## Torque or Moment of Force

**Definition**

If a body is capable of rotating about an axis, then force applied properly on this body will rotate it about the axis (axis of rotating). This turning effect of the force about the axis of rotation is called torque.

Torque is the physical quantity which produces angular acceleration in the body.

**Explanation**

Consider a body which can rotate about O (axis of rotation). A force F acts on point P whose position vector w.r.t O is r.

Diagram Coming Soon

F is resolved into F1 and F2. θ is the angle between F and extended line of r.

The component of F which produces rotation in the body is F1.

The magnitude of torgue (π) is the product of the magnitudes of r and F1.

Equation (1) shows that torque is the cross-product of displacement r and force F.

Torque → positive if directed outward from paper

Torque → negative if directed inward from paper

The direction of torque can be found by using Right Hand Rule and is always perpendicular to the plane containing r & F.

Thus

Clockwise torque → negative

Counter-Clockwise torque → positive

**Alternate Definition of Torque**

π = r x F

|π| = r F sin θ

|π| = F x r sin θ

But r sin θ = L (momentum arm) (from figure)

Therefore,

|π| = F L

Magnitude of Torque = Magnitude of force x Moment Arm

*Note*

If line of action of force passes through the axis of rotation then this force cannot produce torque.

The unit of torque is N.m.

**Couple**

Two forces are said to constitute a couple if they have

1. Same magnitudes

2. Opposite directions

3. Different lines of action

These forces cannot produces transiatory motion, but produce rotatory motion.

**Moment (Torque) of a Couple**

Consider a couple composed of two forces F and -F acting at points A and B (on a body) respectively, having position vectors r1 & r2.

If π1 is the torque due to force F, then

π1 = r1 x F

Similarly if π2 is the torque due to force – F, then

π2 = r2 x (-F)

The total torque due to the two forces is

π = π1 + π2

π = r1 x F + r2 x (-F)

π = r1 x F – r2 x (-F)

π = (r1 – r2) x F

π = r x F

where r is the displacement vector from B to A.

The magnitude of torque is

π = r F sin (180 – θ)

π = r F sin θ ……………….. {since sin (180 – θ) = sin θ}

Where θ is the angle between r and -F.

π = F (r sin θ)

But r sin θ is the perpendicular distance between the lines of action of forces F and -F is called *moment arm* of the couple denoted by d.

π = Fd

Thus

[Mag. of the moment of a couple] = [Mag. of any of the forces forming the couple] x [Moment arm of the couple]

Moment (torque) of a given couple is independent of the location of origin. right here

Tangential and Centripetal Acceleration Next Post:

Centre of Mass