1.1 Torque or moment of a force

  Rotational Dynamics

The torque or moment of force is a measure of the turning effect of force about the axis of rotation. It is denoted by the symbol $\overrightarrow \tau $

Mathematically, $$\overrightarrow \tau = \overrightarrow r \times \overrightarrow F $$ or $$\tau = rF\sin \theta $$
where $\theta $ is the angle between vectors $\overrightarrow r $ and $\overrightarrow F $

In other words , torque is a vector cross product of the radius vector and force.

So, from the property of vector cross cross product, $$\tau \bot r\quad \& \quad \tau \bot F$$
Consider s horizontal force $\overrightarrow F $ is applied on a rigid body at point $P$ which is at a distance $r$ from the hinge $O$ as shown in the figure.

Method 1 for calculating toque

$$\begin{equation} \begin{aligned} \overrightarrow \tau = \overrightarrow r \times \overrightarrow F \\ \tau = rF\sin (90^\circ - \theta ) \\ \tau = rF\cos \theta \\\end{aligned} \end{equation} $$

The torque is in the anti clock wise direction as given from thumb rule.

Method 2 for calculating torque

Position vector of a particle $P$ is, $$\begin{equation} \begin{aligned} \overrightarrow r = r\cos \theta \left( { - \widehat j} \right) + r\sin \theta \left( {\widehat i} \right) \\ \overrightarrow r = r\sin \theta \widehat i - r\cos \theta \widehat j \\\end{aligned} \end{equation} $$

Force acting on point $P$ is, $$F\widehat i$$

So, torque is given by, $$\begin{equation} \begin{aligned} \overrightarrow \tau = \overrightarrow r \times \overrightarrow F \\ \overrightarrow \tau = r\left( {\sin \theta \widehat i - \cos \theta \widehat j} \right) \times F\widehat i \\\end{aligned} \end{equation} $$

\[\overrightarrow \tau = \left| {\begin{array}{c} {\widehat i}&{\widehat j}&{\widehat k} \\ {r\sin \theta }&{ - r\cos \theta }&0 \\ F&0&0 \end{array}} \right|\]

$$\begin{equation} \begin{aligned} \overrightarrow \tau = \widehat i(0 - 0) - \widehat j(0 - 0) + \widehat k(0 - ( - Fr\cos )) \\ \overrightarrow \tau = rF\cos \theta \widehat k \\\end{aligned} \end{equation} $$

Note:

• Torque is a vector quantity
• SI unit of a torque is $N-m$
• Dimensional formula of torque is $\left[ {{M^1}{L^2}{T^{ - 2}}} \right]$
• Direction of torque is determined by the right hand thumb rule
• By convention, torque in the anti clock wise direction is taken as positive and torque in the clockwise direction is taken as negative [Digram of thumb stating the direction as positive or negative]