Question

The acceleration due to gravity on the Moon's surface is known to be about one-sixth the acceleration due to gravity on the Earth.

Given that the radius of the Moon is roughly one-quarter that of the Earth, find the mass of the Moon in terms of the mass of the Earth.

Answer 1

To find the mass of the Moon in terms of the mass of the Earth, we can use the formula for gravitational acceleration and the given ratios between the Moon and Earth's acceleration and radii. Using these values, we can solve for the mass of the Moon in terms of the mass of the Earth.

Step-by-step explanation:

To find the mass of the Moon in terms of the mass of the Earth, we can use the formula for gravitational acceleration: g = GM/r². Given that the acceleration due to gravity on the Moon is one-sixth that of Earth and the radius of the Moon is one-quarter that of Earth, we can set up the following equation:

(1/6) * (9.8 m/s²) = GM/(1/4 * R)²

Simplifying, we get: 1/6 * 9.8 = GM/(1/16)

Now, we can solve for the mass of the Moon (M) in terms of the mass of the Earth (m):

M = (1/6 * 9.8 * R²)/(1/16 * G)

Substituting the values for R and G, we get:

M = (1/6 * 9.8 * (1/4)²)/(1/16 * 6.67 × 10⁻¹¹)

M ≈ 0.123m

Answer 2

Step-by-step explanation:

If the acceleration due to gravity on the Moon's surface is
`a_M` and the acceleration due to gravity on the Earth is
`a_E`, we can write that:

`a_M=(1)/(6)a_E`

`(a_M)/(a_E)=(1)/(6)`

If the radius of the Moon is
`r_M` and the radius of the Earth is
`r_E`, we can write that:

`r_M=(1)/(4)r_E`

`(r_M)/(r_E)=(1)/(4)`

By Newton's 2nd Law we know that F=ma and using Newton's law of universal gravitation we can calculate the gravitational force an object with mass m experiments from a planet with mass M being at a distance r from it. We will assume our object is on the surface so this distance will be the radius of the planet.

Since the force the object experiments is the force of gravitation we can write, for Earth:

`F=ma_E=(GM_Em)/(r_E^2)`

which means:

`a_E=(GM_E)/(r_E^2)`

`M_E=(a_Er_E^2)/(G)`

And for the Moon:

`F=ma_M=(GM_Mm)/(r_M^2)`

which means:

`a_M=(GM_M)/(r_M^2)`

`M_M=(a_Mr_M^2)/(G)`

We can then write the fraction:

`(M_M)/(M_E)=(a_Mr_M^2)/(G)(G)/(a_Er_E^2)=(a_M)/(a_E)((r_M)/(r_E))^2=(1)/(6)((1)/(4))^2=0.01`

Which means:

`M_M=0.01M_E`