Planet X has six times the diameter and seven 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?
The gravitational acceleration at the surface of a planet is given by the formula:
g = GM / R^2
where:
g is the gravitational acceleration,
G is the universal gravitational constant,
M is the mass of the planet, and
R is the radius of the planet.
Let's assume the radius of Earth is Re and the radius of Planet X is RX. We are given that Planet X has six times the diameter (or twice the radius) of Earth. This means RX = 2Re.
We are also given that Planet X has seven times the mass of Earth. This means the mass of Planet X is MX = 7Me, where Me is the mass of Earth.
Now, let's calculate the ratio of gravitational acceleration:
gX / ge = (GX * MX) / (Ge * Me)
Substituting the expressions for GX, MX, Ge, and Me:
gX / ge = (G * MX / RX^2) / (G * Me / Re^2)
Mass cancels out:
gX / ge = RX^2 / Re^2
Substituting RX = 2Re:
gX / ge = (2Re)^2 / Re^2
= 4
Therefore, the ratio of gravitational acceleration at the surface of Planet X to the gravitational acceleration at the surface of Earth is 4:1 (gX : ge).
To find the ratio gX : ge of gravitational acceleration at the surface of planet X to the gravitational acceleration at the surface of the Earth, we can use Newton's law of gravitation, which states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The formula for gravitational acceleration at the surface of a planet is given by:
g = (G * M) / r^2
where:
g is the gravitational acceleration
G is the universal gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2)
M is the mass of the planet
r is the radius of the planet
Now, let's compare the gravitational acceleration of planet X to that of Earth.
Given:
The diameter of planet X is 6 times that of the Earth, so its radius (rX) will be 3 times that of Earth (rE).
The mass of planet X is 7 times that of Earth (MX).
Let's assume the gravitational acceleration at the surface of Earth is ge, and the gravitational acceleration at the surface of planet X is gX.
Using the formula for gravitational acceleration, we have:
gE = (G * ME) / rE^2
gX = (G * MX) / rX^2
Substituting rX = 3 * rE and MX = 7 * ME, we get:
gE = (G * ME) / (rE^2)
gX = (G * (7 * ME)) / ((3 * rE)^2)
Simplifying:
gE = (G * ME) / (rE^2)
gX = (G * 7 * ME) / (9 * rE^2)
Now, let's find the ratio gX : ge:
gX / ge = [(G * 7 * ME) / (9 * rE^2)] / [(G * ME) / (rE^2)]
The G, ME, and rE^2 terms cancel out, leaving:
gX / ge = (7 * ME) / (9 * ME)
Simplifying further:
gX / ge = 7 / 9
Therefore, the ratio gX : ge of gravitational acceleration at the surface of planet X to the gravitational acceleration at the surface of Earth is 7:9.