Sometimes people say that water is removed frot + cv2xwtf0i 4 u5vpp/a swkt1(t3i7dhj*xv3m clothes in a spin dryer by centrifugal force throwing the water out kicftvdp3vas4i+p j* 1(x0w2/3t x75twu h tvward. What is wrong with this statement?

Will the acceleration of a car be the sz a,gr 77w:s rywkpv8 yjc08z3ame when the car travels around a sharp curve at a constant 60 km/h as when it travels w,pvk w rs 7ja7r0gyc83yz8:z around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a h;q lpt*s.,alt illy road. Where does the car exert the greatest and least force st.ta,ql*l;ps 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 for -2knl7i:4p fii)lja b 9p5ewices acting on a child riding a horse on a merry-go-round. Which of these forces provides eb 5liijaiwn7:)4kpp9-li 2f the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle without the water spill3l1 f6p;ih el;xz yra,ing out, even at the top zlhx6air 3y,p1e;;lf of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? There are,xj;*o cnr, ji at least three controls in the car which can be,i nr*j,cxjo; used to cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes ks6* zv7kpesx1 9xa7h;w 5bsdflying over the crest 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 hills a1e s7dz7s6h;9pv*k5 xkbx w. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inward when rounding a curve at high spee)+(zcgxh t lt5d?

Why do airplanes bank when they turn? How would you compute tv)8 th3h vq8ngd w3:glhe banking angle giq)3g83 vlnvh8w:dhgt ven its speed and radius of the turn?

A girl is whirling a ball on a string around-zb :oyz5luzy*)nf j4 her head in a horizontal plane. She wants to let go at precisely the right tznub*zy o:)zj5-f y4lime 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 gravitati5 wd:o25(oz04yc 9punwymg hdonal force on the Earth? If so, how large a forceom5nhud5d wp( 0wyz o4:yg92c ? Consider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, 1tvm iosl *zyd +9o4cm2nh/j0in what ways would the Moon’s orbit be di m1inv l24jt/*cmods0zh+y 9offerent?

Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the kua4 e7;fuzk/ Earth? Which accelerates mor ;zkka /4fuu7ee?

The Sun's gravitational pull on the Earth is much larger than the Moon's. 6 n vavlz+p4:dYet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull6ld pv a4+:znv from one side of the Earth to the other.]

Will an object weigh more at th y0(uixev -l1xe equator or at the poles? What two effects are at work? Dov-y0 ixlx( ue1 they oppose each other?

The gravitational force on the Moon due to the Earth is 9pg 5i1gauf d4uxn7 hiz+s )9eonly about half the force on the Moon due to thegix d+9)zusfu ag ne149 p5hi7 Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal accelerati wj;5y2 e v.yz(a8pluq,8geazon of Mars in its orbit around the Sun larger or smplyyaa8;,ev z25(u jw eqg.8zaller than the centripetal acceleration of the Earth?

Would it require less speed t* 0l pflg5zo/po launch a satellite (a) toward the east or (b) toward the west? Consider the Earth’s rol5g/p0z *lofp tation direction.

When will your apparenw -do usj9d7q4hh 9(n*9sy patt weight be the greatest, as measured by a scale in a moving elevator: when the elevator (a) accelerates downward, (b) accelerates upward, (c) is in free fall, (d) moves upward at constant speed? In which case would your weight be the least? When wouws dp yn dqj9oa*7-( hhu9s49tld 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 outer space could be adver0pe sg6q 9+hpasely 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 feet, and (c) any other aspects ofq6ah0p+se9 pg gravity you can think of.

Explain how a runner experiences “free fall” or “apparent weightlessness” betn (u-ku4uj.qdtq ly354( w yps4tyv0gween steps. 4ykwq4jstlt uqpu(g.3vy 054n(ud -y

The Earth moves faster in its orbit around the Sun in January tiqm 2wo3np ekuem22/. han in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main factor is the tilt of the Earth’s axis relative to the plane of its orbio 2n2i/u 3 .mkempqwe2t.]

The mass of Pluto was not known until it was disco up/nk:z-pt4zk + g*rpf+ ll8h3i+ essvered to have a moon. Explain how this discov4 pg t3+sk un*rf8i-zz/+s+elpkh:plery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth ifi)f+o*uiby4d me)k;s 5 s7ods 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 of the Earth to the other.]A child sitting 1.1o4f sskd 7f)+ eu5*d)i obm;yi0 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 1981wqqf ,v-fd-g4rxrzq *v3r) k (onp9u0 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 y l .xae8d-w f08(ubodaround the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth's orbit is a cira wdl088f u-.xoy(bd ecle 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 ub f-p9tg4t;ubniformly in a horizontal circle (at tt94gub fp;b -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 sy un ;bl )ph+3rsptvpv):bo43urface, and circles the Earth about once every 90 minl)rnp 3py pv:4vh b+3 ob;s)ututes. 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 surface. $\approx$    g

  What is the magnitude of tbx *9*3lq zymc/z4 eexhe acceleration of a speck of clay on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wh/3x9mcq l4*z z* bexeyeel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at 1to9a9 ms1lzg a uniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4.1 a9sz9tl mg1o00 $\mathrm{~m} / \mathrm{s}$ 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 rtu7vycyz,8rnd o:y5r o /a):tadius 77 m on a:d :nyy cot,r yv 5a)7t8uoz/r flat road if the coefficient of static friction between tires and road is 0.80? Is this result independent of the mass of the car?   

  How large must the coefficient of static fri d*i ,cfymldez-e(o/+ction be between the tires and the road if a e *m dlyodze,ci(f/-+car is to round a level curve of radius 85 m at a speed of 95km/h ?   

  A device for training as6 jj8ng8 nu/bdi w+b4gtronauts 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 onj 8ngwn4ubig+ /db68j her back is 7.85 times her own weight, how fast is she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the axis of a rotp*epzsbt1x* . 6fm0a/9qrost ating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 36 rpm is reached and the coin slides off. What is the coefficient of static zx*tsb6 *e1os.r 9mt0q f/app friction between the coin and the turntable?   

  At what minimum speed must a roller coaster be traveling when upside down at hvz,p vc:g+a7the top o 7,+acpz:ghv vf a circle
(Fig. 5-34) so that the passengers will not fall out? Assume a radius of curvature of 7.4 m .   

  (II) A sports car of mass 950 kg (including the driver) sp,-ch u0376g5m ung tinj9fgcrosses the rounded top of a hill -punc9h j g53,i60uggmf7 snt(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-m-diam8t(l; g. dfe 4:awtms kkaf+,beter Ferris wheel need to make for the passengers to feel “weightless” at the top4ml+ds a,f8.atbw: t e fgk;k(most point?    rpm

  A bucket of mass 2.00 kg is whirled in f7to4c- z *diow(e(v ga vertical circle of radius 1.10 m. At the lowest v7tgeico*dw z fo((- 4point of its motion the tension in the rope supporting the bucket is 25.0 N.
(a) Find the speed of the bucket. $\approx$   
(b) How fast must the bucket move at the top of the circle so that the rope does not go slack?   

  How fast (in rpm) must a centrifuge rotate if a particle 9.00 cm from the axisyy8,b*+aoseh of rotation is to ex yo,yhbs 8a*+eperience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated in a cylindr/ov(e +d1mnrl ically walled "room." (See Fiovrle(n+m/ 1 dg. 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 i o3y *z-ze*rdxls b8o;r8n 0ion a circle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m or nzz8;odi 0e-s*rb8 yx lo3*) 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 ignoring the d*uo/f 1lw062wlpr+7ro. voz7sg harweight of the ball which revolves on a string 0.600 m long. In particula+ rsro ul*.fzovpo wa177gd2lw 0/h6rr, 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 radiusrpb rjnwcd8+27 4 zig) of 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 massespp3e*0izb7utb3c ,m)q3h w d3t 9 lrbu $ 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 maneuverhhzuq1;b *-c q79i cuf 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 tangentia y z ugn8mqx7xo zv1w;oyg813*l and centripetal components of the net force exerted on the car (by the ground) in Eygw1g8m ;o 3vxu8o*nzq 1xy7z 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 acc+,pbn+uy n ff+9gm53b qo png;elerates uniformly from the pit area, going from rest to 320km/h in a semicircular arc with a radius of 220 m. Determine the tangential and radial acceleration of the car when it is halfway through the turn, assuming constant tangential acceley;f+ no3ng5gm, 9pb pqnf b+u+ration. 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 o7i gc/txp:k.fsg8x8cp lr:te (m6 h6nf radius 2.90 m . At a particular instant, its acceleration is 1.05 g.ks8cth8ix(:f6pln7p e gcxr m6/:t $\mathrm{~m} / \mathrm{s}^{2}$ , 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 aw4fpq6u 1f*/x ph x( 3xqjmht0 spacecraft 12,800 km (2 Earth radii) above tu 4*qxphxm p1/qwfj0(63th f xhe Earth’s surface if its mass is 1350 kg.   

  At the surface of a certain planet, the nc(r w-u vw5+vgravitational acceleration g has a magnitude of 12 m/c wvv-r5w(un+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 M3 kv, rj1pe9qr s1r8cpoon. The Moon's radius is 1.74 rp 1 k8,vepr31 9rqjcs$\times 10^{6} \mathrm{~m} $ and its mass is 7.35 $\times 10^{22} \mathrm{~kg}$ .   

  A hypothetical planet 2pknl yfr. 19chas a radius 1.5 times that of Earth, but has the same mass. What is the n fk21rlcp.9y acceleration due to gravity near its surface?   

  A hypothetical planet has a mass 1.66 times tph:a t,s4etk8hat of Earth, but the same radius. What isk4tt sp:a he,8 g near its surface?   

  Two objects attract eacg9epkc3 nj(a) h 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 kz-)mtcg59* voikbq7w q7sb/ value of g , the acceleration of gravity, at (a) 3200 mi *7q- v)kkqg/tm bbsc7o5 9zw    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from thh hh4f t uo1ky34d7zj,e Earth’s center to a point outside the Earth where thegravitational accelerthku7d,yfz 14j h3 h4oation 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 ourhz;) k:f )a4xqvvtwg/ Sun packed into a sphere about 10 km in radius. Estim/wvqa ) t4xfg:) kv;zhate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an average star like our Sun but :8xhe90t,akkl-e qe w pi(v y4is now in the last stage of its evolution, is the size of our Moon but has the mass of our4 e -0:lviek(k e9p q,x8wayth Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts feel weightless wh ckj7,p 6ouw-gile orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface,-7c jgupo6w k in terms of g.   

  Four 9.5-kg spheres are located at the c samzzs(t/o7m2 hf), horners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force exerted on o7)sz2s t (hof azhm,/mne sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred yeibniv-r58(5vyiv - ueni)1 vrars most of the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiteriy5-n1(u5-i rvi8b)nvirevv , 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 acceleratid*g. :/3fsg mrk20bytq) e vlmon of gravity at the surface of Mars is 0.38 of what it is o 20tr/ s)qmlg*.kb e:3vfmdygn 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 for tn1. d(0cojk :z jv0bf8aabh 1q)n fp:pk7zd+fhe period of the Earth and its distance from the Sun. [No:zfo :(h080 qzjd )nvk1pk bnbf1jpca +7.d afte: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a saaa h ty3( :de83uwq 92+hhihyu2wnekx 2oe8p1tellite moving in a stable circular orbit about the Earth w3+ e2(yuw1ipeohha 8q2h9e28:kx tadyh3 n uat a height of 3600 km.    $\times 10^3$

  The space shuttle releaseb 1qbp xcwhg + ulhr21.0xr1n5s a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release ocu wgr h1+hrp2n.x0cqxl1 bb1 5curs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupoe wllodm ()mrq j3x4v,1-2clants 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 on12-e 3jr,o l(m)cl4lm xdvwqo e revolution. (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit the Earth in a circue9c/x uloe.j gch ze+ 9g+h-f6lar “near-Earth” orbit. A “near-Earth”.+ -c +x ehuee9/6gfz9jo lgch 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?    s ~    min

  At what horizontal velocity wzsi 0aad7xy. /ould a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbit arya.sd/0 ax7iz ound the Earth?   

  During an Apollo lunar landing mission, the commz jg-m71q+b3si iq kktx ok u5oo1..5dand module continued to orbit the Moon at an altitude of about 100 km. Hoj.do15gz3koxi o bq. +quti7m5- 1kksw long did it take to go around the Moon once?    s

  The rings of Saturn are cot6wdep *uvt+ ce54 mg:mposed of chunks of ice that orbit the planet. The inneegvc+ 5p6:mtwudte4*r 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 1f(nieq n.bxn n; gy3fp/r):(e0h:tkxhh 93 tn5.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to her real weight (a) at the pgnn3 if b;( er)nx:: .0 3 qnnhthfhye9x(tk/top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75- eyi0 b4;dn,sx3v fcfkbgo3u4gx4:3rkg astronaut 4200 km from the center of the Earth's Moon in a space ve3e b04vg3,rn ixf4f3osdybx :; ugck4hicle (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 consi9jq/wx :d7wwk sts of two stars of equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years tojkdww:9/7x qw orbit about a point midway between them. What is the mass of each?    $\times 10^{29}$

  What will a spring scale read for the weight of a 55-kg woman in an;1 +o 0 mdauyov1/v2i9a dzb*mpxm s3j elevator that moves mo1vzxj; +*m2 3mdo/baudv0ia ysp 91
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspended from the ceiling of an si)j j7dnwdkvj86k.35e 4 uhpelevator. The cord can withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitudei4 njkupk.6 )5vh38s7dd wjje and direction)? a =   

Show that if a satellite orbitspuit/k5,ioct-m0o 8 s very near the surface of a planet with period T , the density (mass/volume) of the planet itis,8 0/ioom tuc 5pk-s $\rho=m / V=3 \pi / G T^{2}$ .
(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 d1dot 4qymi7r0y3b(vi + , tkwMoon (27.4 d) to determine the period of an artificial satellite orbidyk0+i1 rdm,iowvq tb3 (4 7tyting very near the Earth’s surface.    s

  The asteroid Icarus, though only a ferf59-kg *rctnp 60yde w hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean df0drgt6n9c 5erk *-p yistance from the Sun?    $\times 10^{11}$

  Neptune is an average distaj*q/z5b,ih k fnce 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 t 5 0smvz;quup8 g5pzs.he Sun roughly 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? [Hiu;5pgq8 szmp 0.zu5s vnt: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center of 2w0lwa-hev seyv3 vq ir)6s3m/:x;cethe 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 distanmx z5s 0*zc 5xda,djj6f2q*kkce for the four largest moons of Jupiter (thx skd*kf,02zca d*j56mxq5 zjose 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 oscm.pth1 nj /(f the Earth from the known period and distance of the Moon. /s.mc tp(n1j h   $\times 10^{24}$

  Determine the mean distance from Jupiter for each ohlzb+.f7h1o ph te8kx qj d8)9f Jupiter’s moons, using Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compare to the values in the Table.j8bto1xqpl7 dh+. )8 kefh h9z
$r_{\text {Farip }}^{r}$ =    $\times 10^3$
$r_{\text {Guarymale }}$ =    $\times 10^3$
$r_{\text {Culimio }}$ =    $\times 10^3$

  The asteroid belt between Mars and Jupiter coh 51x;jglrlfm8 95/2uctk:i-x om g r6edxf6 onsists of many fragments (which some space scientists think came from a planet that once orbited l8ol1k;jx2mtrm-g x/6e6o9rdic5 g :hufxf5 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 ak d 98ao9u*nzartificial "planet" in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the u98ak 9 noda*zinside 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 im)qdf: b)dc abl h o; ygz ttx7k/wr/*++e7n/dn an arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the e); *cdab/dt l tnrdm7 oz /b/fq h)yk7gw+x+: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 surfabvetl 8og4 q.;ce will the acceleration of gravity be half what it is on the sutlb.; o8v eg4qrface?    $\times 10^6$

  On an ice rink, two skate0neqhy7 :ra1 p q.o7cmrs of equal mass grab hands and spin in a mutual circle once.1emycp: rq7ano70q h 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 per day, the apparent acceleration of gravity atn1e-)oasd- cau ; qc,g the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the 1;q-gn, uca a cs-)deomagnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth 0ghc)xv,wkyb* z m i ;lz;e2b1will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth am vbzz ;,w;1y)xg e l2hb*c0iknd Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, i2l1; 0um i jkbi1k6a mbvoi*,but when you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the*,0;oi m1ki am vjb2b1ul6 iik acceleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tubgkt1s+bj7dd bt0ct8nu2)r p 6e that will rotate about its center (like a tubular bicycle tirendtkjd0p c8us t1t6rgb2 +7)b ) (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 aircraft in a vertical loop (Fig. 5–43).8 qj+ +dvgsg5cx hj6;u
(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 massh f;pnxiv;)7 -qaab-s of a planet in terms of its radius r, the acceleration due to gravity at its surface and the n)saxif vbq;-p-7a ;hgravitational constant G.

A plumb bob (a mass m hanging on a string) is deflec4jx d4.l7nr/8 3tyqx0jnfy qoted from the vertical by anof0 /l yyt43xrdnqj7x.8q4nj angle $\theta$ due to a massive mountain nearby (Fig. 5-44). (a) Find an approximate formula for $\theta$ in terms of the mass of the mountain, $m_{\mathrm{M}}$ , the distance to its center, $D_{\mathrm{M}}$ , and the radius and mass of the Earth. (b) Make a rough estimate of the mass of Mt. Everest, assuming it has the shape of a cone 4000 m high and base of diameter 4000 m . Assume its mass per unit volume is 3000 kg per $ \mathrm{m}^{3}$ . (c) Estimate the angle $\theta$ of the plumb bob if it is 5 km from the center of Mt . Everest.

A curve of radius 67 m is banked for a design speed of If tqk4av*- hn;ocvh/9 pw he coefficient of static friction is 0.30 (wet pavement), at what range of k o-pnh4c*;v9w ah/vqspeeds can a car safely handle the curve?

  How long would a day be if the Earth were rotating so / .kv). )a;gf2yo (xgjuakomdfast that objects at the equatoroyx(kj.ad); f ma2vk). ogg/u were apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain ,t5l em.cup ac2 h:-xva 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 cgy 4ygr 8*tw2aurve of radius 235 m. A lamp suspended from the ceiling s2*rwa4 gyg 8ytwings 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. Th r7gkaxw)n,1wk i 6:auus, 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 i1wu :w6nk )aagxk, 7r's. Calculate the number of g 's a person would experience at the equator of such a planet. Use the following data for Jupiter: mass =1.9 $\times 10^{27} \mathrm{~kg}$ , equatorial radius =7.1$ \times 10^{4} \mathrm{~km}$ , rotation period =9 $\mathrm{hr} 55 \mathrm{~min}$ . Take the centripetal acceleration into account.    g's

  Astronomers using the Hubble Space Telescope deduced the presence of an exsu/ k.+len.u3bs 4t vhtremely massive core in the distant galaxy M87, so dense that it could be a black hole (from which no light escapes). They /4+tbhnl s3.kus .ue vdid 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 shown insy-cj vx7w.*j 1tquwbbn )/f; pwk,4y 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 compaw4y)buq / fw* 7yck,j-1nws; .vtxpjb re? (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) utilizesuu3 oidi;c,b + a group of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each,u;u+i ic3obd 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 km, wny3sw covp7 .y9)5k2tp tpz. is meant to orbit the asteroid Er 72 p zpky9yct3.wtn wp5.)ovsos 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 spacezgy g73vpkv(p g 2w2(c shuttle pursuing a satellite in need of repair. You are i py3vwgpg72 g(cvz(k2n a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it.
(a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km?$\approx$    h
(b) By how much must you reduce your orbital radius to catch up in 7.0 hours?$\approx$    $\times 10^3$

  The comet Hale-Bopp has a pe +ihfd:-hbr/pk4fxwnl 2c2 e:riod of 3000 years. (a) What is its mean distance from the Sun? xpd h-i wl2hnb:r2 k/cf4f+:e  
(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 jzw(m/t uw+b) x05d;e /uwymxactually "feel" yourself gravitationally attracted to someone near you. Make reasonable assumptions, liuw/xdw) tzb50/ +ej(uw mxm;yke F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Fig.6jg0 f3e5lrq i 5-46) at a distance offj r3lqe650g i about 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 cornftegsf b-jdhf go1 /5x;80h35swdxr r w6 ,ja,ers of a square 0.50 m on each side. Find the magnitude and direct ;w0saf,x31,do6d -hrsff/g5 jhbgw85tjex rion of the gravitational force on a fifth 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 5s( ao.ol ea jye5q v1.zyv1v29500 kg 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 expldlgy.ofx i1k/vkum5(*;v bohn (*1perienced by the tip of the 1.5-cm-long sweep second hand on yourivoglk*yp n/f*d bx( (115 ;.kvmhulo wrist watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight a.h u5yzw9t2gnjn/rr35ladv6round in a circle below you on a 0.25-m piece of fir/vy.zj 2tdng6 55a 3nulw9hrshing 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 highwpx o mjg07/.sway is designed so that a car traveling at spgxs 7wp/j m.0oeed $ 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 Acedag ;dktsihsy: 2,(bq-c*(t yslrv*)la, 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 friction for q* s ia)y td:blg -ctk*,sv;2d(sr(hyce 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.$\approx$    $\times 10^2$