Question

Consider two bodies with equal masses of \( 10^{12} \) kg each and R distance apart. Let G be the gravitational constant and \( V_0 \) be a constant with dimensions of energy. Which of the following represent(s) gravitational potential energy (V) between the bodies, such that Newton's law of gravitation is valid?

• A. \( V = -\frac{G}{R} 10^{24} \)
• B. \( V = -\frac{G}{R} 10^{24} + 1000 V_0 \)
• C. \( V = \frac{G}{R^2} 10^{24} \)
• D. \( V = 10^{12} GR \)

Hint:

Gravitational potential energy between two masses is negative and inversely proportional to the distance between them.

Answer 1

The Correct Option is A, B

The problem is related to calculating the gravitational potential energy between two bodies using Newton's law of gravitation. The gravitational potential energy \( V \) between two masses \( m_1 \) and \( m_2 \), separated by a distance \( R \), is given by the formula:

\(V = -\frac{G m_1 m_2}{R}\)

where \( G \) is the gravitational constant.

In this question, both masses \( m_1 \) and \( m_2 \) are \( 10^{12} \) kg each. Thus, the gravitational potential energy can be calculated as follows:

Substitute the given values into the formula:

\(V = -\frac{G \times 10^{12} \times 10^{12}}{R}\)

Simplify the expression:

\(V = -\frac{G \times 10^{24}}{R}\)

This matches the expression given in the first option:

\(V = -\frac{G}{R} 10^{24}\)

Next, let's evaluate if other options can also be considered:

The second option is:

\(V = -\frac{G}{R} 10^{24} + 1000 V_0\)

This option includes an additional term \( 1000 V_0 \), which represents an added constant energy \( V_0 \) with dimensions of energy. Such an addition does not affect the validity of the gravitational potential energy formula, as potential energy is only defined up to an arbitrary constant. Therefore, this option is valid as well.

The third option is:

\(V = \frac{G}{R^2} 10^{24}\)

This is incorrect because the potential energy formula involves \( \frac{1}{R} \) rather than \( \frac{1}{R^2} \).

The fourth option is:

\(V = 10^{12} G R\)

This formula neither uses the correct variables nor follows the inverse relationship with the distance, and hence it is incorrect.

In conclusion, the correct representations of gravitational potential energy for the given scenario are:

• \(V = -\frac{G}{R} 10^{24}\)
• \(V = -\frac{G}{R} 10^{24} + 1000 V_0\)

These options are consistent with the principle that potential energy can be defined up to an arbitrary constant.