Universal Gravitational

Question

Gravity on Planet X is three times weaker than on earth. It has a radius roughly, 25% that of earths radius. What is the mass of Planet X as a percentage of earths mass?

Answer 1

REQUIREMENT of this problem is to return mX as a percentage of mEarth (to 1 sig fig)

Fg = G * m1 * m2 / r^2, so

Fearth = G * m * mEarth / rEarth^2, and

Fx = G * m * mX // rX^2, which means

Fearth / Fx = (G * m * mEarth / rEarth^2) / (G * m * mX / rX^2), and cancelling out common terms

Fearth / Fx = (mEarth / rEarth^2) * (rX^2 / mX) and rearranging masses and radii to be together

Fearth / Fx = (mEarth / mX) * (rX / rEarth)^ 2 plugging in (Fearth = 3 * Fx, rX = 1/4 rEarth)

3 / 1 = (mEarth / mX) * (1/4)^2

= (mEarth / mX) * (1/16) multiplying both sides by 16

48 = mEarth / mX inverting both sides

1/48 = mX / mEarth or

mX = (1/48) * mEarth or

mX = 2.08% of mEarth or

mX = 2% of mEarth (ANSWER to 1 sig fig)

Answer 2

Newton's universal law of gravitation:

and, Newton's second law:

but, in the case of the acceleration due to gravity, we use a = g meaning F = mg.

Setting these equal, we get:

But if m₁ = mass of the planet, then m₂ and "m" are the same thing (the mass of some object sitting on the surface of the planet) so:

g = G(m_planet)/r² meaning m_planet = gr²/G

On earth g = 9.81 m/s² so on this planet g = 9.81/3 = 3.27 m/s²

The radius of earth is 6.3781x10⁶ m so on this planet, r = (0.25)(6.3781x10⁶) = 1.595x10⁶ m

G = 6.674x10⁻¹¹ m³⋅kg⁻¹⋅s⁻²

Plugging those values into m_planet = gr²/G we get:

m_planet = (3.27)(1.595x10⁶)²/6.674x10⁻¹¹ = 1.25x10²³ kg