[Hecht2 5.P.028.] At what center-to-center distance from the Earth would a 0.7-kg mass weigh 0.7 N?

Question

(Universal Gravity and Satellite Orbits)

___ meters

This question is from my Web Assignment from http://www.webassign.com/.

Answer 1

Fg=G*M*m / r^2

.7N= 6.67x10^-11*5.97x10^24*0.7kg / r^2
r^2= 6.67x10^-11*5.97x10^24*0.7kg / 0.7N
r= 19954924.2m

Answer 2

stmc has given you a bogus equation to start with. F = W = mg = GMm/r^2 is what you should start with, not the Fg =... stmc gave you.

Your problem is to find a radius (r) where g = 1 m/sec^2; this follows from W = mg(r) = .7 N = .7 kg = m and that can only happen when g(r) = 1 which is a function of r.

So we have mg(R) = GMm/R^2; where R = Earth's radius and g(R) = 9.81 m/sec^2 the acceleration due to gravity at Earth's surface. And we have mg(r) = GMm/r^2; where r = Earth's radius where g(r) = 1.00 from the givens above.

The ratio g(R)/g(r) = (1/R^2)/(1/r^2) = r^2/R^2; so that r^2 = R^2 (g(R)/g(r)) and r = R sqrt(9.81/1) = sqrt(9.81)R, which means your body's weight will equal the value of its mass when m is at a point about 3.1 = sqrt(9.81) times the radius of the Earth from the center of the Earth..

Lesson learned: The masses are not germane (except to specify the radius' g level through the W = mg relatiionship). That is, both M and m cancel out when the ratio is formed. In fact G cancels out as well; so even it does not have to be known for this problem.