Torque in rotational motion

Force and Torque

We know about the definition of force in linear motion as an external agent to change the state of rest or of uniform motion in a straight line according to Newton’s first law of motion. Similarly, the rotating force is required to change the state of rest or of uniform motion in a circular motion and that rotating force is called torque.

Torque is the moment of force. In other words, The rotating effect of force can be said as a torque. for e.g: applying force to open or close the door.

Torque and expression of torque

Torque in rotational motion can be defined as the vector product of the perpendicular distance of a point of force from the axis of rotation and the force.

In Vector form: \boxed{\overrightarrow{\tau} = \overrightarrow{r} \times \overrightarrow{F}}

    \[ \tau = rF \sin\theta \hat{n} \]

where \theta is the angle between \overrightarrow{r} and \overrightarrow{F} and \hat{n} is the unit vector along the direction of \tau which gives the direction of \tau.

The direction of Torque(\overrightarrow{\tau}) is towards the perpendicular direction to both of \overrightarrow{r} and \overrightarrow{F}

The magnitude of \overrightarrow{\tau} is:

    \[ |\overrightarrow{\tau}| \quad \text{or} \quad \tau = rF\sin\theta\]


Some Special Cases:

1) If \theta = 0^0 or 180^0 then \sin0^0 = \sin 180^0 = 0. So,

    \[ \boxed{\tau = 0} \]

2) If \theta = 90^0 then \sin 90^0 = 1. So,

    \[  \boxed{\tau = rF} \]

So, we can define torque as the product of perpendicular force and distance from the axis of rotation.

Couple and torque due to couple

If two equal forces act on a rigid body but in opposite directions such that the body undergoes rotational motion, these forces form a couple.

Since a couple forms a rotational motion, the moment of a couple (or Torque due to a couple) can be calculated by the product of either force with the perpendicular distance between them.

The torque due to couple = Force x perpendicular distance between them.


Suppose a wheel of radius r is rotating about a center ‘O’ by the application of couple as shown in figure.

Torque due to couple
fig: Torque due to couple

Torque at point A: \tau_A = F r

Torque at point B: \tau_B = Fr

The total torque on a body is: \tau = \tau_A + \tau_B

    \[ \tau = Fr + Fr \]

    \[  \tau = 2 Fr\]

    \[ \tau = F (2r)\]

Hence, Torque due to force = Force x perpendicular distance between two forces.

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