Short Answer 
The distance from the Earth where gravitational forces from the Earth and Moon balance is 3.456 x 10^8 m, and the gravitational acceleration at this point is 3.33 x 10^-3 m/s². These results provide insight into the gravitational interactions within the Earth-Moon system.
Step 1: Determine the Distance of Point rM
To find the point where the gravitational forces from the Earth and Moon balance, we need to establish the distance between them, denoted as rM. The relationship can be defined using the formula:
- rM = (sqrt(mE/mM) * r) / (1 + sqrt(mE/mM))
With known masses for Earth (mE = 5.98 x 1024 kg) and Moon (mM = 7.36 x 1022 kg), and the distance r = 3.84 x 108 m, we can substitute these values to calculate:
- rM = (sqrt(5.98 x 1024/7.36 x 1022) * 3.84 x 108) / (1 + sqrt(5.98 x 1024/7.36 x 1022))
This results in rM = 3.456 x 108 m.
Step 2: Calculate Resultant Gravitational Acceleration
The next step involves determining the gravitational acceleration a at the point rM. Utilize the gravitational force formula:
- F = G * mA * mE / (rM2)
- Where G is the gravitational constant (G = 6.67 x 10-11 Nm2/kg2).
The gravitational acceleration is then expressed as:
- a = G * mE / (rM2)
Plug in the values:
- a = (6.67 x 10-11 Nm2/kg2 * 5.98 x 1024 kg) / (3.456 x 108 m)2
This simplifies to a = 3.33 x 10-3 m/s².
Step 3: Summary of Results
Upon completing the calculations, we have obtained two crucial results based on the gravitational interactions between the Earth and the Moon:
- The distance from the Earth where gravitational forces balance is rM = 3.456 x 108 m.
- The gravitational acceleration at this point is a = 3.33 x 10-3 m/s².
These calculations help in understanding the gravitational dynamics in the Earth-Moon system.