Sometimes people say that water is removed 6./. -trv+g kpv: r:pxmelncsfrom clothes in a spin dryer by centrifugal force throwing the water oursn p+../vx c6:r: pmlkvgte- tward. What is wrong with this statement?

Will the acceleration of a car be the same when n 5jylp/af.m4c dwv2le7 9ph+the car travels around a sharp curve at a constant 60 km/h as whe+ 5ja/ d9vlnmp 4c hf2lpey.7wn it travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constxm23ha:eyz(q ant 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 nearx:y 2eh(qamz 3 the bottom of a hill?

Describe all the forces act:n3 .df qrt3x besc,+ding on a child riding a horse on a merry-go-round. Which of these forces provides the centripetal ac+srqf.3xdtb3,n ed:c celeration of the child?

A bucket of water can be whirled in a vertical circle without t- q,hvmvkn0uua 0(m6che water spilling out, even at the top of the circle when the bucket is upside down. Explain.(vk umaq6, 0- mnh0uvc

How many “accelerators” do you have in your car? There are at 9ru x;qhs6gihb2l 1j 7tor;.w least three controls in the car which can be used to cause the car to accegqiblh ;;jwt 27.hux9 61rorslerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crest ofl *w,qa6sf2 wedbkd49w8tn)b 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 decreaql ws2wdabd) knf86 w9t *,be4se as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean inwaeuwziq(x 7- m2 av12nmrd when rounding a curve at high speed?

Why do airplanes bank when they turn? How wou0o( 0,r.h f da(slkffjsof7:sld you compute the banking angle given its speed and r7,fkh.(dsjsf (:fros0la o 0fadius of the turn?

A girl is whirling a ball on a stri.e2 kj t)d7gusng around her head in a horizontal plane. She wants to let go at precisely the right time so that the ball will hit a target on t2ug j)se.t k7dhe other side of the yard. When should she let go of the string?

Does an apple exert a(m r,jjf7 ijy,lo)xgv7 p2j8a gravitational force on the Earth? If so, how large a force? Consider r x,i y)g7a (87,2ovjpjmfjjlan apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in 9)bj f rbt7.:upvg49kpbk g-qwhat ways would the Moon’s orbit be diff9 :ugjbf 7.tpbb49krk -pq g)verent?

Which pulls harder gravitationally, the Earth on the Moon, or the Moon o 2 ;em- f8y0+hm 9jlulqcq:hvqn the Ear0fl:;l 8qc 2y-jmqehv+m huq9th? Which accelerates more?

The Sun's gravitational pull on the Earth is much larger than the Moon's. Yn .e wimumk,(-et the Moon's is ma(,inw em-ukm.inly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.]

Will an object weigh more at the equator or at the (pwmcthm i v0wv7ywc8. 9ne fbi85,(upoles? What two effects are at work? Do they oppose each other?w bv cpct( i,i8(975wm y 8.vw0fenmuh

The gravitational force on the Moon due to the Earth is only about half the ,z tfe. m:.xekforce on the Moon due to the mk.,x.e tezf:Sun. Why isn’t the Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around the Sun larger or ;o19oxggro x00k yty+smaller than the centrip+ogxr ykt0x1goo 9 0;yetal acceleration of the Earth?

Would it require less speed to launch a satellite (a4b- ume;b),nmrg*rsbi 9 fd (x) toward the east or (b) toward the west? Consider fmnb(sx-;b )9 ri r4uge,*bmdthe Earth’s rotation direction.

When will your apparent 0p vvb6 bjt;/q9no(nusi .0 tqweight 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 uppu 6vo;nb9t(sq jt0n0/bq. iv ward 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 periodt5rab)j- qjvv q24s3 xs in outer space could be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface (Fig. 5–32). Explain how this simulates gravity. Consider (a) how objects fall, (b) the force we feelv-4)5vqxsb j3 t2arq j on our feet, and (c) any other aspects of gravity you can think of.

Explain how a runner experiences “free fall” or “appareni ln5lg /xkj;2xpb-.ht weightlessness” betweex.2 xnpjb-l5l/g kh;in steps.

The Earth moves faster in its orbit around the Sun in January th1bj )j 8(aogctan in July. Is the Earth closer to the Sun in January, or in July? Explaigtj8 )o1j(ca bn. [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 orbit.]

The mass of Pluto was not known until it was dii 1k g9;l:fqw+/cpdfz) kzdk0scovered to have a moon. Explain how this disc ;1izfg:k+kp9)f0/dqc zdkw lovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger than the Moqksdkv cc7/f:0 jaq 7dg84l8 zon'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 1.10 m from the center of a merry-go-round moves with kc:a 7sjl d 80f4zgq7c d8vkq/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 1yi,w8+, hsd h9ovg)lp980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 the S2 0 d d8t0:zs):if pgglwywdc4un, and the net force exerted on the Earth. What exerts this force on the Ear4gcdtd)l0:i8wfw:py 0g 2z dsth? 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 exertesun. z+b gi 93zoth+w-46yeapd on a 2.0-kg discus as it rotates uniformly in a horizontal to+be4y6+gs3uz. zniw ah 9p-circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbsz5yk q5bw(lu0 g 3)qyit 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 gravitati5l wqkyyz(0s5b3uq)g onal acceleration at the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clay on the edg.tdixt; f b6v;cibaxjmu)wy-.l 2+ *ne of a potter’s wheel turning at 45 rp *x;2b-i6a cuinwxd. bl.v+f m;tyjt)m (revolutions per minute) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a yr1b , xyya./nzc3 oi*k, b,+7iogtqhuniform rate in a vertical circle ofob/3 q,+abzricyk xt7hon .,y,y*g1 i radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

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

  What is the maximum speed with whicgbg;v ;x cc.; rb*khu:t3oy-w h a 1050-kg car can round a turn of radius 77tu*:gocx;r h.3bb vy-gc;;w k 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 the car?   

  How large must the coefficient of static friction be c04 3xfd5; a(cl /u(pdzokpihbetween the tires and the road if a car is to round a level curve of radius 85 m 4 (;h5k oiaz30lducxfcp p/d (at a speed of 95km/h ?   

  A device for training astronauts and jet 12qul;.k h t*ysnz 9dn5kd0ssfighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force l zd 5hssu* k021 .snqd9kny;tfelt by the trainee on her back is 7.85 times her own weight, how fast is she rotating?    rev/s    m/s
Express your answer in both $ \mathrm{m} / \mathrm{s}$ and $\mathrm{rev} / \mathrm{s}$ .

  A coin is placed 11.0 cm from the lh;bmm xp: +;caxis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains x; clmbm;ph+: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 traveling when upsidxg0 s :bsu w533wlg6jde down at the top of :ws s303l w db6ux5jgga 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 driverlj+juk -:9j 24j0soyu4vvzf1qo pnv1) crosses the rounded top of a hillkvz4jv 0l:ynjj4-1o us2 op9f1 q+vuj (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 mitnjou1kotu75;t /h1sl0d,jknute would a 15-m-diameter Ferris wheel need to make for the passengers to feel “weightless” at the topmost point5jho,7 ;1 ttlt0kojd 1us/nk u?    rpm

  A bucket of mass 2.00 kg is whirled in a jf ;jhf+bj29 wvertical circle of radius 1.10 m. At the lowest point of its 9+ jfjfh;j2w bmotion 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 centbd cul+ *9dps+rifuge rotate if a particle 9.00 cm from the axis of ulpcd++ 9*sd brotation is to experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people areng;ih7g,,rgi7 7ny xe rotated in a cylindrically walled "room." (See Fig. niy7,engh rg, 7;gxi7 5-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rotated in a circle on a frictionless air-9* +() (5 efpeetffd tmhbpug;hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a central hole as shtp9( gd(fe hf*mbe) ; +u5ftepown 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 wei hbzmp0fgvor hv4fk, 51 n5s3.ght of the ball which revolves on a string 0.600 m long. In pa.1 ro,z h534mfkhgv5sfbpv n0rticular, 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 perfectl)w2k0 aa4fn*yt3zbzkr6u h z1y 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 massek./sqk c- 8a 8g:zpujba -ecs+s $ m_{1}$ and $m_{2}$ , are connected to each other and to a central post by cords as shown in Fig. 5-37. They rotate about the post at a frequency f (revolutions per second) on a frictionless horizontal surface at distances $r_{1}$ and $r_{2}$ from the post. Derive an algebraic expression for the tension in each segment of the cord.

  A pilot performs an evasive maneuver byr lecdub*6 2o1 diving vertically at 310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripetal components of the net force exerted o* xztwejtxk *:v,4x3mhzf8 4u n the car (by the ground) in Example 5-8 when ik j * htvm,w8xu:4 xe4f*zt3xzts speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 ac2 sljzpcj,8 u5celerates 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 acceler l jcz5,us8pj2ation. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90 m sisi- ego q* d/0w19pd. At a particular instant, its a0*-dps/ gi wiqd1os9ecceleration 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 gravity on a e.vu- igsjo7- spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 -g e7sju-iv. okg.   

  At the surface of a cn3-5:j h l49s/ itc ;.l 1psacuwomtnfertain planet, the gravitational acceleration g has a magnitude of 12 ms h-t w nin/ljcfs.4 5pa 9;l31tcmuo:/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 *f+8 qdv3kj,blh(xxxb rfx-5 Moon. The Moon's radius is 1.7fxxkh-lqbd 8 ,rxb 5( j*3vx+f4 $\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 4xrddf 0i*t-c Earth, but has the same mass. What is the acceleration due to gx 4dr-0iftd*cravity near its surface?   

  A hypothetical planet has anbr:a5tc)vi s b-xb)3qbb vx) l342eg mass 1.66 times that of Earth, but the same radius. What is g near its surface? bbn3- b23)r lv)te 4c 5isqxb:gx)vba  

  Two objects attract eac3ezttycq, a/x ww;5 k +/ul.qqlq1ok7h other gravitationally with a force of 2.5$ \times 10^{-10} \mathrm{~N}$ when they are 0.25 m apart.
Their total mass is 4.0 kg . Find their individual masses. m$_1$ =    m$_2$ =   

  Calculate the effective value of g , the acceleration of gravity, at (a) 32 g vb u0)i4t99s7rdb/oyd 0kfd00 m o0by d9vt) g9 rd0ksf/i 74dbu   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside t/2hx7te m0pjzg h(9puhe Earth where thegravz0ph 2g(7thpue / m9jxitational acceleration due to the Earth is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has five times the mass of ouriwon4-sqpbx)i b64g i( 1e ul1 Sun packed into a sphere about 10 km in radius. Estiiug1bn(xw4s-614biqi pl )eomate the surface gravity on this monster.    $\times10^12$

  A typical white-dwarf star,0tmjs.-x;yyv:k9 hq h which once was an average star like our Sun but is now in the last s0x mjsykv.:hy9; thq -tage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on this star?    $\times 10^7$

  You are explaining why astronauts x-y:-)m + xu48mvs blsf-nhgcfeel weightless while orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weakerbxm-cgs-:)hy4-mv lsuf8xn+ 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 corner 5cd3qfh:ed,c0p4mx. 9 zrgs ss of a square of side 0.60 m. Calculate the magnitude and directio95s, .f xdscz3m4ch0qdg:pren 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 mjse,:*om w+kp5 (,appxws w 1 ei6v1hgost of the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assumi v(ejpw p k,w+sspg,16ex i:ow5hm a*1ng all four planets are in a line (Fig. 5-38). The masses are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , and their mean distances from the Sun are 108,150,778 , and 1430 million km , respectively. What fraction of the Sun's force on the Earth is this? $F_{Earth-planets}$ =   $\times 10^{17}$ ${F_{Earth-plantes}}$ \ ${F_{Earth-sun}}$=    $\times 10^{-5}$

  Given that the acceleration of gravity at the surface of Mars is 0.38 of qz;xxl u (nd9h t(q-r(7 w8kliwhat it is on Earth, and that Mars’ radius i8-(lkxq7qlzd ui tw9r n((x ;hs 3400 km, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the xc / +mqgxt(y9st)fq)r.v ;qp period of the Earth and its distance from the Sun. [Note: Compare your answer to that obtained using )q ttr;sxpqmfgc x.q) 9+v(y/Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satmx+7cx celncd3 e;x88ysy h -7ellite moving in a stable circular orbit about the Earth at a height ofxc+3h87cxsy7cydn8-m xel e; 3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circu yxunw jii:-kx- p/.l:lar orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the releay/pj-:l i-xuiw: .knx se occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotj/1t+9i vgkgc ate if occupants are to experience simulated gravity of 0.60 g? Assume the spaceship’s diameter is 32 m, and give your answer as the time needed for one revolution. (See Question 21, Fig 5–32.) cgvi/1g9tkj +   

  Determine the time it takes for a satellite to orbit the Earth in a cc ddu.tcs-p-8+4afwc3 6 ckuv ircular “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 wu cc ks vt--c8.d3du4c+6apfof the satellite?    s ~    min

  At what horizontal velocity would a satellite have to be launchec7 vo+/6a 7, favrqxdir;ukx7d from the top of Mt7ud,a7/fk7x;x6+v rqc ivo ar . Everest to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command module c5n7s3s )ne ylbontinued to orbit the Moon sn3b5)ynl se7at an altitude of about 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit the planet. Tux* :.wza;bc( lz mr)phe inner radius of the rings is*zz ;a crpxu. w)lmb(: 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 r/ru q31;q sgzh52wgrretay86 otates once every 15.5 s (see Fig. 5–9). What is theuew/538q ar ghtysr2;6r 1g qz ratio of a person’s apparent weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent w5do)l)l4pfr1q)b i mh eight of a 75-kg astronaut 4200 km from the center of the Earth's Moon ih1ml )4i pdqrlfo)) 5bn a space vehicle (a) moving at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star system consists of two stars l zq(hjt 5w1k, ky1u8xof equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of eltxjq851ky ,1k u hw(zach?    $\times 10^{29}$

  What will a spring scale read for the weight of a 55-kg woman in an elevator ;(sno/fm190sg awt uo that 9n1os/sgfauwot ( 0 m;moves
(a) upward with constant speed of 6.0m./s   
(b) downward with constant speed of 6.0m./s   
(c) upward with acceleration of 0.33 g,   
(d) downward with acceleration 0.33 g   
(e) in free fall?   

  A 17.0-kg monkey hangs from a cord suspended from the ceiling of a cv)upv8-1jzy n elevator. The cord can withstand a tension of 220 N and breaks as the zy c pj)u8vv1-elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surf*0 op+qm*y xweace of a planet with period T , the den0wm qp*+xo *yesity (mass/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 the Moon (27.4 d) to detert0i16 vhaqww5s ;r9e/7xnxvvqd b (e/4 js co7mine the period of an artific /v6d ro9 40is7ja1vve;n qe/wqch7x( xbs5wtial satellite orbiting very near the Earth’s surface.    s

  The asteroid Icarus, though only a few hundred meters across, or;ow+.nk bru7jk/ jx:uve9ej1 bits the Sun like the planets. Its period is 410 d. What is its mean distanojek9 :wn.u7+kb1xu;evr /j jce from the Sun?    $\times 10^{11}$

  Neptune is an average distance of 43, w n+v4vjg0xx mzri:.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 ever8 xd ktf4f xc bics9d(ex,/s1zq-;5tdy 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 neax4ff-/9dt(bzscskc1 58xqi ; de,td xrest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center9yqe:h2uz b o0 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 mass, period, and mean dm yi 5),i ag*ov65knztistance for the four largest moons of Jupiter (those discovered by Gyga 6 kmo5 ,iv*nt)iz5alileo 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 oflo *4cp 2ju+es the Earth from the known period and distance of the Moon. *c j uso+2ep4l   $\times 10^{24}$

  Determine the mean distance from Julh sg4d h11e5fpiter for each of Jupiter’s moons, using Kepler’s third law. Use the dil1d5fgh1h s4 estance of Io and the periods given in Table 5–3. 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 fragment xtgsd5m+p-/5q,kio rs (which some space scientists think came from a planet that once ,it r/pgo5k 5xqd +-smorbited 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 descr fxxb+7ubf 4 fg+qc-s(j2-fv l je,:m9r1m ymkibes an artificial "planet" in the form of a band completely encircling a sun (Fig. 5-40). 4blm:ub,jr jx-cf 7m9( kexy1++ qf gffvm2-sThe 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 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 qp p fgkg/mp,giz0q::xf78,d a gorge by swinging in an arc from a hanging vine (Fig. 5–41). If his arms are capable of exerting a force of 1400 N on the vine, what is the maximum speed he can tolerate at the lowest point of his swing? His mas,xfmf/pg 0q q:,78 izpgkgpd:s is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the acceleration o su0w9tn 0:tpbf gravity be half what it 0nsp9 0b:tutw is on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab han bz8;umd3/h9cdreqew+ x3mqtp3x;l 9 ds and spin in a mutual circle once every 2.5 s. If we assume their arms are each 0.80 m long and their individuabm3+;; m3q /xdue3 e899czp drlx thqwl masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparen2ch,elny :sp59edo:d 2f( shqt acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this efdqh p2: 2on9cd5 lse: ,eyhs(ffect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the yx wf(kqjql2)p u*f6 /yfj b10Earth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forces?pq2fluyfjq0j/ )w fk6(*b1x y    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a bathro9l ty bi1n1ma*v/n 3etom scale in an elevator, it says your mass is 82 kg. What is the accel1*ei t lvnm/ t91byna3eration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space statio7x:. kn e oezq2a) ltgxj47(cqn consists of a circular tube that will rotate about its center (like a tubular bicycle tire) (j lozq) 2e.a7n xkex 4q7:(gctFig. 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. 5–4qd +a58.sn* ybrjm(iz3).
(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 terms3 b;idxj*o.cdk *gw9e of its radius r, the acceleration due to gravity a*3*c;j .9gwi ddoxbekt its surface and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from the verz.g4ep5taq .41 bwn hiurz. n)tical by an anglepthz..1 uqn.aibnz4r4 g) 5w e $\theta$ 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 klbzn,g. fz-h;tv3n7u kwk8 / If the coefficient of static frictfk8/w,bkn.t v ;l3g7k zu- znhion is 0.30 (wet pavement), at what range of speeds can a car safely handle the curve?

  How long would a day be if the Earth were rotating so fast tym)jbgt6mp; d4f.hy ;hat objects at the equat)t j.gyp6yh;;b mm 4fdor were apparently weightless?    $\times 10^3$s

  Two equal-mass stars maintain a constant di7/mo txl7u:h.hkt o6ostance 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 c+zbg 3giebvnq32 h*q7onstant speed rounds a curve of radius 235 m. A lamp suspended from the ceiligq23vb h7zeg qn*+bi3ng swings out to an angle of 17.5$^{\circ}$ throughout the curve. What is the speed of the train?   

  Jupiter is about 320 times as masp4.uh2 6i dtrbsive 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 wupd h2i.b6tr4ould 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 presenjc+7 7qx :ogtsq5;p ztce of an extremely massive core in the dixc tqtjp;o:+7g s7q5z stant 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 the hill and valley )w5s;*,8 hsk ufgyuc9dm z0b pshown 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 larg, sbwhz)u u8mcy59pds ; k0f*gest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 satelliteoi6/w hn hr;t,f m**oxs 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 o ;xmfwoh **t6n rh,/icentimeters. 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 to,9+tf b ytjkkn+j+(y raveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km . Ej++tkt 9 y,kfjybo+(nros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in the space shuttle pursuing a sat4m--t tjpm05wk 8gox bellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth) 4gtk5jt w m8-mpxo0b-, 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 p;9;h76q +ozcqvm2; i+p kkanogxn r/p+ gzm 4teriod of 3000 years. (a) What is its mean distance from the Sunv62rokp x9m+mc4azn g pz;o7 /g;n;qtq+ikh +?   
(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 actuza y+*a kp8)sially "feel" yourself gy s8p +)kaaiz*ravitationally attracted to someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the center of the Milky Way Galaxy (Figz)rwa*hy7:i. s /: 5qd3 rox;gqnzp0ftl hbt0. 5-46) at a distance of about 30,000 light-years fromt q)w0fls n5b*g 3:hq0azz.r;7iy td:ho/xrp 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 corners of a square4m;u:a t( qwsa 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 midpo(aa:smt 4;wuq int of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 550j p c;tzaf.)k(0 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 experienced by the tip of the 1.5-cmfll d-;+-gjsn 1e t7cm-long sweep second hand on your wrist watch? d+f -n1m; jl7-ctgels   $\times 10^{-4}$

  While fishing, you get bored and start+ emj39qhe yu.o l071g lrebs8 to swing a sinker weight around in a circle below you on a 0.25-m piece of fishing line. The weight makes a complete circle every 0.50 s. What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5–10.] y mbu s1o8heljgqrel 30+7.e9   $^{\circ}$

A circular curve of rai2p woywk 8035h2 mhstm,tsa.dius R in a new highway is designed so that a car travelt ,8s .skh3mtaiwym0hop 25 w2ing at speed $ v_{0}$ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from $v_{\min }$ to $v_{\max }$ . Derive formulas for $ v_{\min }$ and $ v_{\max } $ as functions of $\mu_{s}, v_{0}$ , and R .

  Amtrak’s high speed train, the Acela,4dz;ino q6) : ilp.ej.hp fqf5 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 rou: ; fl4.fd qpn)jz. oei6qhp5inds 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.$\approx$    $\times 10^2$