If the units M and L are increased three times, then the units of energy will be increased by

Options

A. 3 times
B. 6 times
C. 27 times
D. 81 times

Explanation

To solve the problem of how the units of energy change when the units of mass (M) and length (L) are increased three times, we will use the dimensional formula for energy and apply the changes accordingly.

### Step-by-Step Solution:

1. **Identify the Dimensional Formula of Energy**:
The dimensional formula for energy (E) is given by:
\[ [E] = [M][L^2][T^{-2}] \]
This means energy is dependent on mass (M), length (L), and time (T).

2. **Increase the Units of Mass and Length**:
According to the problem, both the mass and length units are increased three times. Therefore, we can express the new units as:
\[ M' = 3M \quad \text{and} \quad L' = 3L \]
The time unit remains unchanged, so:
\[ T' = T \]

3. **Substitute the New Units into the Dimensional Formula**:
Now, substituting the new units into the dimensional formula for energy:
\[ [E'] = [M'][L'^2][T^{-2}] = (3M)(3L^2)(T^{-2}) \]

4. **Calculate the New Energy Expression**:
Simplifying the expression:
\[ [E'] = 3M \cdot 3L^2 \cdot T^{-2} = 9 \cdot [M][L^2][T^{-2}] = 9E \]
This shows that the energy increases by a factor of 9 due to the increase in mass and length.

5. **Final Calculation**:
Since the problem states that both M and L are increased three times, we need to consider the square of the length increase:
\[ [E'] = 3 \cdot 3 \cdot E = 27E \]

### Conclusion:
Thus, when the units of mass and length are increased three times, the units of energy will increase by a factor of **27**.