Sometimes people say that water is rjfr t4 d q.zx9imo2u9:emoved from clothes in a spin dryer by centrifugal force throwing the watud9tq4 fjxoz:.ri9 m2 er outward. What is wrong with this statement?

Will the acceleration of a car be the same when the car travoldfc-z:+ g - nb y*1n3(fcuxiels around a sharp curve at a constant 60 kmgx--d +zcc* f3u(fb iyln:1no/h as when it travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speehowsbq-0i -5qp(. m gtd along a hilly road. Where does the car exert the greatest andhw5.q b- qmo(g0i pt-s least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a child riding a horse on a merry-t)(;lhg z1a5dkl; zevgo-round. Which of th;ze (v51gtkhlz; a)ldese forces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a veab8k*b f (:a+aee gt(.nz yvt6rtical circle without the water spilling out, even at the ba yg a( an(t6k+z8fb :*teev.top of the circle when the bucket is upside down. Explain.

How many “accelerators” do you have 6k(z cw agn tqq/o7(/uin your car? There are at least three controls in th ( 7u(6tozkgnq/qcaw /e car which can be used to cause the car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes flying over the crwyrn397h q1 faest 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 r7aw3hfynq 91 hill. Explain this decrease using Newton’s second law.

Why do bicycle riders leaw*k3jrytoj u:wex68( td; h7j qf/cv )n inward when rounding a curve at high speed?

Why do airplanes bank when thh*h-+o6 5bseet,wn ghey turn? How would you compute the banking angle given its wth+h5g*,o eh- bes n6speed and radius of the turn?

A girl is whirling a ball on a fct; v(e+/ ojr q( oi ogveupteni;/+),e/ zi7string around her head in a horizontal plane. She wants top //)fre+qojv/ io;t+ ev(z i, egci(;ot eu7n let go at precisely the 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 gravitational force on the Earth? If so, how largeut(e0 n3o 5ebr a force? Conb0e3 e5(ontr usider an apple
(a) attached to a tree, and (b) falling.

If the Earth’s mass were double what it is, in what ways would the Moon’s) cmu *ojva5pe.6dcblr a3n*4 orbit be difmnp4 oc53)cva*ued.6jl* r abferent?

Which pulls harder gravitationally, the Earth on the Mo. 9d1 a jja..w3)fsppl x;hvyson, or the Moon on the Earth? Which acceleratesawa.s3 )1 vl;s.jjpx 9dhy.pf more?

The Sun's gravitational pull on the Earth is much larger than thzbhbocd79l ht0cp, (rqmw /9, e Moon's. Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational 9z(7t/bpdhc,,0mrbhl qco w9 pull from one side of the Earth to the other.]

Will an object weigh more at the10q98ysrkpp)bi9c0 - mgigs a equator or at the poles? What two effects are at work? Do they oppose each ig0 10ipyb9-a km9s8sp )crq gother?

The gravitational force on the Moon due to theq +k7mg m4f3pf Earth is only about half the force on the Moon due to the Sun. Why isn’t th3g q m4p+mkff7e Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in its orbit around+6m gksvauxz6.bct5q i;pfa yqz1.6a9 1wf2 b the Sun larger or smaller than the centripetal acceleration of the q6yq;fa6bc5w1 tak6 .xb+sf9z ip1zm v.ua 2gEarth?

Would it require less speed to launch a satetn)0-tfuj eb w rx1p;n:cy53gllite (a) toward the east or (b) toward the west? Consider th)wrn5n3c;:j -p utt ybge10xfe Earth’s rotation direction.

When will your apparent wer+;a4ol9w aetw5h u xzd.+ x-iight 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 would it be the same a xw9othr5+d-uwa. l4aex+i;z s 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 affected by xba0 3os3o x(pweightlessness. One way to simulate gravity is to shape the spaceship like a cyx(p3 30absoxo lindrical 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 of gravity you can think of.

Explain how a runner experiences “free fal +z .m2jb)9qzlp - te4t8 h-yz6xblqxul” or “apparent weightlessness” betweenj 6htyx29-8bq)tl ebpzzxmulz-4 q +. steps.

The Earth moves faster inlum1 v7i::: j3wlnde l its orbit around the Sun in January than in July. Is the Earth closer to the Sun in January,:1:el d ilujwmn3l:v7 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 orbit.]

The mass of Pluto was not knowknbhk2 u9e7 -un until it was discovered to have a moon. Explain how this discovery enabled an estimate of Plkke9-7un bu2h uto’s mass.

  The Sun's gravitational pull on the1vkz2f v0rd,p;o di1 i Earth is much larger than the Moon's. Yet th1z; vk1dfrp, vd o20iie 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 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 a,,wjfj1z e p0*e9izr pvdf)1 1980 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ pulls out of a dive by moving in an arc of radius 6.00 km . What is the plane's acceleration in $g^{\prime}$ 's?    g's

  Calculate the centripetal acceleration of the mk)fv;ki7e oi jb7e+a8. 5ftcq6ac 2bEarth in its orbit around the Sun, and the net force exerted on the Earth. What exerts this force on the Eart; e+b6c5ej)catv28 7i. m oibf kqak7fh? Assume that the Earth's orbit is a circle of radius 1.50$ \times 10^{11} \mathrm{~m}$ . [Hint: see the Tables inside the front cover of this book.]    $\times10^{22}$

  A horizontal force of 210 N is exerted on ui8l)r:we xhrnb93eo4 25ime af/j/t u/ab2 fa 2.0-kg discus as it rotates uniformly in a horizontal circle (at e:mhre n9t53e u/8)/2 bf42aauw/fijirxol barm’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 su(-jmzmdyd8bxwm. -iv qtog6* 4m 8;c crface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your . i;m x(6-mcmdo 8vcqy4-z twmb8g*djanswer 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 clay on the edgxh)zi l )+c -kq+a pwunt3(j9xe of a potter’s wheel turning at 45 rpm (revolutions per minut ) +k -)qn3hxpl9jxzu+c(twiae) if the wheel’s diameter is 32 cm?   

  A ball on the end of a string is revolved at a uniform ratqc1igv4d9 e* re in a vertical circle of radius 72.0 cmvi4qdcge19 *r , 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 a 1050-kjduy)-gi eqkf.. :ps 5g car can round a turn of radius 77 m on a flat road if the j -qy.e .pdus5i)k:fgcoefficient 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 b :jsr4yb*/+ny.ip gkof static friction be between the tires and the road if a car is to round a level curve of radius 85 m ynpyr sbg4b/ i+j*.k:at a speed of 95km/h ?   

  A device for training astronauts and jet fighter * wbmn: et sz:icj2/h (svrn/7pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back is 7.8*enmh/s:vrs 2 ( in7cw/tz:bj 5 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 *6sdlhi6dv:p cm from the axis of a rotating 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 coi:sv6dl 6idp*h n 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 dowom69m93n-z )n/mchanw vj ts*n at the top of nta)jmvzwn*-s3/96 nh o9 mcma 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) crosses the rounded z f mg/16p goa1um08kkvb+5bktop of a hill (radi 0/bg8uvkg mo5k6mazf1p+kb1 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-diameter Ferris wheel neeu)yksyu 1r 3::8:eh3jkwd6t0ezz g e pd to make for the passengers to feel “w3:6pgs1hu8zey 3e e kurk0y: zw :)tjdeightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirled in a vertical circle ofd tj2 + /l6beq7lxzlladb25l3--wkb u radius 1.10 m. At thez/b+ulb -3a x d5k 2bj72wdt ql-l6lle lowest point 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 tvx6;o l30 vsdnhe axis of rotation is to experieno6lnvvx;d0 s3ce an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carnival, people are rotated .zll0smnk/ 9 uwrbaq1js+b :6in a cylindrically walled "room." (See0/s 9 +szlrmjwq.kan6b:1ubl Fig. 5-35.)
The room radius is 4.6 m , and the rotation frequency is 0.50 revolutions per second when the floor drops out. What is the minimum coefficient of static friction so that the people will not slip down? People on this ride say they were "pressed against the wall." Is there really an outward force pressing them against the wall? If so, what is its source? If not, what is the proper description of their situation (besides "scary")? [Hint: First draw the free-body diagram for a person.]   

A flat puck (mass M ) is rotated inlfr n4t (bj-a: a circle on a frictionless air-hockey tabletop, and is held in thina4:-r (lfbt js orbit by a light cord connected to a dangling block (mass m ) through a central hole as shown in Fig. 5-
36. Show that the speed of the puck is given by

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

  Redo Example 5-3 , yyczpo sy9 * u,2s9j+wprecisely this time, by not ignoring the weight of the ball which revolves on9s ju9 oy w+py,z*yc2s a string 0.600 m long. In particular, find the magnitude of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ , and the angle it makes with the horizontal. [Hint: Set the horizontal component of $\overrightarrow{\mathrm{F}}_{\mathrm{T}}$ equal to m $a_{\mathrm{R}}$ ; also, since there is no vertical motion, what can you say about the vertical component of $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ ?] $\overrightarrow{\mathbf{F}}_{\mathrm{T}}$ =    the angle =   

  If a curve with a radius of 88 m is perfectly banked for a car traver*jbyp eke 0ggnbsxqq0;55jb2,h(,7 (xkt ajling 75 $\mathrm{~km} / \mathrm{h}$ , what must be the coefficient of static friction for a car not to skid when traveling at 95 $\mathrm{~km} / \mathrm{h}$ ?   

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

Two blocks, of masses ml9pzq4 2yzy , l./seproer7 5$ 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 verticji etcj8 n rqbu+)6r0-ally 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 /xak-hdl:;ng6w . gm don the c-n k xl6agdwhg;.:m/ dar (by the 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 acceler) sl 2: zrfe)/7e7y2kalzx cyyates uniformly from the pit area, going from rest to 320km/h i fra7csl:yz)y 2)e/ke l2y7xz n 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 provide this acceleration with no slipping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90 m . At a partirz4.; pkn yo4mcular instant, its acceleration m; o. 4rpnk4yzis 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 o 41ghlorss52ud5mz 1 12,800 km (2 Earth radii) above the Eart5dr524u1los1gzm s ohh’s surface if its mass is 1350 kg.   

  At the surface of a certain planet, the gravitational acceleration gtkyayqf)f4)ml; u f ,. has a magnitude oftf)qa)4.uf yk l;yf,m 12 m/s$^2$ A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet   
(b) the weight of the brass ball on the Earth and on the planet? $W_{Earth}$   $W_{planet}$  

  Calculate the acceleration due to graa eddb:32 6lnfe, j2vuvity on the Moon. The Moon's radius is 1.7e 2vu ,3:de2fl6dnjba4 $\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-mc.i x ukx6r3 of Earth, but has the same mass. What is the acceleration due to gravity x-x3mr6i k. cunear its surface?   

  A hypothetical planet has a mass 1.66 times that of Earth, but the same radius.*p b3eex;p, 6hkq6ljv qk m6v3 What is g near its surfa3b6q kp emv6j*v x;p6hqk3l ,ece?   

  Two objects attract each other gravitationally with a force of 8 ;p4i s;vzf 5u* ogw8;wzkxvk2.5$ \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) 3 czbbgw n-(h27200 m nb( gbc 2wh7z-   , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth n 2o1l(tfziv6pfoi 9u*ap(auw9t e67 where thegravitational1 76o(9tualav tn fw*6iopz92(fpui e 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 o; u:4x-bgx *fi va,ibef our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on thisb-if ,xxb;4egav:* ui monster.    $\times10^12$

  A typical white-dwarf star, which once was an average stad)5/ce5 y,qf plg(ro jr like our Sun but is now in the last stage of its evolut/d5er)5lyco p ,qfj(gion, 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 astrcf m2atp1t )* -ba,zmconauts feel weightless while orbiting in the space shuttle. Your friends respond that map *am ct) c1,f-tzb2they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s surface in terms of g.   

  Four 9.5-kg spheres are located at the corners of a squa ej8+ v h,jltknj8gjej326s )mre of side 0.60 m. Calculate the magnitud 3tleg jhjk 2+s,)vj8j68emnje and direction of the total gravitational force exerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of -fgn-1w*ss p1c ,asy b1eg*et the planets line up on the same side of the Sun. Calculate the total force on the Earth du eg1sg -nsebs1 cy *t1,f-paw*e 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 sur.y xj2kh 4of4yface of Mars is 0.38 of what it is on Earty4h4k2.yxof jh, 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 fon+odx r yu8py;p*zvn3serr4(*k g*-o r the period of the Earth and its distance f * *; sn yxg k3puen(oyrrpvd+*r4-8ozrom the Sun. [Note: Compare your answer to that obtained using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stable ci jfsk +wa68tl4rcular orbit about the Earth at a height 8f+jwalt s 64kof 3600 km.    $\times 10^3$

  The space shuttle releases a sjowhtmtim8- 8) n n7x(atellite into a circular orbit 650 km above the Earth. i7j w8n8 htmnm)-( xotHow fast must the shuttle be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spacesyo+l dk 9x1f)5,iln:gy a(k tthip rotate if occupants are to experience simutl,k5dknt xil o ag)+9 :yyf(1lated 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.)   

  Determine the time it takes for a satel*o xt*(ofr 0g m w*4d5gh-dps b.y6llblite to orbit the Earth in a circular “near-Earth” orbit. A “near-Earth” orbit is one at a height above psobrh5mlf*0.b*gg (ddo *lty4-w 6xthe 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 launched from 1vpr0u::hk-laqb+fw i d(e s2the top of Mqbr:d- p01+e(uwisvlh: f2akt. Everest to be placed in a circular orbit around the Earth?   

  During an Apollo lunar landing missio t/vu.0dlh0x on, the command module continued to orbit the Moon at an altitude of about 100 km. Howtu/dxol. 00hv long did it take to go around the Moon once?    s

  The rings of Saturn are composed of chunks of ice that orbit the plane ; 6rxw)yo;myet. The inner;;y) xm eryo6w 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 15.5 s 06 ihw0a m s:zqjakuzz(,.0qp (see Fig. 5–9). What is the ratio ofi ,hp:qkj s6a q0z0(w u.zam0z a person’s apparent weight to her real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75-kg astronaut 4200 km from 5;p c97xeosswthe center of the Earth's Moonwco p;se7s x95 in a space vehicle (a) moving at constant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star qstseda0 vip- :gvfl i,k6;t9dxt 3)1system consists of two stars of equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years tl)v;iskt6 pfa gi -d,ttxq93s 0d:ev1o 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 wciv 8);9 q2 wpdiilss9oman in an elevator that iwsvipci d)s92l 8q9; 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 su1gm+lvp7k8ujh+ min- *-w da rspended from the ceiling of an elevator. The cord can withstand a tension of 220 N and breaks as the eleva -gadlh7m*vm-8urp ++nj1 wkitor 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 wkrp .dljvw1t a86 6nz in4uc( f *flfr;,ay)s6ith period T , the density (mass/volume) of ca r6 4fzft6ski*d8yflr.naj ;nvlp,w)u (16the 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 (lv,iobb f vx11zz29hdetermine the period of an artificial satellite orbbl zbxz,19vvo(f2 h i1iting very near the Earth’s surface.    s

  The asteroid Icarus, though2a;oax)mo l+ 1duvb 3y5: csyj only a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its melas uy);5 yocjdma3vb +o12x :an distance from the Sun?    $\times 10^{11}$

  Neptune is an average dis 6 nc97/k neic7td*gb rca9y;wg 1gac.tance of 4.5 $\times 10^{9} \mathrm{~km}$ from the Sun. Estimate the length of the Neptunian
year given that the Earth is 1.50 $\times 10^{8} \mathrm{~km}$ from the Sun on the average.    $\times 10^2$

  Halley’s comet orbits the Sun roughly once ever.w1rctx9 4utl y 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 pl w.9 ttlcrxu41anet’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/(+qerev8 ul u the center 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 massfpw/h 8mm 3e7x:x 2ep /wcbxt7dg3,fq, period, and mean distance for the four largest moons of Jupiter (those discovered by Galileo xd:3 2txxheg c/p7w/,b8m3f7mqe p fwin 1609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the Earth from the known perimdj.6taha5fdrc*v ow5* cs 8 -od and distance of the Moon. .c 5hd5*w6d tar*oasc8vjmf-   $\times 10^{24}$

  Determine the mean distance from Jupiter for eachd j*ybj4duh97 of Jupiter’s moons, using Kepler’s third law. Use the b 4d7yhudjj*9distance 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 Jupitvuhf/2c ioj3. jn*x:ser consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destroye u3 :cfxnj.s*vhi/j 2od).
(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 artificiax3oxq+(d6 farx0 b)cel "planet" in the form of a band completely encircling a sun (Fig. 5-40). The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band ie 3 aqd(bxc0x+o6)xf rs the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. 5– x- n,q4 raqnk 9/hh/c1s3s41palf lux41). 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 mass is 80 kx-l3qfs n4 a1lh/rsa ucq9/h ,1p4xnkg, and the vine is 5.5 m long.   

  How far above the Earth’s surface will the accelerati wgjywbtq6m5 ,e (9sa +aoi1l swn2:m/on of gravity be half what itma+i,2b:61o5wsjw slqm nt wa /( g9ye is on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass grab hands and spin in a mutualiu1 0ak s6sn8h circle once every 2.5 s. If we assume nsi1k6uh 0 8satheir 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 apparedhur3 /5ygex 8nt acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotate. Estimate the magnitude of this effect. Wh3gy rx5e8 /hduat fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth willjxmpra* n/4c, kk19e 0df 8v zrp7ut*g a spacecraft traveling directly from the Earth to the Moon experience zero net force because the n 7pukcm4rda,jr g p1e8k/*t 0fvz9x*Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg, but when you stand on a batf (c2fwk s/sig e29gg(hroom scale in an elevator, it says your mass is 82 kg. What is the acce2 ( 9eg/ifgs(kc w2sgfleration of the elevator, and in which direction?     ----待选择----upwartddown 

  A projected space station consists of a circula;;9ad2k 6;( puxxsmlh6haqcor tube that will rotate about its center (like a tubular bicycle tire) (Fig. 5–42). The circle formed by the tube has a diameter of about 1.1 kmh (6sa okx9hap;d2u;6mx cq;l. 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 loo c)cucqvcf:xc :+6 rm,p (Fig. 5–43).
(a) If the jet is moving at a speed of at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed 6.0 g’s.    $\times 10^3$
(b) Calculate the 78-kg pilot’s effective weight (the force with which the seat pushes up on him) at the bottom of the circle    $\times 10^3$
(c) at the top of the circle (assume the same speed).
   $\times 10^3$

Derive a formula for the mass of a planet in terms of its radiupythrbt:j 6m iih-j7s2 .ny5 6s r, the acceleration due to gravity at its surhhyn6mr. it:j5 ij-t27 bs py6face and the gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected from the vertical bc:vxiym+5i mt k ox3uu 3+an;;y an 3ym+;+:iua;kxt 3x cmouvni 5 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 the coefgypei*(8/p mds7 sb1c ficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely handle t bcp(d1pe*7m/igs8y she curve?

  How long would a day be gtj9sc w2 89v:9 e,l4x8 bmmagr mb6m:pas:qyif the Earth were rotating so fast that objects at the equator were apparentlama ecpt: 8m 2lwb4g,s9y:g9q x rsm:6j9mv b8y weightless?    $\times 10^3$s

  Two equal-mass starsvglke.ey;f8) m xt3/m maintain a constant distance apart of 8.0 $\times 10^{10} \mathrm{~m} $ and rotate about a point midway between them at a rate of one revolution every 12.6 yr . (a) Why don't the two stars crash into one another due to the gravitational force between them?
(b) What must be the mass of each star?    $\times 10^{26}$

  A train traveling at a constant speed rounds a curve of radius 23y.gx ix6:s; gmrvv7r +0xt vl55 m. A lamp suspended from the ceiling swing:ix+v g r6;vlgmxx5v0. t7 rsys 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 bev1qv 6 6vfj( hqv,x(vuh3f+eeen claimed that a person would be crushed by the force of gravity on a planet the size of (vvvxueq6 +ev6f3 1jh( q,vhfJupiter 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 Telescope deduced the presence of an e q6; indeuqt1;xtremely 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 of gas clouds o n d;;qqetui16rbiting 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 hijivfumc0kh3f(px+d s9g0 b00ll and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the car go without losifjs bf0i9 c+hmduvpk0g(x03 0 ng contact with the road at A?

  The Navstar Global Positioning Syster1d1afuxvj+ + m (GPS) utilizes a group of 24 satellites orbiting the Earth. Using “trdur+1 ja1vfx+ iangulation” 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), afte2kb ).y pwxg:mr traveling 2.1 billion km, is meant to orbit the asteroid Eros at a 2bg:wyx m)k p.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 th0 n pg1e3wamojdh/*+ns 9fr 5xe space shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth), bu njo+ n153xga/sp*de 0w r9mhft 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 has9-5 ;.xrrim( cgr cmdl a period of 3000 years. (a) What is its mean distance from the c9m r (r;gc-rm5ix d.lSun?   
(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 :jz0gc 9h g/cbv.lq (kyou could actually "feel" yourselvjkh b g z0c:/qcg(.9lf gravitationally attracted to someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around the cenbi/xr f3zvr3qg 36yo-ter of the Milky Way Galaxy (Fig. 5-46) at a distance of about 30,000 light-years from the center3 /gzioxf 3-y6rbrqv3 $ \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 locatkfdq; /hgyyc k1yth53*;5dhved at the corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force onq;dhk y3d hk;y ct yvfg155*h/ 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 5500 kg orbits tjf8bw 6;hsbv(s7 +hf 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 1p7 6-ryq0qwz0n. qbia-dfc b/.5-cm-long sweep second hand on your wrist qfpr. n c0-76zy w bqab/qi-0dwatch?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight aroum.)) c/ fzrbj5+lpq yynd 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 anr+y5jzc q.)bp /lfm) ygle that the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R a2)h1jz/ y e 86tvwxep*puh s5 in a new highway is designed so that a car traveling at sp5ea26ve* p1/xj sy uwzphht8)eed $ 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, utilizlcwqq b il7 qvw3mz,1j/9-0g hes 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 force needed on a train passenger oc0wlzqi l9g- jw1hv q,3q7/bmf 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.$\approx$    $\times 10^2$