The dimensions of torque are

The correct answer is Option 2: [ML²T⁻²].

Step-by-step derivation:
1. Definition of Torque: Torque ($\tau$) is defined as the cross product of the position vector ($r$) and the force vector ($F$). The magnitude of torque can be written as:
$\tau =r·F·sin\left(\theta \right)$
Since $sin\left(\theta \right)$ is a dimensionless quantity, the dimensions of torque are determined by the product of the dimensions of distance and force.

2. Dimensions of Distance ($r$):
Distance is a fundamental quantity of length:
$\left[r\right]=\left[L\right]$

3. Dimensions of Force ($F$):
According to Newton's second law, force is mass ($m$) times acceleration ($a$):
$F=m·a$
- The dimension of mass is $\left[M\right]$.
- Acceleration is the rate of change of velocity, which has the dimension of length divided by time squared: $\left[a\right]=\left[L{T}^{-2}\right]$.
Therefore, the dimensional formula for force is:
$\left[F\right]=\left[ML{T}^{-2}\right]$

4. Calculating the Dimensions of Torque ($\tau$):
Now, substitute the dimensions of distance and force into the formula for torque:
$\left[\tau \right]=\left[r\right]·\left[F\right]$
$\left[\tau \right]=\left[L\right]·\left[ML{T}^{-2}\right]$
Combine the terms of the same base (Length):
$\left[\tau \right]=\left[M{L}^2{T}^{-2}\right]$

Thus, the dimensions of torque are [ML²T⁻²].