`(x^2)/("mass")` has dimensions of kinetic energy. Then x has the dimensions of

Options

A. Pressure
B. Torque
C. Moment of Inertia
D. Impulse

Answer

To solve the problem, we need to determine the dimensions of \( x \) given that \( \frac{x^2}{\text{mass}} \) has the dimensions of kinetic energy.

### Step-by-Step Solution:

1. **Understand the dimensions of kinetic energy**:
Kinetic energy (KE) is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is mass and \( v \) is velocity. The dimensions of kinetic energy can be derived as follows:
- The dimension of mass \( [m] \) is \( M \).
- The dimension of velocity \( [v] \) is \( [L][T]^{-1} \) (length per time).
- Therefore, the dimension of \( v^2 \) is \( [L^2][T^{-2}] \).

Thus, the dimension of kinetic energy is:
\[ [KE] = [M][L^2][T^{-2}] \]

2. **Set up the equation**:
According to the problem, we have:
\[ \frac{x^2}{\text{mass}} \text{ has dimensions of kinetic energy} \]
This can be expressed as:
\[ \frac{x^2}{[M]} = [M][L^2][T^{-2}] \]

3. **Rearranging the equation**:
To find the dimensions of \( x^2 \), we multiply both sides by \( [M] \):
\[ x^2 = [M][M][L^2][T^{-2}] = [M^2][L^2][T^{-2}] \]

4. **Finding the dimensions of \( x \)**:
To find the dimensions of \( x \), we take the square root of both sides:
\[ x = \sqrt{[M^2][L^2][T^{-2}]} = [M][L][T^{-1}] \]

5. **Interpret the result**:
The dimensions of \( x \) can be interpreted as:
\[ x \text{ has dimensions of momentum, which is } [M][L][T^{-1}] \]

### Conclusion:
Thus, \( x \) has the dimensions of momentum.