2.6 Torque (Moment of Force)

In mechanics, while force causes an object to accelerate linearly, torque (or moment of force) is what causes an object to acquire angular acceleration. It is the rotational equivalent of a linear force. Torque is a vector quantity that measures the tendency of a force to rotate an object around an axis or pivot point.

Scalar And Vector Quantities→

Definition of Torque

Torque is the measure of the effectiveness of a force in causing or changing the rotation of an object. It depends on three factors:

1. The magnitude of the applied force.
2. The distance from the axis of rotation to the point where the force is applied (the lever arm).
3. The angle at which the force is applied relative to the lever arm.

Formula for the Magnitude of Torque

The magnitude of torque ($\tau$) is given by:

$\tau =rF\sin \theta$

Where:

• $\tau$ = magnitude of torque (N·m)
• $r$ = distance from the axis of rotation to the point of force application
• $F$ = magnitude of the applied force
• $\theta$ = angle between the position vector $\vec{r}$ and the force vector $\vec{F}$

SI Unit: Newton-metre (N·m)
Dimensions: $[M{L}^2{T}^{-2}]$ — same as work/energy, though torque and work are physically distinct quantities.

Vector Nature of Torque (Cross Product)

Torque is formally defined by the cross product of the position vector $\vec{r}$ and the force vector $\vec{F}$:

$\vec{\tau }=\vec{r}\times \vec{F}$

The direction of the torque vector indicates the axis of rotation and the sense of rotation (clockwise or counter-clockwise).

The Right-Hand Rule

The direction of $\vec{\tau }$ is determined by the right-hand rule:

1. Point the fingers of your right hand along $\vec{r}$ (from pivot to point of force application).
2. Curl your fingers toward $\vec{F}$.
3. Your thumb points in the direction of $\vec{\tau }$.

Sign Convention (2D):

• Counter-clockwise (CCW) rotation → torque is positive (vector points out of the page)
• Clockwise (CW) rotation → torque is negative (vector points into the page)

Characteristics of Torque

1. Dependence on Angle

From $\tau =rF\sin \theta$:

• Maximum torque: when $\theta =90^{\circ }$ (force perpendicular to lever arm), since $\sin 90^{\circ }=1$
• Zero torque: when $\theta =0^{\circ }$ or $180^{\circ }$ (force parallel or anti-parallel to lever arm), since $\sin 0^{\circ }=\sin 180^{\circ }=0$

2. The Lever Arm (Moment Arm)

The lever arm (or moment arm) is the perpendicular distance from the axis of rotation to the line of action of the force, equal to $r\sin \theta$. A longer lever arm produces a larger torque for the same applied force.

Torque in Rotational Dynamics

Rotational Equilibrium

For an object to be in rotational equilibrium, the net torque about any axis must be zero:

$\sum \vec{\tau }=0$

This means the sum of clockwise torques equals the sum of counter-clockwise torques, resulting in zero angular acceleration.

Equilibrium→

Newton's Second Law for Rotation: $\tau =I\alpha$

Just as a net force causes linear acceleration ($F=ma$), a net torque causes angular acceleration ($\alpha$):

${\tau }_{net}=I\alpha$

Where:

• $I$ = moment of inertia (rotational analogue of mass, units: kg·m²)
• $\alpha$ = angular acceleration (rad/s²)

This is the rotational analogue of Newton's second law. A larger torque or smaller moment of inertia produces greater angular acceleration.