Co-planar Force, Moment of a Force, Clockwise and Anticlockwise Moments and Torque

Co-planar Force, Moment of a Force, Clockwise and Anticlockwise Moments, and Torque

Force, being a vector quantity, can be added up and subtracted according to vector laws.When a number of forces act on a body then the body can be at rest or equilibrium. For the body to be in translational equilibrium, the resultant of all the forces must be zero.

$$ \sum F = 0 $$

A rigid body can also be in rotational equilibrium. The turning effect of force is called torque. For rotational equilibrium, the sum of all the torques acting on the body must be zero,

$$\sum \tau = 0 $$

Coplaner Forces

Forces acting in a single plane or in a same plane called co-planer forces. If only two forces act through a point, they must be co-planar. Three or more non-parallel forces acting through a point may not be co-planer.

Moment of a Force

One of the effects of the force on a body is to produce rotational motion. The rotational effect of the force depends on it’s magnitude and on the perpendicular distance of line of action of force from the axis of rotation. The perpendicular distance of line of force from axis of rotation is called moment arm as shown in the figure. The product of the force and the moment arm is called moment of force or torque.

$$\begin{align*} \tau &= rF \\ \text {Torque is a vector and in vector form,} \\ \vec \tau &= \vec r \times \vec F \end{align*}$$

The direction of torque is determined by the right hand rule. The unit of torque is Nm.

Clockwise and Anticlockwise Moments

As shown in the figure is the point of rotation. Forces F₁ and F₂ are at perpendicular distance of r₁ and r₂ from O. The effect of the force F₁ is called anticlockwise moment and due to F₂ is called clockwise moment.

Principle of Moment

It states that when a body in equilibrium under the action of a number of forces, then the algebraic sum of moments of all these forces about any point is zero. That is, sum of clockwise moment must be equal to anticlockwise moment.

Let us consider a metre scale is in equilibrium under the action of forces F₁ , F₂ , F₃ , F₄ and F₅ at points P,Q,R, S and T respectively as shown in the figure. The fulcrum supports metre scale at point O. The forces F₁ and F₂ produce anticlockwise while the forces F₃, F₄ and F₅ produce clockwise moments. Then,

$$\begin{align*} \text {Sum of anticlockwise moments about O} &= F_1 \times OP + F_2 \times OQ \\ \text {and sum of clockwise moments about O} &= F_3 \times OR + F_4 \times OS + F_5 \times OT \\ \text {According to the principle of moment} \\ \text {sum of clockwise moments} &= \text {sum of anticlockwise moment} \\ \text {or,} F_1 \times OP + F_2 \times &= F_3 \times OR + F_4 \times OS + F_5 \times OT \\ \end{align*}$$

Parallel Forces

When a number of forces act on a body and their directions are parallel, they are called parallel forces. Parallel forces are of two types i.e. like parallel forces and unlike parallel forces. Those parallel forces which act in same direction are called like parallel forces. Those parallel forces which act in different direction are called unlike parallel forces.

Torque due to Couple

Two equal and opposite force acting on a body such that their lines of action do not coincide, a constitute couple as shown in figure.

The action of a couple is to produce rotational motion of a body. As shown in the figure, both forces produce anticlockwise rotation on the rod AB about O. So, the moment of a couple is the sum of the moment of the two forces.

$$\begin{align*} \text {Torque about O} &= \text {Moment of force F at A} + \text {Moment of force F at B} \\ &= F \times AO + F \times OB \\ &= F (AO + OB) \\ &= F \times AB \\ \tau &= F \times AB \end{align*}$$

Some examples of couple are turning a tap, turning a cork screw, tuning a screw driver and a compass needle in the earth’s magnetic field.