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 find the gravitational potential energy of a galaxy, we need to consider the gravitational potential energy between multiple point masses that make up the galaxy. Here’s a step-by-step solution:

### Step 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:
\[ U = -\frac{G m_1 m_2}{r} \]
where G is the gravitational constant.

### Step 2: Considering Multiple Masses
In a galaxy, there are many point masses (stars, planets, etc.). If we have n point masses, the total gravitational potential energy of the system can be calculated by summing the potential energies of all pairs of masses.

### Step 3: Total Gravitational Potential Energy
The total gravitational potential energy (U_total) of n point masses can be expressed as:
\[ U_{\text{total}} = \sum_{i=1}^{n} \sum_{j=i+1}^{n} -\frac{G m_i m_j}{r_{ij}} \]
where \( r_{ij} \) is the distance between the masses \( m_i \) and \( m_j \).

### Step 4: Considering the Condition at Infinity
According to the problem, the gravitational potential energy of two point masses at infinity is taken to be zero. This means that when the masses are infinitely far apart, their potential energy does not contribute to the total energy of the galaxy.

### Step 5: Conclusion
Thus, the gravitational potential energy of a galaxy, which consists of many point masses, can be calculated using the above formula. The potential energy will be negative, indicating that work must be done to separate the masses to infinity.

### Final Expression
The final expression for the gravitational potential energy of a galaxy with n point masses is:
\[ U_{\text{total}} = -\sum_{i=1}^{n} \sum_{j=i+1}^{n} \frac{G m_i m_j}{r_{ij}} \]