Planetary Gravitational Acceleration?

Question

a planet has a mass of 6.2*10^25 kg and a radius of 2.42*10^6m. Determine the gravitational acceleration of the planet. Also determine the value of the gravitational acceleration of the planet that is three planetary radii away from the surface of the planet.

Answer 1

This is a Newton's Law of Universal Gravitation problem.

Given:
F = GMm / r²
F = ma

Solve for a:
ma = GMm / r²
a = GM / r²

Plug in the values and solve:
a = (6.674x10^-11)(6.2x10^25)/(2.42x10^6)²
a = 707 m/s² (surface acceleration)

Rather than doing the whole calculation again, you know that the acceleration of gravity is inversely proportional to the square of the distance. So if you quadruple the distance (r + 3r, per the problem wording), gravity decreases to 1/16.

a = 707/16 = 44.2 m/s² (acceleration at distance 4r)

Answer 2

For the first part of your question, you need an expression for gravitational acceleration as a function of planetary mass and radius. You can get this by remembering that weight is the amount of gravitational force between an object and the planet it is on. One way of writing weight is W=mg, where g is the local acceleration due to gravity. The second way comes from using Newton's law of gravity:

F= G m1 m2/r^2, where G is the Newtonian Grav Constant, m1 and m2 are the masses of the object and planet, and r^2 is the square of the radius of the planet. Equating these two descriptions of gravity, you get:

g=GM/R^2, where M and R are the mass and radius of the planet. Since G=6.67x10^-11 in MKS units, the value of g on this planet is:

g= 6.67X10^-11 x 6.2x10^25/[2.42x10^6]^2=706 m/sec/sec, which is really high compared to the Earth.

I would ask you to check the numbers you post, I did a quick calculation of the density of the planet these numbers imply, and they imply an unreasonably large density.

For the second part of your problem, I am guessing you mean what would be the gravitational acceleration caused by this planet at a point that is 3 planetary radii away from the original planet. In this case, use the formulation of Newton's Law given above, and recognize that if the accel due to gravity has a certain value on the surface of the planet (which is one radius from the center of the planet), then the grav acceleration 3 planetary radii away will be 1/9 of this value, since gravity decreases as 1/r^2, and if the distance triples, the force of gravity will decrease by 1/3^2 or by 1/9. But, I ask you to check the values you were given. Once you are sure of the numbers, the procedures above should help you plug them in and get the values you need. Good luck!

Answer 3

Both questions are answered using Newton's Law of Gravity: the acceleration of gravity g near a planet is equal to

g = G M / r²

For your first problem (in which I assume you want the local acceleration of gravity at the surface), plug in Newton's Gravity Constant G, the mass of the planet M, and the radius of the planet for r.

For the second, same parameters except you triple the distance.