Question

Write the expression for the moment of force about a given axis.

Answer 1

$\mathcal{MOMENT\:\:OF\:\:FORCE}$

The moment of a force gives us the turning effect of the force about the fixed axis.

It is measured by the product of magnitude of force and perpendicular distance of the line of action of force from the axis of rotation.

Torque ia represented by a Greek letter $\tau$

Expression:

Suppose the line of action of force $\sf{\vec{F}}$ makes an angle $\alpha$ with X-axis. Two rectangular components of F are:

$\sf{F_x=Fcos\alpha}\\\sf{F_y=Fsin\alpha}$

If x, y are the co-ordinates of the point P, where $\sf{\vec{OP}=\vec{r}\:and\:\angle{XOP}=\theta}$,

Then $\sf{x=rcos\theta}$

$\sf{and\:y=rsin\theta}$

Substituting these values in $\sf{\tau_z=\left(xF_y-yF_x\right)}$, we get

$\sf{\tau=\left(rcos\theta\right)Fsin\alpha-\left(rsin\theta\right)Fcos\alpha}$

$\sf{=rF\left[sin\alpha\:cos\theta-cos\alpha\:sin\theta\right]}$

$\sf{\tau=rFsin\left(\alpha-\theta\right)}$

Let $\emptyset$ be the angle which the line of action of $\sf{\vec{F}}$ makes with the position vector $\sf{\vec{OP}=\vec{r}}$

$\therefore\:\theta+\emptyset=\alpha\\\emptyset=\alpha-\theta$

$\boxed{\sf{\tau=rFsin\emptyset}}$

It is the expression for torque in polar coordinates.

Answer 2

Moment of force about a given axis= Force x perpendicular distance of force from the axis of rotation.