The radius of the moon is one-fourth, and its mass is one eighty-first that of the earth. If the acceleration due to gravity on the surface of the earth is 9.8 m/s2, what is its value on the moon’s surface.
To find the acceleration due to gravity on the moon's surface, we need to use the formula for gravitational acceleration:
g = G * (M / r^2)
where:
g is the acceleration due to gravity
G is the gravitational constant (approximately 6.67430 × 10^(-11) m^3 kg^(-1) s^(-2))
M is the mass of the celestial body (in this case, the Earth or the Moon)
r is the distance from the center of the celestial body
Let's start by finding the acceleration due to gravity on the Earth's surface using the given information.
Given:
Acceleration due to gravity on Earth's surface (g_earth) = 9.8 m/s^2
Assuming the mass of the Earth (M_earth) remains constant, we can express the equation for the Earth as:
g_earth = G * (M_earth / r_earth^2)
Now, we need to find the value of g_moon, the acceleration due to gravity on the Moon's surface.
Given:
Radius of the Moon (r_moon) = (1/4) * r_earth
Mass of the Moon (M_moon) = (1/81) * M_earth
We can now express the equation for the Moon as:
g_moon = G * (M_moon / r_moon^2)
Now, substitute the values into the equations:
g_earth = G * (M_earth / r_earth^2)
g_moon = G * (M_moon / r_moon^2)
From the given information, we can conclude that:
r_moon = (1/4) * r_earth
M_moon = (1/81) * M_earth
By substituting these values, we get:
g_earth = G * (M_earth / r_earth^2)
g_moon = G * ((1/81) * M_earth) / ((1/4 * r_earth)^2)
Now, divide the equation for g_moon by the equation for g_earth:
g_moon / g_earth = ((G * ((1/81) * M_earth) / ((1/4 * r_earth)^2)) / (G * (M_earth / r_earth^2))
Simplify the equation:
g_moon / g_earth = ((1/81) * M_earth * r_earth^2) / ((1/4)^2 * r_earth^2 * M_earth)
Cancel out the common terms and simplify further:
g_moon / g_earth = 4 / 81
Finally, substitute the value of g_earth into the equation:
g_moon = (4 / 81) * 9.8 m/s^2
Therefore, the acceleration due to gravity on the Moon's surface is approximately 0.48 m/s^2.