Sometimes people say th6oa9;y m3r f-qs9ld poat water is removed from clothes in a spin dryer by centrifugal force throwing the water outward. What is wrong with this statempq ld3m6;s-f9oroay9 ent?

Will the acceleration of a car be the same when the car travels around a sharp c1wo .8-rzrnst8 i;mdv urve at a constant 60 km/h as when it travels around a gentle curve at-8dr vw8s; i.orzt1 mn the same speed? Explain.

Suppose a car moves at constant speed alon9sca58 s ddp1xg 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 tw1spacsd895d xo hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a child cdiut 0w-mdth)9,e 2oriding a horse on a merry-go-round. Which of these forotcuwi d20) met, hd9-ces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a vertical circle without th(fb ymoz, 0o99huj yi.e water spilling out, even y,f(y9 mo0ohiub jz .9at the top of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have in your car? There are at least 15) urkso3/ne ;n mbzqthree controls in the car which can be used to cause the car to accelerate. What are they1ro)uqk5mn enb z ;3/s? What accelerations do they produce?

A child on a sled comes flying o /iqtj6xjsuqf * 3j6t56uw s.jver the crest of a small hill, as shown in Fig. 5–31. His sled does not leave the ground (he does not ac5 q6is sjwx*j6uu3t /6jf.tqj hieve “air”), but he feels the normal force between his chest and the sled decrease as he goes over the hill. Explain this decrease using Newton’s second law.

Why do bicycle riders lean ini bvsl96p0: mj*o(2c4q 32hfyu usnsoward when rounding a curve at high speed?

Why do airplanes bank when they turn? How would you cp giuella5lr ;):n5j2)p +s w2uq.ory ompute the banking angle given itsr.p 5lpo:ly)u+n) 2wa25ul;i sqrej g speed and radius of the turn?

A girl is whirling a ball on a string around her head in a.b2 vt.f8yl ns horizontal plane. She wants to let go at precisely the v28s. t bylfn.right time so that the ball will hit a target on the other side of the yard. When should she let go of the string?

Does an apple exert a gravi1o84 gd n( w/3holkp/r5otwnjqn74 y htational force on the Earth? If so, how large a force? Consider an apple 7p438rj(hn4owk1tg n/ y oqd wlnh5o/
(a) attached to a tree, and (b) falling.

If the Earth’s mass were doubl m tbl8z4af7x5o ,ci0le what it is, in what ways would the Moon’s orbit be different?lom5iz 48a,7fb xlc0t

Which pulls harder gravitationally, the Earth on the Moon, or4q9 gq8heqm q: the Moon on the Earth? Whihe94:q qg qq8mch accelerates more?

The Sun's gravitationl1s on g5 *8co ;xyh0ubl-1unpal pull on the Earth is much larger than the Moon's. Yet the Moon's is mainly responsible for the tides. Ex1g*pb l;-hoy 5 n1oc8nusuxl0plain. [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 poles? What two egu f58.rtzz 6 6:ptv4 u2fwyzxffects are at work? Do they oppose each other?5:u2fzz6tw48z. guv6f xp t yr

The gravitational force on the Moon due to the*visd :ax y-fd.ww1l9 Earth is only about half the force on the Moon due to the Sun. Why isn’t the Moon pulled away from they-: dv i.fw9x 1dw*las Earth?

Is the centripetal acceleration of Mars in its orbit around thek hzjo:v.rkbd -gm+ am-42h;k Sun larger or smaller than the centripetakzr-j4ako v.mg-hb+h : m d;k2l acceleration of the Earth?

Would it require less speed to launch a sa/ zou31 8mazahtellite (a) toward the east or (b) toward the west? Consider the Earth’s rotati/8za3o amu1h zon direction.

When will your apparent weight be the greatest, as measu 5gs/e -sias5ored 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 casee/ssa5 igs-5o would your weight be the least? When would it be the same as when you are on the ground?

What keeps a satellite up in its orbit around theEarth?

Astronauts who spend long periods in outer space could be adversely affect xc-5 f9/pvalj)kk 9vl 6us/qfed 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 othera j5-9 kc)xlpkf 96/qvfu l/sv aspects of gravity you can think of.

Explain how a runner eh;l f*8*hozs5w iitm2 0dww 8pxperiences “free fall” or “apparent weightlessness” between steps0*;mdw8pit 8woh2 5l* iwhsf z.

The Earth moves faster in its orbit around the Sun in January than in July. fceqjpuboj4( ba:n2cl 6 v*q0c23mz 9Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factmccfc naoujq 23be v6b:(z*pql0249jor 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 +3 /c l0 htq q2t(x0p8tyjrpc8xyo9zy; jqw*lmoon. Explain how this di(q; lhpoxqly/r t0q89y+8 j2wcjy ttz3p0*cxscovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth is much larger th tl/)qd*vc+i aan the Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull l cq+v iadt/)*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 1980 7qbjz 9j 576c/boy;eo9iwlhh$\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 o c3/ lpa;lb fre)3x.jacceleration 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 orbit is a circle ofeo a 3fr;p bl3/cl.jx) 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 uniforkuq 0 ovd;t2*pmly in a horizontq2 *pd 0;tovkual circle (at arm’s length) of radius 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttl r )1w7v f8to5sgylas*e is in orbit 400 km from the Earth's surface, and circles the Ear7r5sw1 vta)y*ogls 8fth 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 surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clay2ksuap.z kzxv7c ckjz ;/e14( on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if the wheel’s diameter is 32 cm? zzjkxa 12e/up ( k7v.;zc4cks   

  A ball on the end of a string is revwmh ty0/u(1hcolved at a uniform rate in a vertical circle of radius 1 t (yuchwhm0/72.0 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 which azhj - 1do .ifro6h-ps1e sqbv7e3l( u8 1050-kg car can round a turn of radius 77 m on a flat road if the coefficient of static friction between t8.j1rls-6bp 3dqvueh1-oi(fhz so 7eires 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 tires and)spdyq7*lzm )pva ;;o the road if a car is to round a level curve vyp;q; ol p7)*a m)szdof radius 85 m at a speed of 95km/h ?   

  A device for training astronauts and jet fighter pi-v7yw,lz/f.nxiwbv z8c 2w k6lots is designed to rotate a trainee in a horizontal circle of radius 12.0 m- i 26znvv7xwwf/yz8w.l,cbk . If the force felt 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 axis of a rotating turntable of variablmt 8se5cb.s5anex-c s5 49t*m cku0 mqe speed. When the speed of the turn4bs k c95u.qate5x5c0s* mm-nte 8mcstable 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 must a roller coaster be traveling when upside down(do1qsy 5ofeb-q05 ei at the top of a circle 5(1o eei5fdsb 0oqq-y
(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 kgeqe .t1zf.+6t xow(qz (including the driver) crosses the rounded top of a hill (radiwfqzqe.to+ .6ze( tx1 us =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-diametersexpyr+ajav 7 6r7m7 th08o g3 Ferris wheel need to make for the passengers to feel “weightlesgyp3r8jvasmor 76t7aeh x+07 s” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled 15:3j jdlx lfvp7 ;8tulm::zaldjc( iin a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the bucket is 2:ftlud;daj 8zpjm xv317(5c ljlil: : 5.0 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 rotz b+k km8:yn4sjjz 6u0ate if a particle 9.00 cm from the axis of rotation is to experience an acceleran4bu 8+zyj0km:6zsjk tion of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotadsh04 t 7 t*lfr0 kmq9.x)cm;qjerp:fted in a cylindrically walled "room." (See Fig. 5-35.)ejx fm;9ht0 f.*l:7 kdt rpr0qqms4 c)
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 frictionlesef4f89gh *lo./cns-goks0m ta8av6 v e ui u7(s air-hockey tabletop, and is held in this orbit by a light cord0 u*f /lat8 ko gmsg8(vinh-uc4o.6afv s7 ee9 connected to a dangling block (mass m ) through a central hole as shown in Fig. 5-
36. Show that the speed of the puck is given by

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

  Redo Example 5-3 , precisely this time, by not ignoring y z,lrh (c2ep1the weight of the ball which revolves on a string 0.600 m long. In particular, find the magnitud1yl hpc e(2rz,e 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 radiut h*vx ft(-oq/s 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 masseza0 ov+zk v7.n1 ( -fcxk3g coakze6.bs $ 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 vs:kbl6v 1h 9gfertically 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 centrip4l 6kfz0e kox*6ei gr f.4d my6-pugt9etal components of the net force exerted on the car (by tf9i60fkp4td oez 6rgyg4ml .u*kx e6-he ground) in Example 5-8 when its speed is 15$ \mathrm{~m} / \mathrm{s}$ . The car's mass is 1100 kg . F$_{tan}$ =    F$_R$ =   

  A car at the Indianapolis 500 accelerayv*a azwcrytxm 4jz -a/4j570tes 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 acceleration. If the curve were flat, what would the coefficient of static friction have to be between the tires and the road to provid t wma4jv 7z-jcay0 z54yr*a/xe this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizont a ;gshys8egho3wwd-x q7xc4 2 ry97q)al circle of radius 2.90 m . At a particular instant, its7 49-) 3ow8 ye h;g2rqshs xcxgqdway7 acceleration is 1.05 $\mathrm{~m} / \mathrm{s}^{2}$ , in a direction that makes an angle of 32.0$^{\circ}$ to its direction of motion. Determine its speed (a) at this moment   
(b) 2.00 s later, assuming constant tangential acceleration.   

  Calculate the force of Earth’s gravity on a spacecraft 12,800 km (2 Eart t m5ch9n2yre *gkv:m6lo+sg/ h radii) above the Earth’s surface if i :* +m nv926/ekrhmycgg ols5tts mass is 1350 kg.   

  At the surface of a certain planet, the gravitl; (dvs j)rd/fational acceleration g has a magnitude of 12 md v /)rdls;f(j/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 accelerationl0kmb oa 4*ug:g, n/qj due to gravity on the Moon. The Moon's radius is 1.74 :q,n0m log*auk jb/4g$\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 thehfd;g tg(5 br9 same mass. What is the acceleration due to grgr htg5b f9;d(avity near its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, but thu0z5y6q dpp dw9f.r 5ge same radius. What is g near its sfd0 u q6w5pz.5p dg9ryurface?   

  Two objects attract each other gravq y:xtefa -89pydm5ys)mn f;9e/bz)ritationally 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 l0(f mjw19i+w7nq+b ni qjh-b of gravity, at (a) 3200 mi-+7qlbj1j 0m+f n n9bhw(qwi    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside lzk w:f-h4;d 2uav51dbn0ugmthe Earth where thegravitational acceleration due to the Eartw lauz5v2b 0dhn-g;m 1fdku4:h is $\frac{1}{10}$ of its value atthe Earth’s surface?   $\times10^7$

  A certain neutron star has five times the mass ps 6 gx-6okcu8of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity 6x-8kpgo csu6 on this monster.    $\times10^12$

  A typical white-dwarf star, which once was an averagebzvn6sk6w d;so :: g+p 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 Sun. What is the surface gravity on this wnoksg:zp 6d:;b +vs6 star?    $\times 10^7$

  You are explaining why astronauts feel weightless whou;s, ytl75 ff vf-p0file orbiting in the space shuttle. Your friends respond that they thought gravity was ju-f fs7 fy;vouf lt0,5pst 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 +0kah8p j a e+jfjqm /-2 lg,:bpdcud7of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force exerted on one 2j- uqlec7mj :/pdg +bhafd+ka0j, p8sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred yeai-fz,ehu . /vars most of the planets line up on the same side of the Sun. Cala ,eif/z-uh.v culate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line (Fig. 5-38). The masses are $M_{\mathrm{V}}=0.815 M_{\mathrm{E}}, M_{\mathrm{J}}=318 M_{\mathrm{E}}, \quad M_{\mathrm{S}}=95.1 M_{\mathrm{E}}$ , and their mean distances from the Sun are 108,150,778 , and 1430 million km , respectively. What fraction of the Sun's force on the Earth is this? $F_{Earth-planets}$ =   $\times 10^{17}$ ${F_{Earth-plantes}}$ \ ${F_{Earth-sun}}$=    $\times 10^{-5}$

  Given that the acceleration of gravity at the surface of Mars is 0.38 of what itriws.;uj) xss)tr/y(t 7dz( x is on Earth, and that Mars’ radius is 3400 km, determine the masrj)i s/u.x;drztsw()x(t7 ys s of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the period of the )2 x l5 1lqkdr+iv3mcp ot)+ulEarth and its distance from the Sun. [Note: Compare your answer to that obtained u l)pdimrc+3u)txq+2k 5llov1 sing Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite movisonmi90t 8sq7wtu-+ri:hzy 8ng in a stable circular orbit about the Earth at a height of 3600 km. h :78tsoi9z8iw0s umqn rt+-y   $\times 10^3$

  The space shuttle releases a satellite into a c zlv;5vnb2puzyg-a:u ; oq99q9bie5nircular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs?;e -ub599;vgnon9plq qaz2ib5zu :vy    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupants are to experiu drfv029ljgf 36 j:rmence 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 Qjr j60 u: 9mdr23vflgfuestion 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit the Earth 7a yqon823l0 rrt-ah lwt +s;uin a circular “near-Earth” orbit. A “near- oyt+s3l; 8a7-wl 0n atr2uhrqEarth” 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 would a satellite have to be launchedaq/2p3hv2ewgr-i li) -hwxtp 4qq8 z 5 from the top of Mt. Everest to be placed in a circul2pi23haq l/-qv45iz wxw)qtrhp8 g e-ar orbit around the Earth?   

  During an Apollo lunar landing mission, the command module continukzw9s wdjl 01e-ho-q7ed to orbit the Moon at an97 qw-lkeozw0d-j1hs 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 sfba))rhyvj h rnr+/ 6br715lthe planet. The inner radius of the r1 n 6rbr )rhsyrl+5fjav /)bh7ings is 73,000 $\mathrm{~km}$ , while the outer radius is 170,000$ \mathrm{~km}$ . Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn's mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 $\times 10^{26} \mathrm{~kg}$ .


$T_{\text {irner }} $ =   
$T_{\text {outer }} $ =   

  A Ferris wheel 24.0 m in diameter rotates once every 15.5 s (see Fig. 5–9). pc*e 9-7n5rxxab m6/ zh(zegoWhat is the ratio of a person’s apparent weight to her re bxczm(6ez* 7/h nx9-o5ger paal weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from fp n1 xm-p(woi:g w9he3r /gj0the center of the Earth's Moon in a space vehicle (a) movinxg19pm:igjehfwon03w /-r (pg 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-sta1y/0t x9ftg6giy 9 ldpr system consists of two stars of equal mass. They are observed to be separated by 360 mill 9d61yx/iyp ttf0lg9gion 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 scale read for the weight of a 55-kg woman in an elevator zmame98k l:(rg3sn,j;ua xz 2that ,a2a 3zeml9: xj su8mn(gz k;rmoves
(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 frobhv mtp2l7vmlwp*/ ,4m the ceiling of an elevator. The cord *7tp v4/2hmbm ,pwvllcan withstand a tension of 220 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (magnitude and direction)? a =   

Show that if a satellite orbits very near the surface of a planet with i1 pqtn9apbdpfz)1x7.; ayy.period T , the density (mass/volumy1 9q.p.xptd7 ;)a1i bnazpfye) 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 pbdp6w 8i:psqfk 0n 25heriod of the Moon (27.4 d) to determine the period of an artificial satellite orbiting very near the Earth’s i5 p6kb:8p2dn0hs fqwsurface.    s

  The asteroid Icarus, though onlyv hsu,7ec /2 w-w-.z),dxmdoco0 sc id a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is w0o,od2d/ 7chz.mwu ses)dc,-vc ix-its mean distance from the Sun?    $\times 10^{11}$

  Neptune is an average dggi 3a-jsx5to n; )za0istance 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 - nb(zheo9v-x the 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 planezveo9b-n- x( ht’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.]    $\times 10^6$

  Our Sun rotates about the centen9b 1unx9ozhjx54jp,/ t my h8r 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 distance for the four larges uh-sl-e uk1i(f* irfkfm*1 /tt moons of Jupiter (those discovered by Galileo in 16 rhu f/mtu e*kif1lifsk (1-*-09).
(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 ofuiwgv4w6 l4.d the Earth from the known period and distance of the Moon. d gwwu46i 4lv.   $\times 10^{24}$

  Determine the mean distance from Jupiter for each of Jupiter’s moonzfq sn90 kpi;.s, using Kepler’s third law. Use the distance of Io and the periods given in Table 5–3. Compaz 0i;spkf n.9qre 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- :)in oq)la:hl ho1oj Jupiter consists of many fragments (which some space scien:o )o)l:n o lq1a-hjihtists think came from a planet that once orbited the Sun but was destroyed).
(a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun?$\approx$    y
(b) Can we use these data to deduce the mass of this planet?

  A science-fiction tale describes an artificial "planet" in the form of a bmkp+s.7aftpqp/v- .oz9vzw -and completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year,a+-p9 w. f-svzk pomt/pvz. 7q in Earth days?   

  Tarzan plans to cross a gov 5(uvul*ut*w*q1pu)oncm 64 o yh 4forge 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 i5)cv yuouv*1*qmtf*uou6 n lh4(w op4s the maximum speed he can tolerate at the lowest point of his swing? His mass is 80 kg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the a*5 )w9wv.v v,9xolofyj4 uq+xkkmj kx j,.a.xcceleration of gravity be half what it is on the surface?qmka) .xj4. .f9jk,o +vku x5y x*w,owxvl9vj    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mu(d3c;y jpco zn-5h lq4tual circle once every 2.5 s. If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg4 cczd3(q -noljh5;py, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day, the apparent accelera /p7 npw-2xwdrtion of gravity at the equator is slightly less than it would be if t2wn- wpdpr 7x/he Earth didn’t rotate. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the E.(za2ns ,wmv barth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Eazabm2 ,w v(sn.rth 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 fljy2*u m6w;) fbd0-x5li2fy gti o5lan elevator, it says your mass is 82 kg. What is the accelerati 2;* ) -i 5flltyxlyjfubim6g2odfw0 5on of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circular tube that will rteio+5qy 2 uufnm:q*olowu 1vh* 5-5 rotate about its center (like a tubular bicycle tire) (Fig. 5–42). The circle formed by the tube has a diameter of about 1.1 km. What mus55 eriw nv o+ t**qmu-uhuo lq15:2ofyt 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 (pvrqqi8d2 p2 eq;t6xue3ze1) 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 rj,c7rn8 h;n1cc ;r pu lgiil* ipf6**the mass of a planet in terms of its radius r, the acceleration due to gravity at its surface and the n urch,*r7;jc l rilnp1p6*8f*;ic iggravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from the verticbpueo a,;cq8**gfti 8 a )xk;/3m ncbcal by an anglepu)38 aiemcfxo;bqnc*a ;cg8b/, *k t $\theta$ 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 eeg(4: d/ idw1xgr4z jspeed of If the coefficient of static friction is 0.30 (wet pavement), at what range of 4z4:er ij (d/gd1w xgespeeds can a car safely handle the curve?

  How long would a day be if the Earth were rotating so fapqfr 4u 8u: es5mu1u1h2eq ;;r eqet-est that objects at the equator were apparently weightless?q:q;;pefe m421h 8ru r1e - 5qeetuusu    $\times 10^3$s

  Two equal-mass stars maintain atf)e as:3 hvk0a)fiq+ 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 rounde,*8ufe/ 6ud8dz iukws a curve of radius 235 m. A lamp suspended from the uu d*86iu,zk 8d eef/wceiling 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 massive as the Earth. Thus, it has been claime7vpnvrh ;d*gqp3,n( ujre-z 5d that a person would be crushed by the force of gravity on a planet the sdg,nq* ; (-vp3 r je5pvnu7rzhize of Jupiter since people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of such a planet. Use the following data for 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 Telesco jp+0g3 2bfkvbpe deduced the presence of an extremely massive 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 ofjv+ 2b3 bgp0kf 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 vwluu)4i *ezm5*xf.cb2kz ejj4 8.zl r alley 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 wouldlf jx. r)4lmkjw u*852iz ezu.4* becz 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 satellites o dyk7p8t7ok*tkczp *o;:2ihz rbiting the Earth. Using “tk2h *oz:k 8top7i7ykdztc ;p* riangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 7+h,opuq e8gibillion km, is meant to orbit the asteroid Eros at a height of about 15 km . Eros is roughly qheiopg ,8+7u 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 satellite in need of re; l1:af9slsti*dgk4lpair. You are in a circular o*9;agl:l kd t1i4 sflsrbit 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 3lqk8yhv6 bs;v51bv4 lw x fk7has a period of 3000 years. (a) What is its mean distance from the S16k bhlyvl5xk ;fvqw7s38 4vbun?   
(b) At its closest approach, the comet is about 1 A.U. from the Sun ( from Earth to the Sun). What is the farthest distance?   
(c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler’s second law and estimate areas by a triangle (as in Fig. 5–29, but smaller distance travelled; see also Hint for Problem 59).]   

  Estimate what the value of G would need to be if you could actually "feel";+xwz c.wu ef1 yourself gravitationally attracted to someone near you. 1zuwxw;.f c+ eMake reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the centesn: 9;svtyrv (r of the Milky Way Galaxy (Fig. 5-46) at a distance of about 30,000 light-years from the centerss :v9nvrt y;( $ \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 squalooa1k db3(7:1zq gd gre 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 midpoint of the bottom side of b:137aloqd(ogkzgd 1the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 55fx 8hi5nl+,8l,ylsf7uab if 200 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 experien;9- apxhhg v,fced by the tip of the 1.5-cm-long sweep second hand on your wrist ;,fgph a-hxv9 watch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight ary9j 6e 1v5qvtl(j i2vjound 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 tivjvjeyjq 2(561 l9v makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highwaov:;vp) n m8jey is designed so that a car traveling at spe8 oe:j)mv p;nved $ 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 neg dg+8 zl59jdgeotiating 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 (appg+l5 g 8dzjed9roximately ). (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.$\approx$    $\times 10^2$