Where is the Zero Gravitational Field Strength Point Between Earth and Moon?

[linnus]

Homework Statement

100 ton rocks are "thrown" from the moon to the earth.

(a) Calculate the location of the point, between the Earth and the Moon, where the gravitational field strength is zero. Give your answer in km, from the center of the Earth.
(b) What is the difference in PE between the surface of the Moon (at the point closest to Earth) and the “balance point” you found in part a? (HINT: be sure to take into account the change in PE due to the Moon’s gravity, and the change due to Earth’s.)

mass of earth=5.97*10^24 Kg
mass of moon=7.37*10^22 Kg
radius of moon= 1738 Km
radius of earth= 6378 Km
radius of Moon's orbit= 384,000 Km

I got the first one to be 341,333 Km
I don't know how to do the 2nd one

Homework Equations

I got the first one to be

-GMm/r

The Attempt at a Solution

So, for part B i did
change in PE= PE at object of surface of moon- PE at the balance point
change in PE= ((-GM(moon)m/radius of moon)+(GM(earth)m/(distance from Earth to surface of moon))-((GM(moon)m/distance from the moon to to balance point )+(GM(earth)m/distance from the Earth to the balance point)
is that right?
I got PE= 7.88 x 10^11 joules...

 

[anathema]

Your first answer for part A is correct. For part B, you are on the right track but there are a few errors in your calculation.

First, the formula for potential energy is PE = -GMm/r, where G is the gravitational constant, M is the mass of the larger object (Earth), m is the mass of the smaller object (Moon), and r is the distance between the two objects.

In your calculation, you have used the radius of the Earth and Moon instead of the distance between them. The correct distance is the difference between the radius of the Moon's orbit (384,000 km) and the distance from the Earth to the surface of the Moon (384,000 km - 6378 km = 377,622 km).

Second, you have not taken into account the change in potential energy due to the Moon's gravity. This can be calculated using the same formula but using the distance from the Moon to the balance point (341,333 km) instead of the distance from the Moon's surface.

Third, when calculating the change in potential energy at the balance point, you should use the total mass of the Earth-Moon system (M + m) instead of just the mass of the Earth. This is because at the balance point, the gravitational forces from both objects are balanced.

Therefore, the correct calculation for the change in potential energy is:

change in PE = (-GMm/377,622 km) + (G(M+m)/341,333 km)

Substituting in the given values, we get:

change in PE = (-6.67 x 10^-11 Nm^2/kg^2)(7.37 x 10^22 kg)(5.97 x 10^24 kg)/377,622 km + (6.67 x 10^-11 Nm^2/kg^2)(5.97 x 10^24 kg)(7.37 x 10^22 kg)/341,333 km

= -1.81 x 10^19 J + 1.81 x 10^19 J

= 0 J

This means that there is no change in potential energy between the surface of the Moon and the balance point. This makes sense because at the balance point, the gravitational forces from the Earth and Moon are balanced, so there is no net change in potential energy.

 

[Moetasim]

I would first like to clarify some assumptions and definitions in the problem. The term "thrown" implies that the rocks are launched with a certain initial velocity, and the problem does not mention anything about the direction or speed of this launch. Additionally, the term "gravitational field strength" is typically used to refer to the magnitude of the gravitational force per unit mass, not the location where it is zero. I will assume that the problem is referring to the location where the net gravitational force on the rocks is zero, which is also known as the "Lagrange point" or "balance point."

Now, let's address the first part of the problem. To calculate the location of the balance point, we can use the concept of gravitational potential energy. At the balance point, the gravitational potential energy of the rocks is equal to the sum of the gravitational potential energies of the Earth and the Moon. We can express this mathematically as:

PE(moon) + PE(earth) = PE(rocks)

Where PE(moon) and PE(earth) are the gravitational potential energies of the Moon and Earth, respectively, and PE(rocks) is the gravitational potential energy of the rocks at the balance point.

We can calculate the gravitational potential energy using the formula PE = -GMm/r, where G is the gravitational constant, M is the mass of the object exerting the gravitational force, m is the mass of the object experiencing the force, and r is the distance between the two objects.

Substituting in the appropriate values for the Moon, Earth, and rocks, we get:

(-GM(moon)m/radius of moon) + (-GM(earth)m/distance from Earth to surface of moon) = (-GM(moon)m/distance from the moon to the balance point) + (-GM(earth)m/distance from the Earth to the balance point)

We can rearrange this equation to solve for the distance from the Earth to the balance point:

distance from the Earth to the balance point = (GM(moon)/GM(earth)) * (distance from the moon to the balance point - distance from Earth to surface of moon)

Plugging in the given values, we get a distance of 341,333 km. This is the same value that you obtained, so that part is correct.

Now, for part B, we need to calculate the difference in potential energy between the surface of the Moon and the balance point. To do