How far above the surface of the Earth does an object have to be in order for it to have a weight which is 0.53 times the same weight as it would have on the surface of the Moon? (Ignore any effects from the Earth's gravity for the object on the Moon's surface or from the Moon's gravity for the object above the Earth.)

Question

How far above the surface of the Earth does an object have to be in order for it to have a weight which is 0.53 times the same weight as it would have on the surface of the Moon? (Ignore any effects from the Earth's gravity for the object on the Moon's surface or from the Moon's gravity for the object above the Earth.)

Answer 1

To solve this problem, we need to compare the gravitational forces on the object on the Moon and at a certain height above the Earth's surface.

We can start by using Newton's law of universal gravitation, which states that the force of gravity between two objects is proportional to their masses and inversely proportional to the square of the distance between them. Mathematically, it can be written as:

F = G * (m1 * m2) / r^2

Where:
- F is the force of gravity
- G is the gravitational constant
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects

In this case, we want to find the height above the Earth's surface at which the object has a weight that is 0.53 times its weight on the Moon. Let's denote the weight on the Moon as W_moon and the weight above Earth's surface as W_earth.

Since weight is a measure of the force of gravity acting on an object, we have:

W_moon = F_moon
W_earth = F_earth

We can now compare the two forces of gravity. However, since we are asked to ignore any effects of Earth's gravity on the Moon's surface or from the Moon's gravity on the object above Earth's surface, we can assume the masses of the object and Earth remain constant in both cases. Therefore, we can cancel out the mass terms in the equation.

W_moon = G * (m_object * m_moon) / r_moon^2
W_earth = G * (m_object * m_earth) / r_earth^2

Now we can set up our equation:

W_earth = 0.53 * W_moon

Plugging in the expressions for W_earth and W_moon:

G * (m_object * m_earth) / r_earth^2 = 0.53 * (G * (m_object * m_moon) / r_moon^2)

Since G, m_object, and m_earth are common terms in both expressions, we can cancel them out:

m_earth / r_earth^2 = 0.53 * (m_moon / r_moon^2)

Now we can solve for the height above the Earth's surface, r_earth:

r_earth^2 = (0.53 * r_moon^2 * m_moon) / m_earth

Taking the square root of both sides:

r_earth = sqrt((0.53 * r_moon^2 * m_moon) / m_earth)

Finally, substitute the values for r_moon, m_moon, and m_earth to get the answer.