Question

The gravitational potential energy of a body of mass m on the surface of earth of mass M and radius R is ________.

• A. $\frac{-GMm}{R}$
• B. $\frac{GMm}{R}$
• C. $mgR$
• D. $-mgR$
• E. Zero

Hint:

The negative sign indicates that the gravitational force is attractive and work must be done to move the mass to infinity.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Gravitational potential energy (U) of a system of two masses is defined as the work done by an external agent in bringing the masses from infinity to their current separation distance without any acceleration. By convention, the potential energy is taken to be zero when the separation between the masses is infinite.
Step 2: Key Formula or Approach
The formula for the gravitational potential energy U between two point masses M and m separated by a distance r is:
\[ U(r) = -\frac{GMm}{r} \] The negative sign indicates that the gravitational force is attractive. It also signifies that the potential energy at a finite separation is less than the potential energy at infinite separation (which is zero).
Step 3: Detailed Explanation
1. Apply the general formula to the specific case.
We are asked for the potential energy of a body of mass m \textit{on the surface} of the Earth.
- The mass of the Earth is M.
- The mass of the body is m.
- The distance between their centers when the body is on the surface is the radius of the Earth, R.
So, we substitute \(r=R\) into the general formula.
\[ U = -\frac{GMm}{R} \] This matches option (A).
2. Analyze other options.
- (B) \(\frac{GMm}{R}\): This is positive, which would imply a repulsive force. Incorrect.
- (C) mgR and (D) -mgR: These are related to the potential energy \textit{change} near the Earth's surface, using the approximation \(g = \frac{GM}{R^2}\). The potential energy is \(U \approx mgh\). If we set the surface as \(h=0\), the energy there is zero in this approximate model. However, the question asks for the absolute potential energy, where zero is at infinity. So, these are incorrect.
- (E) Zero: This is the potential energy at infinite separation, not on the surface. Incorrect.
Conclusion:
The correct definition of gravitational potential energy for a mass m on the surface of Earth is \(-\frac{GMm}{R}\). Option (A) is the correct answer. The question was likely cancelled due to an administrative error.
Step 4: Final Answer
The gravitational potential energy is \(-\frac{GMm}{R}\).