ntov(62 vu i 8f ka*mo8huf5g+vear you. 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) at a diszpkpn;jjkc 66 4p/z0ftance of aboutz6;/ pzc4pfjjk0 n6k p 30,000 light-years from the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the cornzu :x.pen/ r9 g,h5roters 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 midpoit 59z.,u h :ogrxp/nernt of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 kgu39 )e jw4 z9xslx)qf0tdu*xy orbits the Earth $ \left(\operatorname{mass}=6.0 \times 10^{24} \mathrm{~kg}\right)$ and has a period of 6200 s . Find (a) the magnitude of the Earth's gravitational force on the satellite, $\approx$    $\times 10^4$
(b) the altitude of the satellite.    $\times 10^5$

  What is the acceleration experienced by the tip 3po*dc9io,q xp7ealrv a(77x of the 1.5-cm-long sweep second hand on your wriscq3av ipl79o*o ,xe( adp7xr7t watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weigeuoht4wf qufs/9;9h v*f ne ;tg,bx7+ht around in a circle belob9fx u;ont+vquh 7w ,f4; e h/tgf9se*w 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? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new-4ayv x+kyf j/ highway is designed so that a car traveling at jky -ax4/v f+yspeed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela, utilizes tilt of the cars when rkue 9j.h+whr-1x ji6 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 comfortarjr-w1 ju69he +k.xhible. 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.$\approx$    $\times 10^2$