If the gravitational potential energy of two point masses infintely away is taken to be zero then gravitational potential energy of a galaxy is

Options

A. Zero
B. Positive
C. Negative
D. Can have any value

Answer

To determine the gravitational potential energy of a galaxy, given that the gravitational potential energy of two point masses at infinity is taken to be zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Gravitational Potential Energy**:
The gravitational potential energy (U) between two point masses (m1 and m2) separated by a distance (r) is given by the formula:
\[ U = -\frac{G \cdot m_1 \cdot m_2}{r} \]
where \( G \) is the gravitational constant.

2. **Reference Point**:
We take the reference point where the gravitational potential energy is zero when the two masses are infinitely far apart (i.e., when \( r \to \infty \)). Thus, at infinity:
\[ U(\infty) = 0 \]

3. **Bringing Masses Together**:
As we bring the two masses closer together from infinity to a distance \( r \), work is done against the gravitational force. The work done (W) in bringing the masses from infinity to a distance \( r \) is equal to the change in potential energy.

4. **Using Conservation of Energy**:
According to the conservation of energy, the initial potential energy (at infinity) plus the work done will equal the final potential energy:
\[ 0 + W = U(r) \]
Rearranging gives:
\[ U(r) = -W \]

5. **Calculating Work Done**:
The work done to bring the two masses from infinity to a distance \( r \) is positive, which means that the gravitational potential energy will be negative:
\[ U(r) = -\frac{G \cdot m_1 \cdot m_2}{r} \]
This shows that the gravitational potential energy is negative when the masses are at a finite distance \( r \).

6. **Conclusion for a Galaxy**:
A galaxy consists of numerous stars and masses that are gravitationally bound together. Therefore, the gravitational potential energy of a galaxy, which is a system of many masses, will also be negative, as it is analogous to the potential energy of two point masses brought together.

### Final Answer:
The gravitational potential energy of a galaxy is negative.

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