Mass of planet x

Question

An object fall from planet x with a=4,9 m/s². The radius of planet X is 0,8 times earth radius. Find the mass of planet X compare to earth.

Answer 1

Hi Icy,

 

To calculate the acceleration due to gravity, you need to apply Newton's Second Law:

 

ΣF = mg,

 

the sum of all forces equals the mass times acceleration, where we specify the acceleration as that due to gravity, so we use g.

 

The force is determined by Newton's Law of Universal Gravitation between the test mass m and the mass of the earth M:

 

(GMm)/R² = mg

 

An interesting property of spherical masses is that they are equivalent to a point mass located at their center. So, the R in this equation is the earth radius, 6.37 X 10⁶ m.

 

In the above equation, we can cancel out the test mass m and write

 

g = (GM)/R²

 

Let's test this. Recall that the mass of the earth M = 5.98 X 10²⁴ kg. So, we can calculate

 

g = (6.67 X 10^-11)(5.98 X 10²⁴)/(6.37 X 10⁶)² = 9.8 m/s², which we recognize as the acceleration due to gravity for an object on the surface of the earth.

 

Now, we have a mysterious planet. Its acceleration due to gravity is a = 4.9 m/s². Its radius is 0.8 times the earth radius. Let's take the above equation and solve for M.

 

M2 = ar²/G

 

where we are calling M2 the mass of the mysterious planet, a is the acceleration due to gravity there and r is its radius. It turns out that a = g/2. r = 0.8R. Of course, G is universal.

 

M2 = (g/2)(0.8R)²/G 

 

Or

 

M2 = (1/2)(0.8)² gR²/G

 

But, M = gR²/G, where M is the mass of the earth.

 

So,

 

M2 = (1/2)(0.8)²M

 

M2 = 0.32M