My name is Alexander FufaeV and here I write about:

Torque Simply Explained

Important Formula

Formula: Newton's 2nd Law of Motion for Rotation

$$\class{green}{M} ~=~ \class{brown}{I} \, \class{red}{\alpha}$$ $$\class{green}{M} ~=~ \class{brown}{I} \, \class{red}{\alpha}$$ $$\class{red}{\alpha} ~=~ \frac{ \class{green}{M} }{ \class{brown}{I} }$$ $$\class{brown}{I} ~=~ \frac{ \class{green}{M} }{ \class{red}{\alpha} }$$

Rearrange formula

What do the formula symbols mean?

Torque

$$ \class{green}{M} $$ Unit $$ \mathrm{Nm} = \frac{\mathrm{kg} \, \mathrm{m}^2}{\mathrm{s}^2} $$

Torque describes the rotational effect of a force. It is analogous to the force in linear motion.

Angular acceleration

$$ \class{red}{\alpha} $$ Unit $$ \frac{\mathrm{rad}}{\mathrm{s}^2} $$

Angular acceleration indicates how fast the angular velocity increases per second. The angular acceleration causes the rotation to become faster and faster.

Moment of inertia

$$ \class{brown}{I} $$ Unit $$ \mathrm{kg} \, \mathrm{m}^2 $$

Moment of inertia of a body acts like a resistance to rotation and is analogous to mass in a linear motion.

The torque $ \class{green}{\boldsymbol{M}} $ is defined as follows:

$$ \begin{align} \class{green}{\boldsymbol{M}} ~=~ \boldsymbol{r} ~\times~ \boldsymbol{F} \end{align} $$

The torque is the cross product of lever-arm length $ \boldsymbol{r} $ and force $ \boldsymbol{F} $.

Because of the cross product the torque vector is always perpendicular to $ \boldsymbol{r} $ and $ \boldsymbol{F} $! Its magnitude $ |\boldsymbol{r}\times \boldsymbol{F}| = \class{green}{M}$ is the area spanned by the vectors $ \boldsymbol{r} $ and $ \boldsymbol{F} $.

Cross product of any two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$.

The magnitude of the torque, according to the definition of the cross product, is given by the following formula:

$$ \begin{align} \class{green}{M} ~=~ r \, F \, \sin(\varphi) \end{align} $$

Here $\varphi$ is the angle enclosed by the vectors $ \boldsymbol{r} $ and $\boldsymbol{F}$. The torque becomes maximum when the two vectors are orthogonal to each other. So if the angle is $\varphi = 90^{\circ}$. Formula 2 then simplifies to the product of the force and lever-arm length.

$$ \begin{align} \class{green}{M} ~=~ r \, F \end{align} $$