Suppose that F is an inverse square force field, that is, F(r) = cr/|r|^3 for so

Question

Suppose that F is an inverse square force field, that is, F(r) = cr/|r|^3 for some constant c, where r = x i + y j + z k. Find the work done by F in moving an object from a point P_1 along a path to a point P_2 in terms of the distances d_1 and d_2 from these points to the origin. An example of an inverse square field is the gravitational field F = - (mMG)r/|r|^3 discussed in Example 4 in Section 13.1. Use part (a) to find the work done by the gravitational field when the earth moves from aphelion (at a maximum distance of 1.52 X 10s km from the sun) to perihelion (at a minimum distance of 1.47 * 10^8 km). (Use the values m = 5.97 * 10^24 kg, M = 1.99 * 10^30 kg, and G = 6.67 * 10^-11 N middot m^2/kg^2.)

Explanation / Answer

F = cr / |r|³

Work = [PP] F•dr = c [PP] r•dr/||r||³

r / |r|³ = (x,y,z)(x²+y²+z²)¹

= ( {(x²+y²+z² ) }/x, {(x²+y²+z² ) }/y, {(x²+y²+z² ) }/z )

= ((x²+y²+z² )) = where =1/d

So F is conservative and work is change in c from P to P

= c/d (c/d) = c/d c/d

b) W =c/d c/d

in this case, c = -GMm = -6.67*10-11 * 1.99*1030 * 5.97*1024 = -7.92*1044 Nm2

so, W = -7.92*1044 [1/(1.52*108) - (1/1.47*108)] =1.77E35J

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