Sometimes people say that water is removed from clothes in a spin dryer by t*78x(jo l 7ix-a-ydlmidj5 o centrifugal force throwing the water outward. What is wrong with this state y7o-o58jxldj (x*tm da-i7ilment?

Will the acceleration of a car be the same when the car travels aro4mfw.;sn23)ee8xl m ib; (gdbn+dn fyund a sharp curve at a constant 60 km/h as when it 24;fel s(dwm )xgb+m ;nbfdni8y3en. travels around a gentle curve at the same speed? Explain.

Suppose a car moves at constant speed along a hilly road.-0z4 b.uyznzk Where does the car exert th n uzkz-y.b04ze greatest and least forces on the road: (a) at the top of a hill, (b) at a dip between two hills, (c) on a level stretch near the bottom of a hill?

Describe all the forces acting on a chilvnqjb0m)mc2u i. qn5,d riding a horse on a merry-go-round. Which of thes 0.2umqnc ,bmvi jn)q5e forces provides the centripetal acceleration of the child?

A bucket of water can be whirled in a t i5k* kvhjp6,:sn ux:vertical circle without the water spilling out, even at the top of the circle when the s*5kptu6nxh,:k: jiv bucket is upside down. Explain.

How many “accelerators” do you have in your c;dw nlz7t:bs t7. jev8l+ez2bar? There are at least three controls in the car which can be used to cause thez7+7d8;zb etwlejntl. :2sbv car to accelerate. What are they? What accelerations do they produce?

A child on a sled comes 2aa u3 fcm,gh3flying over the crest of a small hill, as shown in Fig. 5–31. His sled does not leave the grounh3 g3cfa 2u,mad (he does not achieve “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 inward when rounding a curve at hig3h )hv3kj()u)gz gghzh speed?

Why do airplanes bank when they turg q5s1r1;ttgvn? How would you compute the banking angle gi1; qv1ttrggs5ven its speed and radius of the turn?

A girl is whirling a ball on a string aroundvi/w0fjjj// ur5ab-j2r j c*c 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 the other side of the yard. When should she letbjjuv2w /jcc/ -j/5jr0irf *a go of the string?

Does an apple exert a gravitational force on the Earth? Ifw*,,q0vvug+ 6ideg c l so, how large a force? Cou*egiclg w+vq,0 dv, 6nsider 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 orbixj1ork ,ph h).t be differh .or1)kj,hxp ent?

Which pulls harder gravitationally, the Earth on the Moonsn3h i89ad( vq, or the Moon on the Earth? Which accelerates va 3ih8dq(9ns more?

The Sun's gravitational pull on the Earth is much larger than the Moon's.eo v32jou5rl 1 Yet the Moon's is mainly responsible for the tides. Explain. [Hint: Consider the difference uv12j r3loe o5in 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 effects ar: yg.;5(rfxbx sw apz0vtdf+ yeay* ,6e at g y:rp td y s+0by6.w(vafx5e,a*z;xf work? Do they oppose each other?

The gravitational force on the Moon due to the Earth is only abouzaj53 syypnp bj7o) ;,t half the force on the Moon due to the Sun. Why isn’t tjp3; 5y z7p,bnjo)ys ahe Moon pulled away from the Earth?

Is the centripetal acceleration of Mars in 1r(h tpai 899 houjax2its orbit around the Sun larger or smaller than the centripetal acceleration of the xj 9a 1u9 hhiop(ta82rEarth?

Would it require lesd :fii k,:3yk d9rs,zms speed to launch a satellite (a) toward the east or (b) toward the west? Consider the Earth’s zk,i9yr3 :sf,ddkm :i rotation direction.

When will your apparent weight be the greatest, as measured byy 6oo1ni+8d h knd6-uk 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 ckdo6 u+hn681-okndiyase 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 adn9lu2n 9i w(lnversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cyln29(iw9nllun indrical 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 expe)p/m -e4: wov6iihm4g 9lbjfhriences “free fall” or “apparent weightlessness” between step/i9 vb w4)m6jgf h4:lho-epim s.

The Earth moves faster in its orbit around the Sun in January than in Ju kxdsf;*/gqn 1m rbx.;ly. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons — the main facq/xngm.f;xk; sb1*d rtor 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 mo ,l sq/m/tjcx9 )bax*aon. Explain how al, mb9/tsxac/ x)jq*this discovery enabled an estimate of Pluto’s mass.

  The Sun's gravitational pull on the Earth i3irlw/q5u+ jxs much larger than the Moon's. Yet the Moon's is mainly responsible for the tlixju r/5 +3qwides. 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 19,nun-9c rtc,2b rj 6rz80 $\mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})$ 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 accel* gr3p 1+ecj5 yjwqqrr()kx 1r ykgo+;eration 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 a5; cr qry+o +1q1 je3krkggr)xyj(*wp 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 ecv qrykax(tju ,, ;32on a 2.0-kg discus as it rotates uniformly in a horizontal circle (at arm’s length) of radi,u,jr2cvy at kxq3 (;eus 0.90 m. Calculate the speed of the discus.   

  Suppose the space shuttle is in orbit 400 km from the Earth's.m k9 aq+b)tvf c( gul0h4f6fy surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g , the gravitational acceleration av.mhqkf 6lt+cfbga(4 )yf9u0t the Earth's surface. $\approx$    g

  What is the magnitude of the acceleration of a speck of clayy- j)sbc,i.h -/q edlj on the edge of a potter’s wheel turning at 45 rpm (revolutions per minute) if t)yeclj, -b/j. qis-d hhe wheel’s diameter is 32 cm?   

  A ball on the end of a string is y3gsoyz+ l3) prevolved at a uniform rate in a vertical circle of radius 72.0 cm , as shown in Fig. 5-33. If its speed is 4)zp+l33 yys go.00 $\mathrm{~m} / \mathrm{s}$ and its mass is 0.300 kg , calculate the tension in the string when the ball is (a) at the top of its path   
(b) at the bottom of its path.   

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

  What is the maximum speed with which a 1050-kg car can round a.4gkjh,muv vl4.x v an,ea b.. turn of radius 77 m on a flat road if the coefficient of static friction between tires and road is 0.80? Is this result indev 4a.lh v. j n.bgua,xk,4vme.pendent of the mass of the car?   

  How large must the coefficient of static friction be between 8km dj5mp1nelg)5synzfyi y78 5 6 q1kthe tires and the road if a car is to round a level curve of radius 85 m at a spy8 f5m z m ysj qpkk11idl8y567gnne5)eed of 95km/h ?   

  A device for training a+*hacipb 76;d-5x*jsr*kp jt gnn g4c stronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m . If the force felt by the trainee on her back is 7.85 times her own weight, drc-*7 4c+s*jn*gt ag;kpi 6hpnxj5bhow 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 ruh/a+tr enah9je1.f4ulwk,7- arv1 i otating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable unt v n4th1a uhri,9fkael.1/erw+-j7u ail 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 byq-6j 8dk-lkx qjo/jyqi4o 1+e traveling when upside down at the top of koiyqjyk4-q xljjo-d/61+q8 a circle
(Fig. 5-34) so that the passengers will not fall out? Assume a radius of curvature of 7.4 m .   

  (II) A sports car of mass 950 kg (including the driver) crou2c8p ay-c2+lu3wnb-t cvx ( dsses the rounded top oc 3ux +v tac- 8by2c(-2wnpdluf a hill (radius =95$ \mathrm{~m}$ ) at 22 $\mathrm{~m} / \mathrm{s}$ . Determine
(a) the normal force exerted by the road on the car,   
(b) the normal force exerted by the car on the 72-kg driver   
(c) the car speed at which the normal force on the driver equals zero.   

  How many revolutions per minute would a 15-m-dia 7nok ;s w)0h+ob f0kw5ei6rypx) nhup- +l0aimeter Ferris wheel need to make for the passengers to feel “wewy+w os 6)p)nx0+0lb 5p kn-7ui0ehirk hafo;ightless” at the topmost point?    rpm

  A bucket of mass 2.00 kg is whirly:lkt c* .wq7:u5hzb;vr wy6 led in a vertical circle of radius 1.10 m. At the lowest 5.kq ;ry z lwu 7:l*vc6hbwyt: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 s6lc/5y(so5 sg;vy lr a particle 9.00 cm from the axis of rotation is to vcl 5slysg6yro / (;s5experience an acceleration of 115,000 $\mathrm{~g} $'s?    rpm

  In a "Rotor-ride" at a carniva ;qphd tht0e rs,/os271d z5pdl, people are rotated in a cylindrically walled "r7d2 d d/pztp;hrst,1se o5 0qhoom." (See 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 in a c6rf,7n g0 ucuoircle on a frictionless air-hockey tabletop, and is held in this orbit by a light cord connected to a dangling block (mass m ) through a central nu fru0g7c,o 6hole 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 , pz(twcxelq1 c1-hwx .83.vp p trecisely this time, by not ignoring the weight of the ball which revolves on a string 0.6.t8q1tp3ex v1 xhc- w.czwlp( 00 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 gewxvl6d pm n11)iv6n+c6 iv1for 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 masses8(dsq pr(ki)a8m qtr ( $ 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 8tjmo 2gew.2lmaneuver by diving vertically at 310 $\mathrm{~m} / \mathrm{s}$ . If he can withstand an acceleration of 9.0 g 's without blacking out, at what altitude must he begin to pull out of the dive to avoid crashing into the sea?   

  Determine the tangential and centripetal components of the nee)0mrmuv8- prt force exerted on the car (by t8r0 mup- er)vmhe 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 accelerates uniformly f b4pvw ow5s,;f 3ed2ocrom 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 provide this acceleration with no sliws ;pv,bf3e54d2owco pping or skidding? $\approx$   

  A particle revolves in a horizontal circle of radius 2.90 m c)ke9bkq3a ./fcp6g u . At a particular instant, its acceler/ kf9gb 3c auc6qp.ek)ation 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(4 gf*s qb;arw on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 13 *g 4qrs(awf;b50 kg.   

  At the surface of a certain planet, the gravitational 2y -ixcxyn)ur1 97 eauq8 pu(kacceleration g has a magnitude pu8cnyxqu 9 a ()1u2irk 7yex-of 12 m/s$^2$ A 21.0-kg brass ball is transported to this planet. What is (a) the mass of the brass ball on the Earth and on the planet   
(b) the weight of the brass ball on the Earth and on the planet? $W_{Earth}$   $W_{planet}$  

  Calculate the acceleratixo +b 4s+ks7tvon due to gravity on the Moon. The Moon's radius is 1.7kxo btv+s7+4 s4 $\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 thatc;c6a 4fcl358wcdt ej6tt9 kl of Earth, but has the same mass. What is the acceleration due to gravity near its surfacectld38f6 l aec56;k ct9t4cwj?   

  A hypothetical planet has a mass 1.66 times that e (vxbvbq9s3;of Earth, but the same radius. What is g near ivbsq3v (9x; ebts surface?   

  Two objects attract ea* bt,;cwnk7pu+sp,nvm( qz9u(3uuf rch 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 ,,*hb; l5g .*v(qocn dof;e dfs the acceleration of gravity, at (a) 3200 mo. hf5lscq*dfv,(;de*ng;bo    , and (b) 3200 km    , above the Earth's surface.

  What is thedistance from the Earth’s center to a point outside the Earth wher8d n.fw9uqh 6fe thegravitatio 9d .wf6nu8hfqnal 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 our Sun packed into a sphere p -uy*9 o*ik9yhrr;ya about 10 km in radius. Estimate the surface gravity o9 i*- p*hrkyy;rao9uyn this monster.    $\times10^12$

  A typical white-dwarf stxs,y n;lq sd/:hz 0rn4ar, which once was an average star like our Sun but is now in the last stage of its evolutioh sq/,dnyx;: 4lsr z0nn, 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 feel weightless wh0yu+)r xkc eo9e5myv;ile orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn’t so by calculating the acceleration of gravity 250 km above the Earth’s sur rxk)0mo5u+ ye v;c9yeface in terms of g.   

  Four 9.5-kg spheres are located at the cor gkqrzx kse- 5z ua/3+1gd7ds+b0l)sxners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force e0zq zb xleu5-s1+a gxd/kk+ 7sd3g r)sxerted on one sphere by the other three.   $\times 10^{-1} \mathrm{~N} \text { at }$    $^{\circ}$

  Every few hundred years most of th81jhm vd9 j2gf/fg16 8dn 0fkg1ujjqae planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Safj/8 g nujfgjk12 h09j dqfv118m 6dgaturn, 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 accelerati/vko* w)xh:upj 1 5peoon of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars’ radius is 3400 k :51 vjxp*epokohwu/) m, determine the mass of Mars.    $\times 10^{23}$

  Determine the mass of the Sun using the known value for the period ofkwvk( *-w8g-miz t4ts the Earth and its distance from the Sun. [Note: Compare your answer to that obtak-v-kww (4z * t8tmgsiined using Kepler’s laws, Example 5–16.]    $\times 10^{30}$

  Calculate the speed of a satellite moving in a stabl2w :p ghu6-o zxhl ibi, d6j;5gsd52rw6hq4 oye circular orbit about the Earth at a height of q5 i;g 4z:-x626pb, idhg5jolys owd 2hhwr6 u3600 km.    $\times 10^3$

  The space shuttle releases a satellite into a circular orbitfy;:bj3(ey * zmeh z.o 650 km above the Earth. How fast must the shuttle (fzbo :eyym z*e;3h.j be moving (relative to Earth) when the release occurs?    $\times 10^3$

  At what rate must a cylindrical spaceship rotate if occupant)un(yci 5:u.y 4rgth0 oiod5 rs 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.oiy.:u5ud ry4on0) ic5rht (g (See Question 21, Fig 5–32.)   

  Determine the time it takes for a satellite to orbit the 4fpx l0ua tyjc;xr8+6 Earth in a circular “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 4l0r+ ;f8yj6aput xcx your result depend on the mass of the satellite?    s ~    min

  At what horizontal velocity would a satellite have to be launched from gtjru 305-z chthe top of Mt. Everest to be-3gj utchz 50r placed in a circular orbit around the Earth?   

  During an Apollo lunar landing mission, the command mopu lm s y:.zs1e:;6yzyjvh w+0dule continued to orbit thj +zyyez.: 6;lvyhs:p0 u1 wsme Moon at an altitude of about 100 km. How long did it take to go around the Moon once?    s

  The rings of Saturn a4 q6z.g08 5vf dwr stqaaso)h(re composed of chunks of ice that orbit the planet. The inner (o)8s q0 g fqvtha5.awr4z6sdradius 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 eve*uj2 whj 4p.o13 xr:bvjouls1ry 15.5 s (see Fig. 5–9). What is the ratio of a person’s apparent weight to hej*ooj:wbh.4lj v x us1r p3u21r real weight (a) at the top, and (b) at the bottom?
(a)    (b)   

  What is the apparent weight of a 75ydulm, vw1u0 vslad r61* 9xcj9xv.d)-kg astronaut 4200 km from the center of the Earth's Moon in a space vehicle (a) moving at conc1myxusvl)ljw 1ad90*d,vv9 6uxr. d stant velocity,     ----待选择----towards the Moonaway from the Moon 
(b) accelerating toward the Moon at 2.9 $\mathrm{~m} / \mathrm{s}^{2}$ ?     ----待选择----towards the Moonaway from the Moon 

State the "direction" in each case.

  Suppose that a binary-star system consists of two stars of equal ma2ke: :k6 mw)xhb2g2r9zfn vuu ss. They are observed to be separated by 360 million km and take 5.7 Earth years to orbitkb fvmkh6x :9u)2z2nw: e2urg 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 q ij ka);91pyn7,hfnt woman in an elevator that moq,y;ifp jn 1h7k9t na)ves
(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 cof1 w0la1dh-8jk70qobm rlt-krd suspended from the ceiling of an elevator. The cord kr dlta- boj0- 7kh 0lwf11mq8can 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 period Tad.q0dt/k4 e-zix t+ u , the density (mass/vol+d-/tx.d zuet i k40qaume) 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 M pav/ kq2i,(oroon (27.4 d) to determine the period of an artificial satellite orbiting very near the Earth’s surfa ,aqoirp (k2/vce.    s

  The asteroid Icarus, though only a few hundred meters across, orbitl.o(*nfxft4deh qkw -; 2r6zcs the Sun like the planets. Its period is 410 d. What is its mean distance from the Sun?l-n 2.qrwt d (;e4fxhkfzc *6o    $\times 10^{11}$

  Neptune is an average distance of ht, 6irvqgoq ka7*n8besj;it*lv ..0 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 eqljzk, l44bu6lni n 5sh(* 4lrvery 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” thnl, nzql4s5(j4*64krill bh ue Solar System? What planet’s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler’s third law is half the sum of the nearest and farthest distance from the Sun.]    $\times 10^6$

  Our Sun rotates about the center of ).6bt kg x51cqa)lxqi7dy *njthe 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 maw 4jx,).0* rvl ygrugean distance for the four largest moons of Jupiter (those discovered by Ga0 4,y.v juragxrlgw* )lileo in 1609).
(a) Determine the mass of Jupiter using the data for Io.$M_{\substack{\text { Jupiter- } \\ \text { Io }}}$ =    $\times 10^{27}$
(b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent?
$M_{\substack{\text { Jupiter- } \\ \text { Europa }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Ganymede }}}$ =    $\times 10^{27}$
$M_{\substack{\text { Jupiter- } \\ \text { Callisto }}}$ =    $\times 10^{27}$

  Determine the mass of the Earth from the known 4u-o+hhn z2e, vn1u nnperiod and distance of the Moon. + n-non14u2zh uhn,ve   $\times 10^{24}$

  Determine the mean distance from Jupiter for eac wwcz;s+f8 p8lh of Jupiter’s moons, using Kepler’s third law. Use thefz 8l8 w;pwcs+ distance 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 m.lwq 8hl evg3 o30edw(any fragments (which some space scientists think came dowvgll8h0e3 q.w3(e 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 ax,;j cilsoo9 *t 8/l1/7ojfm 9gtut uxn artificial "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 8cj u/gjiltt1*u;fmo9ts,o lo 97xx/sun is exactly like our own, that the distance to the band is the same as the Earth-Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet's year, in Earth days?   

  Tarzan plans to cross a gorgwz 6tn.qxqc9adt9 + n(e 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 sx q nd9tzw.9+(6qanctpeed 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 acceleration ofajd6,y ra).7cg y zu-nkp/gl- gravity be half what it is.ndlj 7cg)-yg,rkuzay 6p/-a on the surface?    $\times 10^6$

  On an ice rink, two skaters of equal mass7-dwnpb0v nw+vlb,- +t en(*rq oi:mt grab hands and spin in a mutual circle once every 2.5 s.nw*v ,bv:wm 0pq(o-en db +7+tnrt-il If we assume their arms are each 0.80 m long and their individual masses are 60.0 kg, how hard are they pulling on one another?    $\times 10^2$

  Because the Earth rotates once per day y3bm3*1l aujrazc6y2d: /s pm, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn’t rotat*a3jly cy:zdmu23a ms1 / 6pbre. Estimate the magnitude of this effect. What fraction of g is this?    $\times 10^{-1}$

  At what distance from the Earth will a spacecraft travelpju z30g,v)dp*ymg7 king directly from the Earth to the Moon experienc g0u3ymjv7d*zp ) ,kgpe zero net force because the Earth and Moon pull with equal and opposite forces?    $\times 10^8$

  You know your mass is 65 kg,khse hn4ml7 (h.cg -9qivnp1 : but when you stand on a bathroom scale in an elevator, it says your mass is 82 kg. What is the acceleration of the elevator, ngn.(h4hsek9lhm 17:qvip-cand in which direction?     ----待选择----upwartddown 

  A projected space station consifn91;aqse ug/+ghif-; ftu z)sts of a circular 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 km. What must be the rotation speed (revolutions perff qghu/uz +itn;-f as)19 ;ge 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 l5,d-tk p21sdeng ,nk soop (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 plg+:w5,t ouw ukm/g, lhdv6b3ganet in terms of its radius r, the acceleration due to gravity at its surface and theu 3 wk,vmu/g,bgl5w: do+6tg h gravitational constant G.

A plumb bob (a mass m hanging on a string) is deflected a - 0cq8:gm-eas 5/iu+qttwrgfrom the vertical by an angla cqg8ea -+gu5stqr: /w0-m tie $\theta$ 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 mqi5qbr3aroa., 1mv.b7 : rdx x is banked for a design speed of If the coefficient of static friction is 0.30 (wet pavm5 1,dbx3br :qaq.o i7ra .rxvement), at what range of speeds can a car safely handle the curve?

  How long would a day be if the Ea 45:9ov xdm1qsm w1 fo-y7rzcrrth were rotating so fast that objects at the equator were apparenqozr 197d-wxs4r 1my c :o5fmvtly weightless?    $\times 10^3$s

  Two equal-mass stars mainicd141d( mm dvtain 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 235 mc:m02vjnkz8*t(d vri . A lamp suspended( d 0km:jzvv8n*r2tic from the ceiling 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 claimed psde.0x1 m/zqxt;z, +oevu0 n that a person would be crushed by the force of zu 0qne/dvm;spxzt.oex , +01gravity on a planet the size of Jupiter since people can't survive more than a few g 's. Calculate the number of g 's a person would experience at the equator of such a planet. Use the following data 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 Hubbl7 u bebh;7+koge Space Telescope deduced the presence of an extremely massive core in the distant gal b7;u 7+ohebgkaxy 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 travef4rx.y sj8o *s6 j/hiazq4xdj;si:.xrses the hill and valley shown in Fig. 5–45. Both the hill and valley have a radius of curvature R. (a) How do the normal forces acting on the car at A, B, and C compare? (Which is largest? Smallest?) Explain. (b) Where would the driver feel heaviest? Lightest? Explain. (c) How fast can the./x4:d64 ixrji.oyj;jx hza *q8s fss car go without losing contact with the road at A?

  The Navstar Global Positioning System (GPS) utilizes a group of 24 satellit bz9x*ohhh/3 pes orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous na hbh3h o/z9*pxvigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical ]. (a) Determine the speed of each satellite.    $\times 10^3$
(b) Determine the period of each satellite. =    hours

  The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion km, d qll4(0 v5q0 jxdsj-o1mjy2sis meant to orbit the asteroid Eros4 1qo(l-y 2qdj0sdmv sjlxj50 at a height of about 15 km . Eros is roughly 40 $\mathrm{~km} \times 6 \mathrm{~km} \times 6 \mathrm{~km}$ . Assume Eros has a density (mass/volume) of about 2.3 $\times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ . (a) What will be the period of NEAR as it orbits Eros? $\approx$    $\times 10^4$sec
(b) If Eros were a sphere with the same mass and density, what would its radius be? $\approx$    $\times 10^3$
(c) What would g be at the surface of this spherical Eros?$\approx$    $\times 10^{-3}$

  You are an astronaut in thx/4 jk;hl bz+tik8i n7e space shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius z + kt7il;hnx8j/k4bi as the satellite (400 km above the Earth), but 25 km behind it.
(a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km?$\approx$    h
(b) By how much must you reduce your orbital radius to catch up in 7.0 hours?$\approx$    $\times 10^3$

  The comet Hale-Bopp has a perioj8q lr-fy kx,-,jn2na d of 3000 years. (a) What is its mean distance from the S8k xj,y2fa -,rn-lqjn un?   
(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 wouldhp7 zdft38 vm/ l9.exg need to be if you could actually "feel" yourself gravitati dv 3/lf9 x8gep.tzm7honally attracted to someone near you. Make reasonable assumptions, like F $\approx 1 \mathrm{~N}$ .    $\times 10^{-5}$

  The Sun rotates around 8 nxw akc0t3qds8,6pkhgtcx(fo*8gg / *;fxkthe center of the Milky Way Galaxy (Fig. 5-46) at a distance of ab8*xt0wch6 g;8n x3 sx,f /dkkgfog cq*tak8p(out 30,000 light-years from the center $ \left(1 \mathrm{ly}=9.5 \times 10^{15} \mathrm{~m}\right)$ . If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun $\left(2 \times 10^{30} \mathrm{~kg}\right)$ , how many stars would there be in our Galaxy?$\approx$    $\times 10^{11}$

  Four 1.0-kg masses are located at the corners of a square 0.50 m on eachjgau g )x2s)si+ 8t*uh side. Find the magnitude and dig )u uij+a)x8s g2*tshrection of the gravitational force on a fifth 1.0-kg mass placed at the midpoint of the bottom side of the square.    $\times 10^{-10}$  ----待选择----downupward 

  A satellite of mass 5500 k zq*1.faonrq lh-ug64g 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 experiem aj9s8ch c/7cnced by the tip of the 1.5-cm-long sweep second hand on your wrist watc 7cc8mc 9hsaj/h?    $\times 10^{-4}$

  While fishing, you get bored and start to swing a sinker weight around in a cizi5y- g)/o7i spqmxrkel 8-l f0*p3fjrcle below you on a 0.25-m piece of fishing line. The weight makes a compl5/fj )ymfpe0 o8 -7x*qlpkzi-is3r glete circle every 0.50 s. What is the angle that the fishing line makes with the vertical? [Hint: See Fig. 5–10.]    $^{\circ}$

A circular curve of radius R in a new highway is 4ybx eh7b;( 3rn(yb ykdesigned so that a car traveling at speebkh4; 73 by(r xy(ynbed $ 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 tirpt/7 tnqiat/:hf8 3 nx ,tq9ilt 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) C/ant /q nqt rtth3pfi,7 98xi:alculate 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.$\approx$    $\times 10^2$