Give the quantities for which the following are the dimensions:
`M^(-1)L^(3)T^(-2)`

Answer

To find the quantities for which the dimensions are given as \( M^{-1}L^{3}T^{-2} \), we can analyze the dimensions step by step.

### Step 1: Identify the Given Dimensions
The given dimensions are:
\[ M^{-1}L^{3}T^{-2} \]
This means we are looking for a physical quantity that has these dimensions.

### Step 2: Recognize the Context
From the video transcript, we learn that these dimensions correspond to the universal gravitational constant \( G \). The gravitational constant is a key quantity in Newton's law of gravitation.

### Step 3: Recall the Formula for Gravitational Force
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{G m_1 m_2}{r^2} \]
Where:
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses,
- \( r \) is the distance between the centers of the two masses.

### Step 4: Rearranging the Formula
We can rearrange the formula to express \( G \):
\[ G = \frac{F r^2}{m_1 m_2} \]

### Step 5: Determine the Dimensions of Each Component
Now, we need to find the dimensions of \( F \), \( r \), \( m_1 \), and \( m_2 \):
- The dimension of force \( F \) is:
\[ [F] = [M][L][T^{-2}] = MLT^{-2} \]
- The dimension of distance \( r \) is:
\[ [r] = L \]
- The dimension of mass \( m_1 \) and \( m_2 \) is:
\[ [m_1] = [m_2] = M \]

### Step 6: Substitute the Dimensions into the Expression for \( G \)
Now substituting these dimensions into the expression for \( G \):
\[ [G] = \frac{[F][r^2]}{[m_1][m_2]} = \frac{(MLT^{-2})(L^2)}{M \cdot M} = \frac{ML^3T^{-2}}{M^2} = M^{-1}L^{3}T^{-2} \]

### Conclusion
Thus, we conclude that the quantity with dimensions \( M^{-1}L^{3}T^{-2} \) is the **universal gravitational constant \( G \)**.

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