Planet X has two times the diameter and eight times the mass of the earth.?

Question

Planet X has two times the diameter and eight times the mass of the earth.

What is the ratio gx : ge of gravitational acceleration at the surface of planet X to the gravitational acceleration at the surface of the Earth?

Answer 1

Force goes up directly with the mass of the planet and down with the inverse square of the radius. So, the mass would multiply by 8 and the radius would divide by 2^2.

Answer 2

When comparing values of the same physical effect (e.g., gravitational acceleration), start with a ratio (e.g., gx/ge). This is the best approach because most of the factors going into the effect will cancel out.

For your problem Rx = 2Re and Mx = 8Me; therefore, gx/ge = (Mx/Me)(Re/Rx)^2 = (8)(1/4) = 2 after all the other factors cancel out. Thus gx = 2 ge ~ 20 m/sec^2 [I'm using ge ~ 10 m/sec^2 for ease of calculation, you can use 9.81 for accuracy if you wish.]

The physics to remember here is that the gravitational acceleration is not a constant...it varies according to mass and the inverse of the square of the distance between masses. Also, relate this to weight W = mg; where m is the other mass, the one we are weighing, in GmM/R^2. Clearly what we weigh depends on where we are (e.g., Planet X or Earth). And, further, it depends on how far away we are from the mass center of the planet.

Answer 3

The acceleration due to gravity at a planet's surface is given by:

a = GM/(r^2)

where G is the universal gravitational constant, M is the mass of the planet (in kg) and r is the radius of the planet (in meters). If you multiply the mass by 8 while changing nothing else, you multiply the surface gravity by 8 also. However, doubling the radius (which is in the denominator) does more than cutting the surface gravity in half; because r is squared in this equation, halving the radius will quarter the surface gravity. When you take these two changes into account (mass x 8 and radius x 2), you find that Planet X's surface gravity would be twice that of Earth.

Answer 4

2

8 times mass over 2 squared = 2