Torque and Equilibrium

Torque: The Turning Effect

Torque (τ), or the moment of a force, is the measure of the force's tendency to produce rotation about a pivot point.

$$\tau = r F \sin(\theta)$$

Torque is determined by force magnitude, distance from pivot, and the angle of application.

$\tau$=Torque magnitude(Nm)

$r$=Position vector (distance from pivot)(m)

$F$=Applied force(N)

$\theta$=Angle between force and position vector(degrees)

$\theta = 90^\circ$

→

$\tau = rF$ (Maximum torque)

$\theta = 0^\circ$

→

$\tau = 0$ (Force passes through pivot)

Moment Arm: The perpendicular distance ($l = r \sin\theta$) from the pivot to the line of action of the force.

Vector Form: Torque as a cross product: $\vec{\tau} = \vec{r} \times \vec{F}$. Direction given by the right-hand rule, perpendicular to the plane of $\vec{r}$ and $\vec{F}$.

Proportionality: $\tau \propto r$ and $\tau \propto \sin\theta$ — doubling the lever arm doubles the torque; maximum torque occurs at $\theta = 90°$.

Dimensional Check: $[\tau] = [r][F][\sin\theta] = \text{m} \times \text{N} = \text{Nm}$. Torque has the same dimensions as energy (J), but they are physically distinct — torque is never expressed in joules.

Two Equivalent Views: Either decompose $\vec{F}$ into $F\sin\theta$ perpendicular to $\vec{r}$, or decompose $\vec{r}$ into $r\sin\theta$ perpendicular to $\vec{F}$. Both give $\tau = rF\sin\theta$.

Conditions of Equilibrium

Total static equilibrium requires that a body has neither linear nor angular acceleration, satisfying two distinct mathematical conditions.

$$\sum \vec{F} = 0, \quad \sum \vec{\tau} = 0$$

The sum of all external forces and all external torques must be zero.

$\sum \vec{F}$=Net external force(N)

$\sum \vec{\tau}$=Net external torque(Nm)

$\sum F = 0$ only

→

Translational equilibrium; object may still rotate.

$\sum \tau = 0$ only

→

Rotational equilibrium; object may still translate.

Principle of Moments: For a body in rotational equilibrium, the sum of clockwise torques equals the sum of counter-clockwise torques.

Axis Independence: If an object is in equilibrium, the net torque is zero about any arbitrary axis chosen for calculation.

Static vs Dynamic: A body at rest is in static equilibrium; a body moving with constant velocity is in dynamic equilibrium. Both satisfy the same conditions.

Choosing a smart pivot point simplifies torque equations by eliminating unknown forces whose lines of action pass through that pivot.

Strategy: Place the pivot where the most unknown forces act — their torques become zero, reducing the number of unknowns.

Coplanar Forces: For 2D problems, resolve forces into x and y components. Apply $\sum F_x = 0$, $\sum F_y = 0$, and $\sum \tau = 0$.

Sign Convention: Counter-clockwise torques are positive, clockwise torques are negative. Choose one convention and stick to it.

Negative Reactions: If solving yields a negative force, the assumed direction was wrong — the force actually acts opposite to your assumption. Don't discard it; flip the direction.

Center of Gravity and Stability

The Center of Gravity (CG) is the point where the entire weight of a body appears to act, determining its stability limit.

$$x_{cg} = \frac{\sum w_i x_i}{\sum w_i}$$

The CG position is the weighted average of the positions of all constituent masses.

$x_{cg}$=Position of CG(m)

$w_i$=Weight of individual component $i$(N)

$x_i$=Position of component $i$(m)

Stable Equilibrium: A small displacement raises the CG, creating a restoring torque that returns the object to its original position.

Unstable Equilibrium: A small displacement lowers the CG, and the object topples further away from its original position (e.g., a ball on top of a hill).

Neutral Equilibrium: Displacement neither raises nor lowers the CG, so the object stays in its new position (e.g., a ball on a flat surface).

Uniform Bodies: For a uniform object, the CG is at the geometric center (e.g., midpoint of a uniform beam, center of a uniform disc).

Base of Support: An object remains stable as long as the vertical line from its CG falls within its base.

Couples and Pure Rotation

A couple consists of two equal, opposite, and parallel forces whose lines of action do not coincide, producing pure rotation.

$$\tau_{couple} = F \times d$$

The torque of a couple is the product of one force and the perpendicular distance between them.

$F$=Magnitude of one of the forces(N)

$d$=Arm of the couple (perpendicular distance)(m)

Zero Net Force: A couple always results in $\sum F = 0$, meaning it cannot produce translational motion.

Constant Torque: Unlike a single force, the torque of a couple is the same about any point in the plane.

Proportionality: $\tau_{couple} \propto d$ — widening the separation between forces linearly increases the torque while keeping net force zero.