Class 11 > Unit # 04: Rotational & Circular Motion > Torque

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Unit No. 4 Rotational and Circular Motion - Torque and Rotational Dynamics

Define Torque and derive a relation between torque, angular acceleration and moment of inertia.

Torque

"Torque is the turning effect of a force applied to an object about an axis of rotation. Also known as the moment of force, it is responsible for producing angular acceleration in a rotating body."

Formula and Definition

Torque is defined mathematically as the cross product of the position vector (moment arm) $\vec{r}$ and the applied force $\vec{F}$:

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

In scalar form, magnitude is given by $\tau = r F \sin\theta$.

Direction, Unit, and Dimension

• Direction: Determined by the right-hand rule. Counter-clockwise torque is taken as positive (directed outward from the paper), while clockwise torque is negative (directed inward).
• SI Unit: Newton-meter ($\text{N}\cdot\text{m}$) or $\text{kg}\cdot\text{m}^2/\text{s}^2$
• Dimensional Formula: $[\text{M}][\text{L}]^2[\text{T}]^{-2}$

Derivation of Relation Between Torque, Moment of Inertia, and Angular Acceleration

Consider a single particle of mass $m$ rotating in a circle of radius $r$ at the end of a light string. Suppose a tangential force $F$ acts on the mass, producing linear acceleration $a$ along the circular arc.

According to Newton's Second Law of Motion:

$F = m a$

Multiplying both sides by the radius $r$ (the moment arm):

$r \cdot F = r \cdot (m a)$

Since torque is defined as $\tau = rF$, and linear acceleration is related to angular acceleration $\alpha$ by $a = r\alpha$:

$\tau = m \cdot (r\alpha) \cdot r = m r^2 \alpha$

For a rigid body consisting of multiple particles located at various distances $r_i$ from the axis of rotation, we sum the torque contributions from all individual particles:

$\tau_{\text{net}} = \sum \tau_i = \sum (m_i r_i^2) \alpha$

Since the moment of inertia for a system of particles is defined as $I = \sum m_i r_i^2$, substituting $I$ into the equation yields the rotational equivalent of Newton's second law:

$\tau = I \alpha \quad \text{(Proved)}$

Conclusion:
This fundamental equation demonstrates that the net torque acting on a rigid body is directly proportional to its moment of inertia and the resulting angular acceleration.

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