13 GRAVITATION 13.1 IDENTIFY and SET UP: Use the law of gravitation, Eq (13.1), to determine Fg EXECUTE: FS on M = G mSmM r 2SM (S = sun, M = moon); FE on M = G mE mM r 2EM (E = earth) 2 FS on M ⎛ mSmM ⎞⎛ r EM ⎞ mS ⎛ rEM ⎞ ⎟= = ⎜ G ⎟⎜ ⎜ ⎟ FE on M ⎜ r SM ⎟⎜ GmE mM ⎟ mE ⎝ rSM ⎠ ⎝ ⎠⎝ ⎠ rEM , the radius of the moon’s orbit around the earth is given in Appendix F as 3.84 × 108 m The moon is much closer to the earth than it is to the sun, so take the distance rSM of the moon from the sun to be rSE , the radius of the earth’s orbit around the sun 13.2 FS on M ⎛ 1.99 × 1030 kg ⎞⎛ 3.84 × 108 m ⎞ =⎜ ⎟⎜ ⎟⎟ = 2.18 11 FE on M ⎜⎝ 5.98 × 1024 kg ⎟⎜ ⎠⎝ 1.50 × 10 m ⎠ EVALUATE: The force exerted by the sun is larger than the force exerted by the earth The moon’s motion is a combination of orbiting the sun and orbiting the earth Gm1m2 IDENTIFY: The gravity force between spherically symmetric spheres is Fg = , where r is the r2 separation between their centers SET UP: G = 6.67 × 10−11 N ⋅ m /kg The moment arm for the torque due to each force is 0.150 m (6.67 × 10−11 N ⋅ m /kg )(1.10 kg)(25.0 kg) = 1.27 × 10−7 N (0.120 m) From Figure 13.4 in the textbook we see that the forces for each pair are in opposite directions, so Fnet = EXECUTE: (a) For each pair of spheres, Fg = (b) The net torque is τ net = Fgl = 2(1.27 × 10−7 N)(0.150 m) = 3.81 × 10−8 N ⋅ m 13.3 (c) The torque is very small and the apparatus must be very sensitive The torque could be increased by increasing the mass of the spheres or by decreasing their separation EVALUATE: The quartz fiber must twist through a measurable angle when a small torque is applied to it IDENTIFY: The gravitational attraction of the astronauts on each other causes them to accelerate toward each other, so Newton’s second law of motion applies to their motion SET UP: The net force on each astronaut is the gravity force exerted by the other astronaut Call the astronauts A and B, where m A = 65 kg and mB = 72 kg Fgrav = Gm1m2 /r and ΣF = ma EXECUTE: (a) The free-body diagram for astronaut A is given in Figure 13.3a and for astronaut B in Figure 13.3b © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-1 13-2 Chapter 13 Figure 13.3 ΣFx = max for A gives FA = m Aa A and a A = FA = FB = G aA = m AmB r FA F And for B, aB = B mA mB = (6.673 × 10−11 N ⋅ m /kg ) (65 kg)(72 kg) (20.0 m) = 7.807 × 10−10 N so 7.807 × 10−10 N 7.807 × 10−10 N = 1.2 × 10−11 m/s and aB = = 1.1 × 10−11 m/s 65 kg 72 kg (b) Using constant-acceleration kinematics, we have x = x0 + v0 xt + 12 axt , which gives x A = 12 a A t and xB = 12 aB t x A + xB = 20.0 m, so 20.0 m = 12 (a A + aB )t and t= 13.4 13.5 2(20.0 m) 1.2 × 10 −11 m/s + 1.1 × 10 −11 m/s = 1.32 × 106 s = 15 days (c) Their accelerations would increase as they moved closer and the gravitational attraction between them increased EVALUATE: Even though the gravitational attraction of the astronauts is much weaker than ordinary forces on earth, if it were the only force acting on the astronauts, it would produce noticeable effects IDENTIFY: Apply Eq (13.2), generalized to any pair of spherically symmetric objects SET UP: The separation of the centers of the spheres is 2R EXECUTE: The magnitude of the gravitational attraction is GM /(2 R) = GM /4 R EVALUATE: Eq (13.2) applies to any pair of spherically symmetric objects; one of the objects doesn’t have to be the earth IDENTIFY: Use Eq (13.1) to find the force exerted by each large sphere Add these forces as vectors to get the net force and then use Newton’s 2nd law to calculate the acceleration SET UP: The forces are shown in Figure 13.5 sin θ = 0.80 cosθ = 0.60 Take the origin of coordinate at point P Figure 13.5 EXECUTE: FA = G FB = G mB m r2 m Am r =G (0.26 kg)(0.010 kg) (0.100 m) = 1.735 × 10−11 N = 1.735 × 10−11 N FAx = − FA sin θ = −(1.735 × 10−11 N)(0.80) = −1.39 × 10−11 N © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation 13-3 FAy = + FA cos θ = + (1.735 × 10−11 N)(0.60) = +1.04 × 10−11 N FBx = + FB sin θ = +1.39 × 10−11 N FBy = + FB cosθ = +1.04 × 10−11 N ΣFx = ma x gives FAx + FBx = ma x = ma x so a x = ΣFy = ma y gives FAy + FBy = ma y 2(1.04 × 10−11 N) = (0.010 kg) a y a y = 2.1 × 10−9 m/s , directed downward midway between A and B 13.6 EVALUATE: For ordinary size objects the gravitational force is very small, so the initial acceleration is very small By symmetry there is no x-component of net force and the y-component is in the direction of the two large spheres, since they attract the small sphere IDENTIFY: The net force on A is the vector sum of the force due to B and the force due to C In part (a), the two forces are in the same direction, but in (b) they are in opposite directions SET UP: Use coordinates where + x is to the right Each gravitational force is attractive, so is toward the mass exerting it Treat the masses as uniform spheres, so the gravitational force is the same as for point masses with the same center-to-center distances The free-body diagrams for (a) and (b) are given in Figures 13.6a and 13.6b The gravitational force is Fgrav = Gm1m2 /r Figure 13.6 EXECUTE: (a) Calling FB the force due to mass B and likewise for C, we have FB = G FC = G m AmB rAB mAmC rAC = (6.673 × 10−11 N ⋅ m /kg ) = (6.673 × 10−11 N ⋅ m /kg ) (2.00 kg)2 (0.50 m) (2.00 kg) (0.10 m) = 1.069 × 10−9 N and = 2.669 × 10−8 N The net force is Fnet, x = FBx + FCx = 1.069 × 10−9 N + 2.669 × 10−8 N = 2.8 × 10−8 N to the right (b) Following the same procedure as in (a), we have m m (2.00 kg)2 FB = G A 2B = (6.673 × 10−11 N ⋅ m /kg ) = 1.668 × 10−9 N rAB (0.40 m) FC = G m AmC rAC = (6.673 × 10−11 N ⋅ m /kg ) (2.00 kg) (0.10 m) = 2.669 × 10−8 N Fnet, x = FBx + FCx = 1.668 × 10−9 N − 2.669 × 10−8 N = −2.5 × 10−8 N The net force on A is 2.5 × 10−8 N, to the left EVALUATE: As with any force, the gravitational force is a vector and must be treated like all other vectors The formula Fgrav = Gm1m2 /r only gives the magnitude of this force © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-4 13.7 Chapter 13 IDENTIFY: The force exerted by the moon is the gravitational force, Fg = the person by the earth is w = mg GmM m r2 The force exerted on SET UP: The mass of the moon is mM = 7.35 ×1022 kg G = 6.67 × 10−11 N ⋅ m /kg EXECUTE: (a) Fmoon = Fg = (6.67 × 10−11 N ⋅ m /kg ) (7.35 × 1022 kg)(70 kg) (3.78 × 108 m) = 2.4 × 10−3 N (b) Fearth = w = (70 kg)(9.80 m/s ) = 690 N Fmoon /Fearth = 3.5 × 10−6 13.8 EVALUATE: The force exerted by the earth is much greater than the force exerted by the moon The mass of the moon is less than the mass of the earth and the center of the earth is much closer to the person than is the center of the moon IDENTIFY: Use Eq (13.2) to find the force each point mass exerts on the particle, find the net force, and use Newton’s second law to calculate the acceleration SET UP: Each force is attractive The particle (mass m) is a distance r1 = 0.200 m from m1 = 8.00 kg and therefore a distance r2 = 0.300 m from m2 = 15.0 kg Let + x be toward the 15.0 kg mass EXECUTE: F1 = Gm1m − x-direction F2 = r12 = (6.67 × 10−11 N ⋅ m /kg ) Gm2 m r22 (8.00 kg)m (0.200 m) = (6.67 × 10−11 N ⋅ m /kg ) = (1.334 × 10−8 N/kg)m, in the (15.0 kg) m (0.300 m) = (1.112 × 10−8 N/kg) m, in the + x -direction The net force is Fx = F1x + F2 x = ( −1.334 × 10−8 N/kg + 1.112 × 10−8 N/kg)m = ( −2.2 × 10−9 N/kg)m Fx = −2.2 × 10−9 m/s The acceleration is 2.2 × 10−9 m/s , toward the 8.00 kg mass m EVALUATE: The smaller mass exerts the greater force, because the particle is closer to the smaller mass IDENTIFY: Use Eq (13.2) to calculate the gravitational force each particle exerts on the third mass The equilibrium is stable when for a displacement from equilibrium the net force is directed toward the equilibrium position and it is unstable when the net force is directed away from the equilibrium position SET UP: For the net force to be zero, the two forces on M must be in opposite directions This is the case only when M is on the line connecting the two particles and between them The free-body diagram for M is given in Figure 13.9 m1 = 3m and m2 = m If M is a distance x from m1, it is a distance 1.00 m − x ax = 13.9 from m2 EXECUTE: (a) Fx = F1x + F2 x = −G 3mM x +G mM (1.00 m − x ) = (1.00 m − x) = x 1.00 m − x = ± x/ Since M is between the two particles, x must be less than 1.00 m and 1.00 m x= = 0.634 m M must be placed at a point that is 0.634 m from the particle of mass 3m and + 1/ 0.366 m from the particle of mass m (b) (i) If M is displaced slightly to the right in Figure 13.9, the attractive force from m is larger than the force from 3m and the net force is to the right If M is displaced slightly to the left in Figure 13.9, the attractive force from 3m is larger than the force from m and the net force is to the left In each case the net force is away from equilibrium and the equilibrium is unstable (ii) If M is displaced a very small distance along the y axis in Figure 13.9, the net force is directed opposite to the direction of the displacement and therefore the equilibrium is stable EVALUATE: The point where the net force on M is zero is closer to the smaller mass © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation 13-5 Figure 13.9 13.10 G G IDENTIFY: The force F1 exerted by m on M and the force F2 exerted by 2m on M are each given by Eq (13.2) and the net force is the vector sum of these two forces SET UP: Each force is attractive The forces on M in each region are sketched in Figure 13.10a Let M be at coordinate x on the x-axis G G EXECUTE: (a) For the net force to be zero, F1 and F2 must be in opposite directions and this is the case G G GmM G (2m) M only for < x < L F1 + F2 = then requires F1 = F2 = x = ( L − x) and 2 x ( L − x) L = 0.414 L 1+ (b) For x < 0, Fx > Fx → as x → −∞ and Fx → +∞ as x → For x > L, Fx < Fx → as L − x = ± x x must be less than L, so x = x → ∞ and Fx → −∞ as x → L For < x < 0.414 L, Fx < and Fx increases from −∞ to as x goes from to 0.414L For 0.414 L < x < L, Fx > and Fx increases from to +∞ as x goes from 0.414L to L The graph of Fx versus x is sketched in Figure 13.10b EVALUATE: Any real object is not exactly a point so it is not possible to have both m and M exactly at x = or 2m and M both exactly at x = L But the magnitude of the gravitational force between two objects approaches infinity as the objects get very close together Figure 13.10 © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-6 13.11 Chapter 13 m , so ag = G 2E , where r is the distance of the object from the center of the earth r2 r SET UP: r = h + RE , where h is the distance of the object above the surface of the earth and IDENTIFY: Fg = G mmE RE = 6.38 × 106 m is the radius of the earth EXECUTE: To decrease the acceleration due to gravity by one-tenth, the distance from the center of the earth must be increased by a factor of 10, and so the distance above the surface of the earth is ( 10 − 1) RE = 1.38 × 107 m 13.12 EVALUATE: This height is about twice the radius of the earth IDENTIFY: Apply Eq (13.4) to the earth and to Venus w = mg SET UP: g = GmE RE2 = 9.80 m/s mV = 0.815mE and RV = 0.949 RE wE = mg E = 75.0 N EXECUTE: (a) g V = GmV RV2 = G (0.815mE ) (0.949 RE ) = 0.905 GmE RE2 = 0.905 g E (b) wV = mg V = 0.905mg E = (0.905)(75.0 N) = 67.9 N 13.13 EVALUATE: The mass of the rock is independent of its location but its weight equals the gravitational force on it and that depends on its location (a) IDENTIFY and SET UP: Apply Eq (13.4) to the earth and to Titania The acceleration due to gravity at the surface of Titania is given by gT = GmT /RT2 , where mT is its mass and RT is its radius For the earth, g E = GmE /RE2 EXECUTE: For Titania, mT = mE /1700 and RT = RE /8, so gT = GmT RT2 = G ( mE /1700) ( RE /8) ⎛ 64 ⎞ GmE =⎜ = 0.0377 g E ⎟ ⎝ 1700 ⎠ RE2 Since g E = 9.80 m/s , g T = (0.0377)(9.80 m/s2 ) = 0.37 m/s EVALUATE: g on Titania is much smaller than on earth The smaller mass reduces g and is a greater effect than the smaller radius, which increases g (b) IDENTIFY and SET UP: Use density = mass/volume Assume Titania is a sphere EXECUTE: From Section 13.2 we know that the average density of the earth is 5500 kg/m3 For Titania ρT = mT π RT3 = mE /1700 π ( RE /8)3 = 512 512 (5500 kg/m3 ) = 1700 kg/m3 ρE = 1700 1700 EVALUATE: The average density of Titania is about a factor of smaller than for earth We can write Eq (13.4) for Titania as gT = 43 π GRT ρT gT < g E both because ρT < ρ E and RT < RE 13.14 IDENTIFY: Apply Eq (13.4) to Rhea SET UP: ρ = m/V The volume of a sphere is V = 43 π R3 M gR = 2.44 × 1021 kg and ρ = = 1.30 × 103 kg/m3 G (4π /3) R3 EVALUATE: The average density of Rhea is about one-fourth that of the earth IDENTIFY: Apply Eq (13.2) to the astronaut SET UP: mE = 5.97 × 1024 kg and RE = 6.38 × 106 m EXECUTE: M = 13.15 mmE r = 600 × 103 m + RE so Fg = 610 N At the surface of the earth, r2 w = mg = 735 N The gravity force is not zero in orbit The satellite and the astronaut have the same EXECUTE: Fg = G acceleration so the astronaut’s apparent weight is zero EVALUATE: In Eq (13.2), r is the distance of the object from the center of the earth © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation 13.16 13-7 IDENTIFY: The gravity of Io limits the height to which volcanic material will rise The acceleration due to gravity at the surface of Io depends on its mass and radius SET UP: The radius of Io is R = 1.815 × 106 m Use coordinates where + y is upward At the maximum height, v0 y = 0, a y = − g Io , which is assumed to be constant Therefore the constant-acceleration kinematics formulas apply The acceleration due to gravity at Io’s surface is given by g Io = Gm/R SOLVE: At the surface of Io, g Io = Gm R2 = (6.673 × 10−11 N ⋅ m /kg )(8.94 × 1022 kg) (1.815 × 106 m) = 1.81 m/s For constant acceleration (assumed), the equation v 2y = v02 y + 2a y ( y − y0 ) applies, so v0 y = −2a y ( y − y0 ) = −2( −1.81 m/s )(5.00 × 105 m) = 1.345 × 103 m/s Now solve for y − y0 when v0 y = 1.345 × 103 m/s and a y = −9.80 m/s The equation v 2y = v 02y + 2a y ( y − y0 ) gives y − y0 = 13.17 v 2y − v 02y 2a y = −(1.345 × 103 m/s)2 2(−9.80 m/s ) = 92 km EVALUATE: Even though the mass of Io is around 100 times smaller than that of the earth, the acceleration due to gravity at its surface is only about 1/6 of that of the earth because Io’s radius is much smaller than earth’s radius IDENTIFY: The escape speed, from the results of Example 13.5, is 2GM/R SET UP: For Mars, M = 6.42 × 1023 kg and R = 3.40 × 106 m For Jupiter, M = 1.90 × 1027 kg and R = 6.91× 107 m EXECUTE: (a) v = 2(6.673 × 10−11 N ⋅ m /kg )(6.42 × 1023 kg)/(3.40 × 106 m) = 5.02 × 103 m/s (b) v = 2(6.673 × 10−11 N ⋅ m /kg (1.90 × 1027 kg)/(6.91 × 107 m) = 6.06 × 104 m/s (c) Both the kinetic energy and the gravitational potential energy are proportional to the mass of the spacecraft EVALUATE: Example 13.5 calculates the escape speed for earth to be 1.12 × 104 m/s This is larger than our result for Mars and less than our result for Jupiter 13.18 IDENTIFY: The kinetic energy is K = 12 mv and the potential energy is U = − GMm r SET UP: The mass of the earth is M E = 5.97 × 1024 kg EXECUTE: (a) K = 12 (629 kg)(3.33 × 103 m/s) = 3.49 × 109 J (b) U = − 13.19 GM E m (6.673 × 10−11 N ⋅ m /kg )(5.97 × 1024 kg)(629 kg) =− = −8.73 × 107 J r 2.87 × 109 m EVALUATE: The total energy K + U is positive IDENTIFY: Apply Newton’s second law to the motion of the satellite and obtain an equation that relates the orbital speed v to the orbital radius r SET UP: The distances are shown in Figure 13.19a The radius of the orbit is r = h + RE r = 7.80 × 105 m + 6.38 × 106 m = 7.16 ì 106 m Figure 13.19a â Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-8 Chapter 13 The free-body diagram for the satellite is given in Figure 13.19b (a) EXECUTE: ΣFy = ma y Fg = marad G mmE r2 =m v2 r Figure 13.19b v= GmE (6.673 × 10−11 N ⋅ m /kg )(5.97 × 1024 kg) = = 7.46 × 103 m/s r 7.16 × 10 m (b) T = 2π r 2π (7.16 × 106 m) = = 6030 s = 1.68 h v 7.46 × 103 m/s EVALUATE: Note that r = h + RE is the radius of the orbit, measured from the center of the earth For this 13.20 satellite r is greater than for the satellite in Example 13.6, so its orbital speed is less IDENTIFY: The time to complete one orbit is the period T, given by Eq (13.12) The speed v of the 2π r satellite is given by v = T SET UP: If h is the height of the orbit above the earth’s surface, the radius of the orbit is r = h + RE RE = 6.38 × 106 m and mE = 5.97 × 1024 kg EXECUTE: (a) T = 2π (7.05 × 105 m + 6.38 × 106 m) = 7.49 × 103 m/s = 7.49 km/s 5.94 × 103 s EVALUATE: The satellite in Example 13.6 is at a lower altitude and therefore has a smaller orbit radius than the satellite in this problem Therefore, the satellite in this problem has a larger period and a smaller orbital speed But a large percentage change in h corresponds to a small percentage change in r and the values of T and v for the two satellites not differ very much IDENTIFY: We know orbital data (speed and orbital radius) for one satellite and want to use it to find the orbital speed of another satellite having a known orbital radius Newton’s second law and the law of universal gravitation apply to both satellites mmp v2 SET UP: For circular motion, Fnet = ma = mv /r , which in this case is G = m r r mmp v EXECUTE: Using G = m , we get Gmp = rv = constant r1v12 = r2v22 r r (b) v = 13.21 2π r 3/2 2π (7.05 × 105 m + 6.38 × 106 m)3/2 = = 5.94 × 103 s = 99.0 −11 2 24 GmE (6.67 × 10 N ⋅ m /kg )(5.97 × 10 kg) r1 5.00 × 107 m = (4800 m/s) = 6200 m/s r2 3.00 × 107 m EVALUATE: The more distant satellite moves slower than the closer satellite, which is reasonable since the planet’s gravity decreases with distance The masses of the satellites not affect their orbits IDENTIFY: We can calculate the orbital period T from the number of revolutions per day Then the period and the orbit radius are related by Eq (13.12) SET UP: mE = 5.97 × 1024 kg and RE = 6.38 × 106 m The height h of the orbit above the surface of the v2 = v1 13.22 earth is related to the orbit radius r by r = h + RE day = 8.64 ì 104 s â Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation 13-9 EXECUTE: The satellite moves 15.65 revolutions in 8.64 × 104 s, so the time for 1.00 revolution is T= 8.64 × 104 s 2π r 3/2 = 5.52 × 103 s T = gives 15.65 GmE 1/3 ⎛ GmET ⎞ r =⎜ ⎜ 4π ⎟⎟ ⎝ ⎠ 1/3 ⎛ [6.67 × 10−11 N ⋅ m /kg ][5.97 × 1024 kg][5.52 × 103 s]2 ⎞ =⎜ ⎟⎟ ⎜ 4π ⎝ ⎠ r = 6.75 × 106 m and h = r − RE = 3.7 × 105 m = 370 km 13.23 EVALUATE: The period of this satellite is slightly larger than the period for the satellite in Example 13.6 and the altitude of this satellite is therefore somewhat greater G G 2π r IDENTIFY: Apply ΣF = ma to the motion of the baseball v = T SET UP: rD = × 103 m EXECUTE: (a) Fg = marad gives G v= mD m rD2 =m v2 rD GmD (6.673 × 10−11 N ⋅ m /kg )(2.0 × 1015 kg) = = 4.7 m/s rD × 103 m 4.7 m/s = 11 mph, which is easy to achieve 2π r 2π (6 × 103 m) = = 8020 s = 134 = 2.23 h The game would last a long time 4.7 m/s v EVALUATE: The speed v is relative to the center of Deimos The baseball would already have some speed before we throw it, because of the rotational motion of Deimos 2π r IDENTIFY: T = and Fg = marad v (b) T = 13.24 SET UP: The sun has mass mS = 1.99 × 1030 kg The radius of Mercury’s orbit is 5.79 × 1010 m, so the radius of Vulcan’s orbit is 3.86 × 1010 m EXECUTE: Fg = marad gives G T = 2π r mSm r2 =m v2 GmS and v = r r 3/2 r 2π r 2π (3.86 × 1010 m)3/2 = = = 4.13 × 106 s = 47.8 days GmS GmS (6.673 × 10−11 N ⋅ m /kg )(1.99 × 1030 kg) EVALUATE: The orbital period of Mercury is 88.0 d, so we could calculate T for Vulcan as T = (88.0 d)(2/3)3/2 = 47.9 days 13.25 IDENTIFY: The orbital speed is given by v = Gm/r , where m is the mass of the star The orbital period is 2π r given by T = v SET UP: The sun has mass mS = 1.99 × 1030 kg The orbit radius of the earth is 1.50 × 1011 m EXECUTE: (a) v = Gm/r v = (6.673 × 10−11 N ⋅ m /kg )(0.85 × 1.99 × 1030 kg)/((1.50 × 1011 m)(0.11)) = 8.27 × 104 m/s (b) 2π r/v = 1.25 × 106 s = 14.5 days (about two weeks) 13.26 EVALUATE: The orbital period is less than the 88-day orbital period of Mercury; this planet is orbiting very close to its star, compared to the orbital radius of Mercury IDENTIFY: The period of each satellite is given by Eq (13.12) Set up a ratio involving T and r T 2π 2π r 3/2 T1 T2 = constant, so 3/2 SET UP: T = gives 3/2 = = 3/2 Gmp Gmp r r1 r2 © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-10 Chapter 13 ⎛r ⎞ EXECUTE: T2 = T1 ⎜ ⎟ ⎝ r1 ⎠ 3/2 ⎛ 48,000 km ⎞ = (6.39 days) ⎜ ⎟ ⎝ 19,600 km ⎠ 3/2 = 24.5 days For the other satellite, 3/2 13.27 ⎛ 64,000 km ⎞ T2 = (6.39 days) ⎜ ⎟ = 37.7 days ⎝ 19,600 km ⎠ EVALUATE: T increases when r increases IDENTIFY: In part (b) apply the results from part (a) SET UP: For Pluto, e = 0.248 and a = 5.92 × 1012 m For Neptune, e = 0.010 and a = 4.50 × 1012 m The orbital period for Pluto is T = 247.9 y EXECUTE: (a) The result follows directly from Figure 13.18 in the textbook (b) The closest distance for Pluto is (1 − 0.248)(5.92 × 1012 m) = 4.45 × 1012 m The greatest distance for Neptune is (1 + 0.010)(4.50 × 1012 m) = 4.55 × 1012 m (c) The time is the orbital period of Pluto, T = 248 y EVALUATE: Pluto’s closest distance calculated in part (a) is 0.10 × 1012 m = 1.0 × 108 km, so Pluto is 13.28 about 100 million km closer to the sun than Neptune, as is stated in the problem The eccentricity of Neptune’s orbit is small, so its distance from the sun is approximately constant 2π r 3/2 2π r , where mstar is the mass of the star v = IDENTIFY: T = T Gmstar SET UP: 3.09 days = 2.67 × 105 s The orbit radius of Mercury is 5.79 × 1010 m The mass of our sun is 1.99 × 1030 kg EXECUTE: (a) T = 2.67 × 105 s r = (5.79 × 1010 m)/9 = 6.43 × 109 m T = mstar = 4π 2r T G = 4π (6.43 × 109 m)3 (2.67 × 10 s) (6.67 × 10−11 N ⋅ m /kg ) = 2.21 × 1030 kg 2π r 3/2 gives Gmstar mstar = 1.11, so msun mstar = 1.11msun 2π r 2π (6.43 × 109 m) = = 1.51 × 105 m/s = 151 km/s T 2.67 × 105 s EVALUATE: The orbital period of Mercury is 88.0 d The period for this planet is much less primarily because the orbit radius is much less and also because the mass of the star is greater than the mass of our sun IDENTIFY: Knowing the orbital radius and orbital period of a satellite, we can calculate the mass of the object about which it is revolving SET UP: The radius of the orbit is r = 10.5 × 109 m and its period is T = 6.3 days = 5.443 × 105 s The (b) v = 13.29 mass of the sun is mS = 1.99 × 1030 kg The orbital period is given by T = EXECUTE: Solving T = 2π r 3/2 for the mass of the star gives GmHD 4π (10.5 × 109 m)3 = 2.3 × 1030 kg, which is mHD = 1.2mS T 2G (5.443 × 105 s) (6.673 × 10−11 N ⋅ m /kg ) EVALUATE: The mass of the star is only 20% greater than that of our sun, yet the orbital period of the planet is much shorter than that of the earth, so the planet must be much closer to the star than the earth is IDENTIFY: Section 13.6 states that for a point mass outside a spherical shell the gravitational force is the same as if all the mass of the shell were concentrated at its center It also states that for a point inside a spherical shell the force is zero mHD = 13.30 4π r 2π r 3/2 GmHD = © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-22 13.56 Chapter 13 IDENTIFY: Apply the law of gravitation to the astronaut at the north pole to calculate the mass of planet G 4π R G Then apply ΣF = ma to the astronaut, with arad = , toward the center of the planet, to calculate the T2 period T Apply Eq (13.12) to the satellite in order to calculate its orbital period SET UP: Get radius of X: 14 (2π R) = 18,850 km and R = 1.20 × 107 m Astronaut mass: m= w 943 N = = 96.2 kg g 9.80 m/s GmM X EXECUTE: MX = R2 = w, where w = 915.0 N mg x R (915 N)(1.20 × 107 m)2 = = 2.05 × 1025 kg Gm (6.67 × 10−11 N ⋅ m /kg )(96.2 kg) Apply Newton’s second law to astronaut on a scale at the equator of X Fgrav − Fscale = marad , so Fgrav − Fscale = 4π mR 915.0 N − 850.0 N = 4π (96.2 kg)(1.20 × 107 m) T2 ⎛ 1h ⎞ T = 2.65 × 104 s ⎜ ⎟ = 7.36 h ⎝ 3600 s ⎠ (b) For the satellite, T = T2 and 4π 2r 4π (1.20 × 107 m + 2.0 × 106 m)3 = = 8.90 × 103 s = 2.47 hours GmX (6.67 × 10−11 N ⋅ m /kg )(2.05 × 1025 kg) 915.0 N = 9.51 m/s , similar 96.2 kg to the value on earth The radius of the planet is about twice that of earth The planet rotates more rapidly than earth and the length of a day is about one-third what it is on earth Gm IDENTIFY: Use g = 2E and follow the procedure specified in the problem RE EVALUATE: The acceleration of gravity at the surface of the planet is g X = 13.57 SET UP: RE = 6.38 × 106 m EXECUTE: The fractional error is − mgh g =1− ( RE + h)( RE ) GmmE (1/RE − 1/( RE + h)) GmE Using Eq (13.4) for g the fractional difference is − ( RE + h)/RE = − h/RE , so if the fractional difference 13.58 is −1%, h = (0.01) RE = 6.4 × 104 m EVALUATE: For h = km, the fractional error is only 0.016% Eq (7.2) is very accurate for the motion of objects near the earth’s surface IDENTIFY: Use the measurements of the motion of the rock to calculate g M , the value of g on Mongo 2π r v SET UP: Take + y upward When the stone returns to the ground its velocity is 12.0 m/s, downward Then use this to calculate the mass of Mongo For the ship, Fg = marad and T = gM = G mM RM The radius of Mongo is RM = c 2.00 × 108 m = = 3.18 × 107 m The ship moves in an orbit 2π 2π of radius r = 3.18 × 107 m + 3.00 × 107 m = 6.18 × 107 m EXECUTE: (a) v0 y = +12.0 m/s, v y = −12.0 m/s, a y = − g M and t = 6.00 s v y = v0 y + a yt gives − gM = v y − v0 y t = −12.0 m/s − 12.0 m/s and g M = 4.00 m/s 6.00 s g R (4.00 m/s )(3.18 × 107 m)2 mM = M M = = 6.06 × 1025 kg G 6.673 × 10−11 N ⋅ m /kg © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation (b) Fg = marad gives G T= mM m r2 =m 13-23 v2 GmM and v = r r 2π r r 2π r 3/ 2π (6.18 × 107 m)3/ = 2π r = = v GmM GmM (6.673 × 10−11 N ⋅ m /kg )(6.06 × 1025 kg) T = 4.80 × 104 s = 13.3 h EVALUATE: RM = 5.0 RE and mM = 10.2mE , so g M = 13.59 10.2 (5.0) g E = 0.408 g E , which agrees with the value calculated in part (a) IDENTIFY: The free-fall time of the rock will give us the acceleration due to gravity at the surface of the planet Applying Newton’s second law and the law of universal gravitation will give us the mass of the planet since we know its radius SET UP: For constant acceleration, y − y0 = v0 yt + a yt At the surface of the planet, Newton’s second Gmrock mp law gives mrock g = Rp2 2( y − y0 ) 2(1.90 m) EXECUTE: First find a y = g y − y0 = v0 yt + a yt a y = = = 16.49 m/s = g t2 (0.480 s) g = 16.49 m/s mp = 13.60 gRp2 (16.49 m/s)(8.60 × 107 m) = 1.83 × 1027 kg 6.674 × 10−11 N ⋅ m /kg EVALUATE: The planet’s mass is over 100 times that of the earth, which is reasonable since it is larger (in size) than the earth yet has a greater acceleration due to gravity at its surface IDENTIFY: Apply Eq (13.9) to the particle-earth and particle-moon systems SET UP: When the particle is a distance r from the center of the earth, it is a distance REM − r from the center of the moon ⎡m mM ⎤ EXECUTE: (a) The total gravitational potential energy in this model is U = −Gm ⎢ E + ⎥ r R EM − r ⎦ ⎣ G = (b) The point where the net gravitational force vanishes is r = 13.61 REM = 3.46 × 108 m Using this + mM /mE value for r in the expression in part (a) and the work-energy theorem, including the initial potential energy of −Gm(mE /RE + mM /(REM − RE )) gives 11.1 km/s (c) The final distance from the earth is the Earth-moon distance minus the radius of the moon, or 3.823 × 108 m From the work-energy theorem, the rocket impacts the moon with a speed of 2.9 km/s EVALUATE: The spacecraft has a greater gravitational potential energy at the surface of the moon than at the surface of the earth, so it reaches the surface of the moon with a speed that is less than its launch speed on earth IDENTIFY and SET UP: Use Eq (13.2) to calculate the gravity force at each location For the top of Mount Everest write r = h + RE and use the fact that h RE to obtain an expression for the difference in the two forces mm EXECUTE: At Sacramento, the gravity force on you is F1 = G 2E RE At the top of Mount Everest, a height of h = 8800 m above sea level, the gravity force on you is mmE mmE F2 = G =G ( RE + h) R E (1 + h/RE ) (1 + h/RE ) −2 ≈ − ⎛ 2h 2h ⎞ , F2 = F1 ⎜1 − ⎟ RE R E⎠ ⎝ © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-24 13.62 Chapter 13 F1 − F2 2h = = 0.28% F1 RE EVALUATE: The change in the gravitational force is very small, so for objects near the surface of the earth it is a good approximation to treat it as a constant IDENTIFY: The 0.100 kg sphere has gravitational potential energy due to the other two spheres Its mechanical energy is conserved SET UP: From energy conservation, K1 + U1 = K + U , where K = 12 mv , and U = −Gm1m2 /r EXECUTE: Using K1 + U1 = K + U , we have K1 = 0, mA = 5.00 kg, mB = 10.0 kg and m = 0.100 kg U1 = − GmmA GmmB ⎛ 5.00 kg 10.0 kg ⎞ − = −(6.674 × 10−11 N ⋅ m /kg )(0.100 kg) ⎜ + ⎟ rA1 rB1 ⎝ 0.400 m 0.600 m ⎠ U1 = −1.9466 × 10−10 J U2 = − GmmA GmmB ⎛ 5.00 kg 10.0 kg ⎞ − = −(6.674 × 10−11 N ⋅ m /kg )(0.100 kg) ⎜ + ⎟ rA2 rB2 ⎝ 0.800 m 0.200 m ⎠ U = −3.7541 × 10−10 J K = U1 − U = −1.9466 × 10−10 J − ( −3.7541 × 10−10 J) = 1.8075 × 10−10 J 2K2 2(1.8075 × 10−10 J) mv = K and v = = = 6.01 × 10−5 m/s 0.100 kg m 13.63 EVALUATE: The kinetic energy gained by the sphere is equal to the loss in its potential energy IDENTIFY and SET UP: First use the radius of the orbit to find the initial orbital speed, from Eq (13.10) applied to the moon EXECUTE: v = Gm/r and r = RM + h = 1.74 × 106 m + 50.0 × 103 m = 1.79 × 106 m Thus v = (6.673 × 10−11 N ⋅ m /kg )(7.35 × 1022 kg) 1.79 × 106 m = 1.655 × 103 m/s After the speed decreases by 20.0 m/s it becomes 1.655 × 103 m/s − 20.0 m/s = 1.635 × 103 m/s IDENTIFY and SET UP: Use conservation of energy to find the speed when the spacecraft reaches the lunar surface K1 + U1 + Wother = K + U Gravity is the only force that does work so Wother = and K = K1 + U1 − U EXECUTE: U1 = −Gmm m/r ; U = −Gmm m/Rm mv 2 = 12 mv12 + Gmmm (1/Rm − 1/r ) And the mass m divides out to give v2 = v12 + 2Gmm (1/Rm − 1/r ) v2 = 1.682 × 103 m/s(1 km/1000 m)(3600 s/1 h) = 6060 km/h 13.64 EVALUATE: After the thruster fires the spacecraft is moving too slowly to be in a stable orbit; the gravitational force is larger than what is needed to maintain a circular orbit The spacecraft gains energy as it is accelerated toward the surface IDENTIFY: In part (a) use the expression for the escape speed that is derived in Example 13.5 In part (b) apply conservation of energy SET UP: R = 4.5 × 103 m In part (b) let point be at the surface of the comet EXECUTE: (a) The escape speed is v = M= 2GM so R Rv (4.5 × 103 m)(1.0 m/s) = = 3.37 × 1013 kg 2G 2(6.67 × 10−11 N ⋅ m /kg ) © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation (b) (i) K1 = 12 mv12 K = 0.100 K1 U1 = − mv 2 − 13-25 GMm GMm ; U2 = − K1 + U1 = K + U gives R r GMm GMm = (0.100)( 12 mv12 ) − Solving for r gives R r 1 0.450v12 0.450(1.0 m/s) = − = − and r = 45 km (ii) The debris r R GM 4.5 × 103 m (6.67 × 10−11 N ⋅ m /kg )(3.37 × 1013 kg) never loses all of its initial kinetic energy, but K → as r → ∞ The farther the debris are from the 13.65 comet’s center, the smaller is their kinetic energy EVALUATE: The debris will have lost 90.0% of their initial kinetic energy when they are at a distance from the comet’s center of about ten times the radius of the comet IDENTIFY and SET UP: Apply conservation of energy Must use Eq (13.9) for the gravitational potential energy since h is not small compared to RE As indicated in Figure 13.65, take point to be where the hammer is released and point to be just above the surface of the earth, so r1 = RE + h and r2 = RE Figure 13.65 EXECUTE: K1 + U1 + Wother = K + U Only gravity does work, so Wother = K1 = 0, K = 12 mv22 U1 = −G mmE GmmE mmE GmmE =− =− , U = −G r1 h + RE r2 RE Thus, −G mmE mmE = mv22 − G h + RE RE ⎛ 1 ⎞ 2GmE 2GmE h v22 = 2GmE ⎜ ( RE + h − RE ) = − ⎟= RE ( RE + h) ⎝ RE RE + h ⎠ RE ( RE + h) v2 = 2GmE h RE ( RE + h) EVALUATE: If h → ∞, v2 → 2GmE /RE , which equals the escape speed In this limit this event is the reverse of an object being projected upward from the surface with the escape speed If h RE , then v2 = 2GmE h/RE2 = gh , the same result if Eq (7.2) used for U 13.66 IDENTIFY: In orbit the total mechanical energy of the satellite is E = − GmE m m m U = −G E r RE W = E2 − E1 SET UP: U → as r → ∞ © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-26 Chapter 13 EXECUTE: (a) The energy the satellite has as it sits on the surface of the Earth is E1 = energy it has when it is in orbit at a radius R ≈ RE is E2 = −GmM E The RE −GmM E The work needed to put it in orbit is RE GmM E RE (b) The total energy of the satellite far away from the earth is zero, so the additional work needed is ⎛ −GmM E ⎞ GmM E 0−⎜ ⎟= RE ⎝ RE ⎠ the difference between these: W = E2 − E1 = 13.67 EVALUATE: (c) The work needed to put the satellite into orbit was the same as the work needed to put the satellite from orbit to the edge of the universe IDENTIFY: At the escape speed, E = K + U = SET UP: At the surface of the earth the satellite is a distance RE = 6.38 × 106 m from the center of the earth and a distance RES = 1.50 × 1011 m from the sun The orbital speed of the earth is 2π RES , where T T = 3.156 × 107 s is the orbital period The speed of a point on the surface of the earth at an angle φ from the equator is v = 2π RE cos φ , where T = 86,400 s is the rotational period of the earth T ⎡m m ⎤ EXECUTE: (a) The escape speed will be v = 2G ⎢ E + s ⎥ = 4.35 × 104 m/s Making the simplifying ⎣ RE RES ⎦ assumption that the direction of launch is the direction of the earth’s motion in its orbit, the speed relative 2π RES 2π (1.50 × 1011 m) = 4.35 × 104 m/s − = 1.37 × 104 m/s to the center of the earth is v − T (3.156 × 107 s) (b) The rotational speed at Cape Canaveral is 2π (6.38 × 106 m) cos 28.5° = 4.09 × 102 m/s, so the speed 86,400 s relative to the surface of the earth is 1.33 × 104 m/s (c) In French Guiana, the rotational speed is 4.63 × 102 m/s, so the speed relative to the surface of the earth 13.68 is 1.32 × 104 m/s EVALUATE: The orbital speed of the earth is a large fraction of the escape speed, but the rotational speed of a point on the surface of the earth is much less IDENTIFY: From the discussion of Section 13.6, the force on a point mass at a distance r from the center of a spherically symmetric mass distribution is the same as though we removed all the mass at points farther than r from the center and concentrated all the remaining mass at the center SET UP: The mass M of a hollow sphere of density ρ , inner radius R1 and outer radius R2 is M = ρ 43 π ( R23 − R13 ) From Figure 13.9 in the textbook, the inner core has outer radius 1.2 × 106 m, inner radius zero and density 1.3 × 104 kg/m3 The outer core has inner radius 1.2 × 106 m, outer radius 3.6 × 106 m and density 1.1 × 104 kg/m3 The total mass of the earth is mE = 5.97 × 1024 kg and its radius is RE = 6.38 × 106 m mE m = mg = (10.0 kg)(9.80 m/s ) = 98.0 N RE2 (b) The mass of the inner core is minner = ρinner 43 π ( R23 − R13 ) = (1.3 × 104 kg/m3 ) 43 π (1.2 × 106 m)3 = 9.4 × 1022 kg The mass of the outer EXECUTE: (a) Fg = G core is mouter = (1.1 × 104 kg/m3 ) 43 π ([3.6 × 106 m]3 − [1.2 × 106 m]3 ) = 2.1× 1024 kg Only the inner and outer cores contribute to the force © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation Fg = (6.67 × 10−11 N ⋅ m /kg ) (9.4 × 1022 kg + 2.1 × 1024 kg)(10.0 kg) (3.6 × 106 m) 13-27 = 110 N (c) Only the inner core contributes to the force and Fg = (6.67 × 10−11 N ⋅ m /kg ) (9.4 × 1022 kg)(10.0 kg) (1.2 × 106 m) = 44 N (d) At r = 0, Fg = 13.69 EVALUATE: In this model the earth is spherically symmetric but not uniform, so the result of Example 13.10 doesn’t apply In particular, the force at the surface of the outer core is greater than the force at the surface of the earth IDENTIFY: Eq (13.12) relates orbital period and orbital radius for a circular orbit SET UP: The mass of the sun is M = 1.99 × 1030 kg EXECUTE: (a) The period of the asteroid is T = 2π a3 / GM Inserting (i) × 1011 m for a gives 2.84 y and (ii) × 1011 m gives a period of 6.11 y (b) If the period is 5.93 y, then a = 4.90 × 1011 m (c) This happens because 0.4 = 2/5, another ratio of integers So once every orbits of the asteroid and orbits of Jupiter, the asteroid is at its perijove distance Solving when T = 4.74 y, a = 4.22 × 1011 m 13.70 EVALUATE: The orbit radius for Jupiter is 7.78 × 1011 m and for Mars it is 2.28 × 1011 m The asteroid belt lies between Mars and Jupiter The mass of Jupiter is about 3000 times that of Mars, so the effect of Jupiter on the asteroids is much larger IDENTIFY: Apply the work-energy relation in the form W = ΔE , where E = K + U The speed v is related to the orbit radius by Eq (13.10) SET UP: mE = 5.97 × 1024 kg EXECUTE: (a) In moving to a lower orbit by whatever means, gravity does positive work, and so the speed does increase ⎛ −Δr ⎞ −3/ ⎛ Δr ⎞ GmE (b) v = (GmE )1/2 r −1/2 , so Δv = (GmE )1/2 ⎜ − =⎜ ⎟ Note that a positive Δr is given as ⎟r ⎝ ⎠ ⎝ ⎠ r3 a decrease in radius Similarly, the kinetic energy is K = (1/2)mv = (1/2)GmE m/r , and so ΔK = (1/2)(GmE m/r )Δr and ΔU = −(GmE m/r )Δr W = ΔU + ΔK = −(GmE m/2r ) Δr (c) v = GmE /r = 7.72 × 103 m/s, Δv = ( Δr/2) GmE /r = 28.9 m/s, E = −GmE m/2r = −8.95 × 1010 J (from Eq (13.15)), ΔK = (GmE m/2r )(Δr ) = 6.70 × 108 J, ΔU = −2ΔK = −1.34 × 109 J, and W = −ΔK = −6.70 × 108 J 13.71 (d) As the term “burns up” suggests, the energy is converted to heat or is dissipated in the collisions of the debris with the ground EVALUATE: When r decreases, K increases and U decreases (becomes more negative) IDENTIFY: Use Eq (13.2) to calculate Fg Apply Newton’s second law to circular motion of each star to find the orbital speed and period Apply the conservation of energy expression, Eq (7.13), to calculate the energy input (work) required to separate the two stars to infinity (a) SET UP: The cm is midway between the two stars since they have equal masses Let R be the orbit radius for each star, as sketched in Figure 13.71 © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-28 Chapter 13 The two stars are separated by a distance 2R, so Fg = GM /(2 R) = GM /4 R Figure 13.71 (b) EXECUTE: Fg = marad GM /4 R = M (v /R ) so v = GM/4 R And T = 2π R/v = 2π R R/GM = 4π R3/GM (c) SET UP: Apply K1 + U1 + Wother = K + U to the system of the two stars Separate to infinity implies K = and U = EXECUTE: K1 = 12 Mv + 12 Mv = ( 12 M ) (GM/4R) = GM 2/4R U1 = −GM /2 R Thus the energy required is Wother = −( K1 + U1 ) = −(GM /4 R − GM /2 R) = GM /4 R 13.72 EVALUATE: The closer the stars are and the greater their mass, the larger their orbital speed, the shorter their orbital period and the greater the energy required to separate them G G IDENTIFY: In the center of mass coordinate system, rcm = Apply F = ma to each star, where F is the gravitational force of one star on the other and a = arad = 4π R T2 2π R allows R to be calculated from v and T T EXECUTE: (a) The radii R1 and R2 are measured with respect to the center of mass, and so SET UP: v = M1R1 = M R2 , and R1/R2 = M /M1 (b) The forces on each star are equal in magnitude, so the product of the mass and the radial accelerations 4π M1R1 4π M R2 From the result of part (a), the numerators of these expressions are = are equal: T12 T22 equal, and so the denominators are equal, and the periods are the same To find the period in the symmetric form desired, there are many possible routes An elegant method, using a bit of hindsight, is to use the GM1M above expressions to relate the periods to the force Fg = , so that equivalent expressions for the ( R1 + R2 ) period are M 2T = ( M1 + M )T = 4π R1 ( R1 + R2 ) 4π R2 ( R1 + R2 )2 and M1T = Adding the expressions gives G G 4π ( R1 + R2 )3 2π ( R1 + R2 )3/2 or T = G G ( M1 + M ) (c) First we must find the radii of each orbit given the speed and period data In a circular orbit, (36 × 103 m/s)(137 d)(86,400 s/d) 2π R vT = 6.78 × 1010 m and , or R = v= Thus Rα = 2π 2π T Rβ = (12 × 103 m/s)(137 d)(86,400 s/d) = 2.26 × 1010 m Now find the sum of the masses 2π (M α + M β ) = 4π ( Rα + Rβ )3 T 2G Inserting the values of T and the radii gives © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation (Mα + M β ) = 4π (6.78 × 1010 m + 2.26 × 1010 m)3 [(137 d)(86,400 s/d)]2 (6.673 × 10−11 N ⋅ m /kg ) 13-29 = 3.12 × 1030 kg Since M β = M α Rα /Rβ = 3M α , M α = 3.12 × 1030 kg, or M α = 7.80 × 1029 kg, and M β = 2.34 × 1030 kg (d) Let α refer to the star and β refer to the black hole Use the relationships derived in parts (a) and (b): Rβ = ( M α /M β ) Rα = (0.67/3.8) Rα = (0.176) Rα , Rα + Rβ = ( M α + M β )T 2G 4π For Monocerotis, inserting the values for M and T gives Rα = 1.9 × 109 m, vα = 4.4 × 102 km/s and for the black hole Rβ = 34 × 108 m, vβ = 77 km/s 13.73 EVALUATE: Since T is the same, v is smaller when R is smaller IDENTIFY and SET UP: Use conservation of energy, K1 + U1 + Wother = K + U The gravity force exerted by the sun is the only force that does work on the comet, so Wother = EXECUTE: K1 = 12 mv12 , v1 = 2.0 × 104 m/s U1 = −GmSm/r1, r1 = 2.5 × 1011 m K = 12 mv22 U = −GmSm/r2 , r2 = 5.0 × 1010 m mv 2 − GmSm/r1 = 12 mv22 − GmSm/r2 ⎛ 1⎞ ⎛r −r ⎞ v22 = v12 + 2GmS ⎜ − ⎟ = v12 + 2GmS ⎜ ⎟ r r ⎝ 1⎠ ⎝ r1r2 ⎠ v2 = 6.8 × 104 m/s 13.74 EVALUATE: The comet has greater speed when it is closer to the sun IDENTIFY and SET UP: Apply Eq (12.6) and solve for g Then use Eq (13.4) to relate g to the mass of the planet EXECUTE: p − p0 = ρ gd This expression gives that g = ( p − p0 )/ρ d = ( p − p0 )V/md But also g = Gmp /R (Eq (13.4) applied to the planet rather than to earth.) Equating these two expressions for g gives Gmp /R = ( p − p0 )V/md and mp = ( p − p0 )VR /Gmd EVALUATE: The greater p is at a given depth, the greater g is for the planet and greater g means greater mp 13.75 IDENTIFY: Follow the procedure outlined in part (b) For a spherically symmetric object, with total mass m and radius r, at points on the surface of the object, g (r ) = Gm/r SET UP: The earth has mass mE = 5.97 × 1024 kg If g (r ) is a maximum at r = rmax , then dg = for dr r = rmax EXECUTE: (a) At r = 0, the model predicts ρ = A = 12,700 kg/m3 and at r = R, the model predicts ρ = A − BR = 12,700 kg/m3 − (1.50 × 10−3 kg/m )(6.37 × 106 m) = 3.15 × 103 kg/m3 R ⎡ AR3 BR ⎤ ⎛ 4π R3 ⎞ ⎡ 3BR ⎤ (b) and (c) M = ∫ dm = 4π ∫ [ A − Br ]r 2dr = 4π ⎢ − ⎟⎟ ⎢ A − ⎥ = ⎜⎜ ⎦⎥ ⎝ ⎠ ⎣ ⎥⎦ ⎣⎢ ⎛ 4π (6.37 × 106 m)3 ⎞ ⎡ 3(1.50 × 10−3 kg/m )(6.37 × 106 m) ⎤ 24 12,700 kg/m3 − M =⎜ ⎟ ⎢ ⎥ = 5.99 × 10 kg ⎜ ⎟⎢ ⎝ ⎠⎣ ⎦⎥ which is within 0.36% of the earth’s mass © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-30 Chapter 13 (d) If m(r ) is used to denote the mass contained in a sphere of radius r, then g = Gm( r )/r Using the same integration as that in part (b), with an upper limit of r instead of R gives the result (e) g = at r = 0, and g at r = R is g = Gm( R )/R = (6.673 × 10−11 N ⋅ m /kg )(5.99 × 1024 kg)/(6.37 × 106 m)2 = 9.85 m/s (f) dg ⎛ 4π G ⎞ d ⎡ 3Br ⎤ ⎛ 4π G ⎞ ⎡ 3Br ⎤ Setting this equal to zero gives =⎜ ⎥ =⎜ ⎟ ⎢ Ar − ⎟⎢A − ⎦⎥ ⎝ ⎠ ⎣ ⎦⎥ dr ⎝ ⎠ dr ⎣⎢ ⎛ 4π G ⎞⎛ A ⎞ ⎡ ⎛ ⎞ ⎛ A ⎞ ⎤ 4πGA r = A/3B = 5.64 × 106 m, and at this radius g = ⎜ ⎟⎜ ⎟ ⎢A − ⎜ ⎟ B⎜ ⎟⎥ = 9B ⎝ ⎠⎝ 3B ⎠ ⎣ ⎝ ⎠ ⎝ 3B ⎠ ⎦ g= 4π (6.673 × 10−11 N ⋅ m /kg )(12,700 kg/m3 ) 9(1.50 × 10 −3 kg/m ) = 10.02 m/s EVALUATE: If the earth were a uniform sphere of density ρ , then g (r ) = setting B = and A = ρ in g (r ) in part (d) If rmax ρV (r ) ⎛ 4πρ G ⎞ =⎜ ⎟ r , the same as r2 ⎝ ⎠ is the value of r in part (f) where g (r ) is a maximum, then rmax /R = 0.885 For a uniform sphere, g (r ) is maximum at the surface 13.76 IDENTIFY: Follow the procedure outlined in part (a) SET UP: The earth has mass M = 5.97 × 1024 kg and radius R = 6.38 × 106 m Let gS = 9.80 m/s EXECUTE: (a) Eq (12.4), with the radius r instead of height y , becomes dp = − ρ g (r ) dr = − ρ gS (r/R) dr This form shows that the pressure decreases with increasing radius Integrating, with p = at r = R, p=− ρ gS R r ∫R r dr = ρ gS R R ∫ r r dr = ρ gS 2R ( R − r ) (b) Using the above expression with r = and ρ = 3(5.97 × 1024 kg)(9.80 m/s ) = 1.71 × 1011 Pa 8π (6.38 × 106 m) (c) While the same order of magnitude, this is not in very good agreement with the estimated value In more realistic density models (see Problem 13.75), the concentration of mass at lower radii leads to a higher pressure EVALUATE: In this model, the pressure at the center of the earth is about 106 times what it is at the surface (a) IDENTIFY and SET UP: Use Eq (13.17), applied to the satellites orbiting the earth rather than the sun EXECUTE: Find the value of a for the elliptical orbit: 2a = + rp = RE + + RE + hp , where and hp are the heights at apogee and perigee, respectively p (0) = 13.77 M 3M , = V 4π R3 a = RE + ( + hp )/2 a = 6.38 × 106 m + (400 × 103 m + 4000 × 103 m) / = 8.58 × 106 m T= 2π a 3/ 2π (8.58 × 106 m)3/2 = = 7.91 × 103 s GM E (6.673 × 10−11 N ⋅ m /kg )(5.97 × 1024 kg) (b) Conservation of angular momentum gives va = rpvp vp va = 6.38 × 106 m + 4.00 × 106 m = = 1.53 rp 6.38 × 106 m + 4.00 × 105 m (c) Conservation of energy applied to apogee and perigee gives K a + U a = K p + U p mva2 − GmE m/ra = 12 mvP2 − GmE m/rp vp2 − va2 = 2GmE (1/rp − 1/ra ) = 2GmE (ra − rp )/ra rp © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation 13-31 But vp = 1.532va , so 1.347va2 = 2GmE (ra − rp )/ra rp va = 5.51 × 103 m/s, vp = 8.43 × 103 m/s (d) Need v so that E = 0, where E = K + U at perigee: mv p − GmE m/rp = vp = 2GmE /rp = 2(6.673 × 10−11 N ⋅ m /kg )(5.97 × 1024 kg)/6.78 × 106 m = 1.084 × 104 m/s This means an increase of 1.084 × 104 m/s − 8.43 × 103 m/s = 2.41 × 103 m/s at apogee: va = 2GmE /ra = 13.78 2(6.673 × 10−11 N ⋅ m /kg )(5.97 × 1024 kg)/1.038 × 107 m = 8.761× 103 m/s This means an increase of 8.761× 103 m/s − 5.51 × 103 m/s = 3.25 × 103 m/s EVALUATE: Perigee is more efficient At this point r is smaller so v is larger and the satellite has more kinetic energy and more total energy GM IDENTIFY: g = , where M and R are the mass and radius of the planet R SET UP: Let mU and RU be the mass and radius of Uranus and let g U be the acceleration due to gravity at its poles The orbit radius of Miranda is r = h + RU , where h = 1.04 × 108 m is the altitude of Miranda above the surface of Uranus EXECUTE: (a) From the value of g at the poles, mU = g U RU2 (11.1 m/s )(2.556 × 107 m) = = 1.09 × 1026 kg G (6.673 × 10−11 N ⋅ m /kg ) (b) GmU /r = g U ( RU /r ) = 0.432 m/s = 0.080 m/s (c) GmM /RM 13.79 EVALUATE: (d) No Both the object and Miranda are in orbit together around Uranus, due to the gravitational force of Uranus The object has additional force toward Miranda IDENTIFY and SET UP: Apply conservation of energy (Eq (7.13)) and solve for Wother Only r = h + RE is given, so use Eq (13.10) to relate r and v EXECUTE: K1 + U1 + Wother = K + U U1 = −GmM m/r1, where mM is the mass of Mars and r1 = RM + h, where RM is the radius of Mars and h = 2000 × 103 m U1 = −(6.673 × 10−11 N ⋅ m /kg ) (6.42 × 1023 kg)(5000 kg) 3.40 × 106 m + 2000 × 103 m U = −GmM m/r2 , where r2 is the new orbit radius U = −(6.673 × 10−11 N ⋅ m /kg ) (6.42 × 1023 kg)(5000 kg) 3.40 × 106 m + 4000 × 103 m = −3.9667 × 1010 J = −2.8950 × 1010 J For a circular orbit v = GmM /r (Eq (13.10)), with the mass of Mars rather than the mass of the earth) Using this gives K = 12 mv = 12 m(GmM /r ) = 12 GmM m/r , so K = − 12 U K1 = − 12 U1 = +1.9833 × 1010 J and K = − 12 U = +1.4475 × 1010 J Then K1 + U1 + Wother = K + U gives Wother = ( K − K1 ) + (U − U ) = (1.4475 × 1010 J − 1.9833 × 1010 J) + ( +3.9667 × 1010 J − 2.8950 × 1010 J) Wother = −5.3580 × 109 J + 1.0717 × 1010 J = 5.36 × 109 J EVALUATE: When the orbit radius increases the kinetic energy decreases and the gravitational potential energy increases K = −U/2 so E = K + U = −U/2 and the total energy also increases (becomes less negative) Positive work must be done to increase the total energy of the satellite © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-32 13.80 Chapter 13 IDENTIFY and SET UP: Use Eq (13.17) to calculate a T = 30, 000 y(3.156 × 107 s/1 y) = 9.468 × 1011 s EXECUTE: Eq (13.17): T = 2π a 3/2 4π 2a , T2 = GmS GmS 1/3 ⎛ GmST ⎞ a=⎜ ⎜ 4π ⎟⎟ ⎝ ⎠ = 1.4 × 1014 m EVALUATE: The average orbit radius of Pluto is 5.9 × 1012 m (Appendix F); the semi-major axis for this comet is larger by a factor of 24 4.3 light years = 4.3 light years(9.461 × 1015 m/1 light year) = 4.1× 1016 m 13.81 The distance of Alpha Centauri is larger by a factor of 300 The orbit of the comet extends well past Pluto but is well within the distance to Alpha Centauri IDENTIFY: Integrate dm = ρ dV to find the mass of the planet Outside the planet, the planet behaves like a point mass, so at the surface g = GM/R SET UP: A thin spherical shell with thickness dr has volume dV = 4π r dr The earth has radius RE = 6.38 × 106 m EXECUTE: Get M : M = ∫ dm = ∫ ρ dV = ∫ ρ 4π r 2dr The density is ρ = ρ0 − br , where ρ0 = 15.0 × 103 kg/m3 at the center and at the surface, ρS = 2.0 × 103 kg/m3 , so b = R M = ∫ ( ρ0 − br ) 4π r 2dr = ρ0 − ρ s R 4π ⎛ ρ − ρs ⎞ ⎞ 3⎛1 ρ0 R3 − π bR = π R3ρ0 − π R ⎜ ⎟ = π R ⎜ ρ0 + ρ s ⎟ and 3 ⎝ R ⎠ ⎝3 ⎠ M = 5.71 × 1024 kg Then g = GM R2 = Gπ R3 ( 13 ρ0 + ρs ) = π RG ⎛ ρ R2 ⎜ ⎝3 ⎞ + ρs ⎟ ⎠ ⎛ 15.0 × 103 kg/m3 ⎞ + 2.0 × 103 kg/m3 ⎟ g = π (6.38 × 106 m)(6.67 × 10−11 N ⋅ m /kg ) ⎜ ⎜ ⎟ ⎝ ⎠ g = 9.36 m/s EVALUATE: The average density of the planet is M M 3(5.71 × 1024 kg) = = = 5.25 × 103 kg/m3 Note that this is not ( ρ0 + ρs )/2 ρav = V πR 4π (6.38 × 106 m)3 13.82 IDENTIFY and SET UP: Use Eq (13.1) to calculate the force between the point mass and a small segment of the semicircle EXECUTE: The radius of the semicircle is R = L/π Divide the semicircle up into small segments of length R dθ , as shown in Figure 13.82 Figure 13.82 © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation 13-33 dM = ( M/L) R dθ = ( M/π ) dθ G dF is the gravity force on m exerted by dM ∫ dFy = 0; the y-components from the upper half of the semicircle cancel the y-components from the lower half The x-components are all in the + x -direction and all add mdM dF = G R mdM Gmπ M dFx = G cosθ = cosθ dθ R L2 π /2 Gmπ M π / Gmπ M Fx = ∫ dFx = ∫ −π /2 cosθ dθ = L2 (2) −π / L2 2π GmM F= L2 EVALUATE: If the semicircle were replaced by a point mass M at x = R, the gravity force would be 13.83 GmM/R = π 2GmM/L2 This is π /2 times larger than the force exerted by the semicirclar wire For the semicircle it is the x-components that add, and the sum is less than if the force magnitudes were added IDENTIFY: The direct calculation of the force that the sphere exerts on the ring is slightly more involved than the calculation of the force that the ring exerts on the sphere These forces are equal in magnitude but opposite in direction, so it will suffice to the latter calculation By symmetry, the force on the sphere will be along the axis of the ring in Figure E13.33 in the textbook, toward the ring SET UP: Divide the ring into infinitesimal elements with mass dM (Gm) dM EXECUTE: Each mass element dM of the ring exerts a force of magnitude on the sphere, a + x2 GmdM x GmdMx and the x-component of this force is = 2 2 a +x a +x (a + x )3/2 Therefore, the force on the sphere is GmMx/(a + x )3/2 , in the − x-direction The sphere attracts the ring with a force of the same magnitude EVALUATE: As x a the denominator approaches x3 and F → 13.84 GMm , as expected x2 IDENTIFY: Use Eq (13.1) for the force between a small segment of the rod and the particle Integrate over the length of the rod to find the total force SET UP: Use a coordinate system with the origin at the left-hand end of the rod and the x′-axis along the rod, as shown in Figure 13.84 Divide the rod into small segments of length dx′ (Use x′ for the coordinate so not to confuse with the distance x from the end of the rod to the particle.) Figure 13.84 EXECUTE: The mass of each segment is dM = dx′( M/L) Each segment is a distance L − x′ + x from mass m, so the force on the particle due to a segment is dF = F = ∫ dF = L GMm dx′ GMm ⎛ = ⎜− ∫ L L L ⎝ L − x′ + x ( L − x′ + x) Gm dM ( L − x′ + x) = GMm dx′ L ( L − x′ + x) 0⎞ L⎟ ⎠ © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-34 Chapter 13 F= 13.85 GMm ⎛ 1 ⎞ GMm ( L + x − x) GMm = ⎜ − ⎟= L ⎝ x L+x⎠ L x( L + x) x( L + x) EVALUATE: For x L this result becomes F = GMm/x , the same as for a pair of point masses IDENTIFY: Compare FE to Hooke’s law SET UP: The earth has mass mE = 5.97 × 1024 kg and radius RE = 6.38 × 106 m EXECUTE: (a) For Fx = −kx, U = 12 kx The force here is in the same form, so by analogy U (r ) = GmE m RE3 r This is also given by the integral of Fg from to r with respect to distance GmE m Equating initial potential energy and RE final kinetic energy (initial kinetic energy and final potential energy are both zero) gives GmE , so v = 7.90 × 103 m/s v2 = RE EVALUATE: When r = 0, U (r ) = 0, as specified in the problem IDENTIFY: In Eqs (13.12) and (13.16) replace T by T + ΔT and r by r + Δr Use the expression in the hint to simplify the resulting equations SET UP: The earth has mE = 5.97 × 1024 kg and R = 6.38 × 106 m r = h + RE , where h is the altitude (b) From part (a), the initial gravitational potential energy is 13.86 above the surface of the earth 2π r 3/2 EXECUTE: (a) T = therefore GM E T + ΔT = Since v = 2π GM E GM E r (r + Δr )3/2 = , ΔT = 2π r 3/2 ⎛ Δr ⎞ ⎜1 + ⎟⎠ r GM E ⎝ 3/2 ≈ 3π Δr v = GM E r −1/2 , and therefore v ⎛ Δr ⎞ v − Δv = GM E (r + Δr ) −1/2 = GM E r −1/2 ⎜1 + ⎟ r ⎠ ⎝ Since T = (b) 2π r 3/ ⎛ 3Δr ⎞ 3π r1/2 Δr ⎜⎝1 + ⎟⎠ = T + 2r GM E GM E −1/2 GM E ⎛ Δr ⎞ and v ≈ GM E r −1/2 ⎜1 − Δr ⎟=v− 2r 3/2 ⎝ 2r ⎠ 2π r 3/2 πΔr , Δv = T GM E Starting with T = 2π r 3/2 GM (Eq (13.12), T = 2π r/v, and v = (Eq (13.10)), find the velocity r GM and period of the initial orbit: v = (6.673 × 10−11 N ⋅ m /kg )(5.97 × 1024 kg) = 7.672 × 103 m/s, and 6.776 × 10 m T = 2π r/v = 5549 s = 92.5 We then can use the two derived equations to approximate ΔT and Δv : 3π (100 m) π (100 m) = 0.05662 m/s Before the cable ΔT = 3πΔr = = 0.1228 s and Δv = πΔr = T (5549 s) v 7.672 × 10 m/s breaks, the shuttle will have traveled a distance d, d = (125 m ) − (100 m ) = 75 m t = (75 m)/(0.05662 m/s) = 1324.7 s = 22 It will take 22 minutes for the cable to break (c) The ISS is moving faster than the space shuttle, so the total angle it covers in an orbit must be 2π radians more than the angle that the space shuttle covers before they are once again in line (v − Δv)t Mathematically, vt − = 2π Using the binomial theorem and neglecting terms of order r (r + Δr ) © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher Gravitation 13-35 vT 2π r (v − Δv)t ⎛ ⎞ Since ΔvΔr , vt − (1 + Δr ) −1 ≈ t ⎜ Δv + vΔ2r ⎟ = 2π Therefore, t = = r r r vΔr ⎞ π Δr vΔr ⎛ r ⎠ ⎝ r + v Δ + ⎜⎝ ⎟ T r r ⎠ vT T2 2π r = vT and Δr = vΔT , t = = , as was to be shown 3π π ⎛ vΔT ⎞ 2π ⎛ vΔT ⎞ ΔT ⎜ ⎟+ ⎜ ⎟ t ⎝ 3π ⎠ T ⎝ 3π ⎠ T (5549 s) = = 2.5 × 108 s = 2900 d = 7.9 y It is highly doubtful the shuttle crew would survive the ΔT (0.1228 s) congressional hearings if they miss! EVALUATE: When the orbit radius increases, the orbital period increases and the orbital speed decreases IDENTIFY: Apply Eq (13.19) to the transfer orbit t= 13.87 SET UP: The orbit radius for earth is rE = 1.50 × 1011 m and for Mars it is rM = 2.28 × 1011 m From Figure 13.18 in the textbook, a = 12 ( rE + rM ) EXECUTE: (a) To get from the circular orbit of the earth to the transfer orbit, the spacecraft’s energy must increase, and the rockets are fired in the direction opposite that of the motion, that is, in the direction that increases the speed Once at the orbit of Mars, the energy needs to be increased again, and so the rockets need to be fired in the direction opposite that of the motion From Figure 13.18 in the textbook, the semimajor axis of the transfer orbit is the arithmetic average of the orbit radii of the earth and Mars, and so from Eq (13.13), the energy of the spacecraft while in the transfer orbit is intermediate between the energies of the circular orbits Returning from Mars to the earth, the procedure is reversed, and the rockets are fired against the direction of motion (b) The time will be half the period as given in Eq (13.17), with the semimajor axis equal to a = 12 (rE + rM ) = 1.89 × 1011 m so t= T π (1.89 × 1011 m)3/2 = = 2.24 × 107 s = 259 days, which is more than 12 (6.673 × 10−11 N ⋅ m /kg )(1.99 × 1030 kg) months (2.24 × 107 s) = 135.9°, and the (687 d)(86,400 s/d) spacecraft passes through an angle of 180°, so the angle between the earth-sun line and the Mars-sun line must be 44.1° EVALUATE: The period T for the transfer orbit is 526 days, the average of the orbital periods for earth and Mars G G IDENTIFY: Apply ΣF = ma to each ear SET UP: Denote the orbit radius as r and the distance from this radius to either ear as δ Each ear, of mass m, can be modeled as subject to two forces, the gravitational force from the black hole and the tension force (actually the force from the body tissues), denoted by F GMm EXECUTE: The force equation for either ear is − F = mω ( r + δ ), where δ can be of either sign (r + δ ) (c) During this time, Mars will pass through an angle of (360°) 13.88 Replace the product mω with the value for δ = 0, mω = GMm/r , and solve for F: ⎡r +δ ⎤ GMm ⎡ ⎥= F = (GMm) ⎢ − r + δ − r (1 + (δ /r )−2 ⎤ ⎦ ⎢⎣ r ( r + δ )2 ⎥⎦ r ⎣ Using the binomial theorem to expand the term in square brackets in powers of δ /r , GMm GMm F ≈ [r + δ − r (1 − 2(δ /r ))] = (3δ ) = 2.1 kN r r This tension is much larger than that which could be sustained by human tissue, and the astronaut is in trouble © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher 13-36 13.89 Chapter 13 (b) The center of gravity is not the center of mass The gravity force on the two ears is not the same EVALUATE: The tension between her ears is proportional to their separation IDENTIFY: As suggested in the problem, divide the disk into rings of radius r and thickness dr M 2M SET UP: Each ring has an area dA = 2π r dr and mass dM = dA = r dr a πa EXECUTE: The magnitude of the force that this small ring exerts on the mass m is then 2GMmx rdr (Gm dM )( x/(r + x )3/2 ) The contribution dF to the force is dF = a ( x + r )3/ The total force F is then the integral over the range of r; 2GMmx a r F = ∫ dF = dr ∫ 2 a ( x + r )3/2 The integral (either by looking in a table or making the substitution u = r + a ) is r a ⎡1 ∫0 ( x + r )3/2 dr = ⎢⎢ x − ⎣ ⎤ 1⎡ ⎤ x ⎥ = ⎢1 − ⎥ 2 a + x ⎥⎦ x ⎢⎣ a + x ⎥⎦ 2 ⎤ 2GMm ⎡ x ⎢1 − ⎥ The force on m is directed toward the center of a a + x ⎦⎥ ⎣⎢ the ring The second term in brackets can be written as Substitution yields the result F = 1⎛ a⎞ = (1 + (a/x) ) −1/2 ≈ − ⎜ ⎟ 2⎝ x ⎠ + (a/x) 13.90 if x a, where the binomial expansion has been used Substitution of this into the above form gives GMm F ≈ , as it should x EVALUATE: As x → 0, the force approaches a constant IDENTIFY: Divide the rod into infinitesimal segments Calculate the force each segment exerts on m and integrate over the rod to find the total force SET UP: From symmetry, the component of the gravitational force parallel to the rod is zero To find the M , positioned at a perpendicular component, divide the rod into segments of length dx and mass dm = dx 2L distance x from the center of the rod EXECUTE: The magnitude of the gravitational force from each segment is Gm dM GmM dx a dF = The component of dF perpendicular to the rod is dF and so the = 2 2 2L x + a x +a x + a2 L ∫ net gravitational force is F = L dF = −L GmMa dx ∫ 2 L − L ( x + a )3/2 The integral can be found in a table, or found by making the substitution x = a tanθ Then, dx = a sec2θ dθ ,( x + a ) = a sec2θ , and so dx ∫ ( x + a )3/2 = ∫ a sec 2θ dθ a sec3θ = and the definite integral is F = a2 ∫ cosθ dθ = a sinθ = a GmM a a + L2 x x2 + a2 , EVALUATE: When a L, the term in the square root approaches a and F → GmM a2 , as expected © Copyright 2012 Pearson Education, Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher