Torque

Torque, denoted by $\tau$, is the equivalent of force in the world of rotational dynamics. Mathematically, it is defined by $$\begin {align*} \tau &= \vec{\mathbf{r}} \times \vec{\mathbf{F}} \\&= r \cdot F \cdot \sin \theta, \end {align*}$$ where $r = \left| \vec{\mathbf{r}} \right|$, $F = \left| \vec{\mathbf{F}} \right|$, and $\theta$ between the force vector $\vec{\mathbf{F}}$ and the level arm vector, $\vec{\mathbf{r}}$.

Indeed, these look like the units for energy. Energy and torque are completely distinct, so in modern-day physics, energy is mentioned with the derived units of Joules whilst torque is referred to with the units $\text{m}\cdot\text{N}$ or, often times, $\text{N}\cdot\text{m}$.

Torque has several interesting properties which show up with relation to other concepts in angular dynamics and will show up a lot as we encounter concepts such as moment of inertia, angular momentum, and others.

Fun fact. Once you learn about angular momentum, denoted by $L$, you will learn that $\tau = \frac {\mathrm{d}L}{\mathrm{d}t}$.

Torque causes an object to rotate about an axis when a force is applied to it.

The most basic formula to represent the torque on an object is given by:

$$\tau = F_{perp} d$$

In this equation, $F_{perp}$ is the magnitude of the force component that's perpendicular to the lever arm (the vector from the axis of rotation to the point where force is applied) and $d$ is the distance from the axis of rotation.