## Equilibrium

A rigid body is considered to be in **equilibrium**, if there is:

- Translational equilibrium; AND
- Rotational Equilibrium.

A body in equilibrium experiences NO acceleration and will remain in equilibrium indefinitely, UNLESS it is disturbed by an external force. (Newton’s First Law)

### Translational Equilibrium

**Translational equilibrium** is obtained when the **resultant force** or vector sum of all forces acting upon the rigid body is **zero**. In simpler terms, this means that:

- forces pointing to left = forces pointing to right; AND
- forces pointing up = forces pointing down

### Rotational Equilibrium (Principle Of Moments)

**Rotational equilibrium** is obtained when the algebraic sum of the torques is **zero**.

However, it can also be interpreted as:

The principle of moments states that for a body to be in rotational equilibrium, the sum of clockwise torques about any point (which acts as a pivot) must equal to the sum of anti-clockwise torques about the same point.

Sum of clockwise torques = sum of anti-clockwise torques

**Recap:** Torque = moment of force

## Summary

In summary, the conditions for object to be in equilibrium:

- The sum of moments about any point is zero. (Rotational equilibrium – Principle of moments)
- The vector sum of forces on object is zero. (Translational equilibrium)

## Self-Test Questions

### Consider the diagram above, whereby a plank is supported by a pivot. Two forces, 120 N and F are acting on the plank. Given that the plank is perfectly balanced (has rotational equilibrium), find F.

**Show/hide answer**

Recall that moment of force is calculated by:

$$r = F \times d$$

Anti-clockwise moment:

$$\begin{aligned} r_{\text{anti-clockwise}} &= F \times d \\ &= 120 \times 6 \\ &= 720 \text{ Nm} \end{aligned}$$

Clockwise moment:

$$\begin{aligned} r_{\text{clockwise}} &= F \times d \\ &= F \times 8 \\ &= 8F \end{aligned}$$

Since there is rotational equilibrium, we know that $r_{\text{anti-clockwise}} = r_{\text{clockwise}}$ from the principle of moments.

$$\begin{aligned} r_{\text{clockwise}} &= 8F \\ r_{\text{anti-clockwise}} &= 8F \\ 720 &= 8F \\ F &= 90 \text{ N}\end{aligned}$$

### State the Principle of Moments and discuss how this principle may be applied to two persons of different weights trying to balance on a see-saw.

**Show/hide answer**

The principle of moments states that for a body to be in rotational equilibrium, the sum of clockwise torques about any point (which acts as a pivot) must equal to the sum of anti-clockwise torques about the same point.

For the see-saw to be balanced, the see-saw must be in rotational equilibrium – which mandates that the clockwise moment about the pivot be equal to the anti-clockwise moment about the same pivot.

Since the two persons have different weights, varying the perpendicular distances to the pivot is the only way for the clockwise and anti-clockwise moment to be equal.

Consider that the two persons are labelled as 1 and 2, with:

- their weights labelled as $W_{1}$ and $W_{2}$, whereby $W_{1} > W_{2}$.
- their distance away from the pivot of the see-saw labelled as $d_{1}$ and $d_{2}$

Recall that moment of force is calculated by:

$$r = F \times d$$

For the see-saw to be balanced:

$$\begin{aligned} r_{\text{clockwise}} &= r_{\text{anti-clockwise}} \\ W_{1} \times d_{1} &= W_{2} \times d_{2} \end{aligned}$$

Since $W_{1} > W_{2}$, $d_{1}$ must be smaller than $d_{2}$ for the see-saw to be balanced. (I.e. The heavier person will have to sit closer to the pivot.)