## Table of Contents

## 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
- Linear acceleration of object is zero

### 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

- Angular acceleration of object is zero
- Lines of action of all forces acting on the object intersect at a point

## Summary

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

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

## Steps To Solve Equilibrium Problems

Solving equilibrium problems involves a systematic approach. Here are the steps:

### Steps To Solve Equilibrium Problems

1. **Isolate Body of Interest:**

- Identify the specific object or system you are analyzing.

2. **Identify Forces:**

- Consider all forces acting directly on the isolated body.
- Include gravitational forces, applied forces, tension, normal forces, friction, etc.

3. **Draw Free Body Diagram (FBD):**

- Sketch a clear and accurate diagram representing the isolated body.
- Use arrows to denote forces with the correct magnitude and direction.
- Clearly label each force.

4. **Coordinate System:**

- Establish a coordinate system for analysis. This is often chosen based on the geometry or symmetry of the problem.

5. **Apply Equilibrium Conditions:**

- For translational equilibrium, sum all forces in both x and y directions and set them equal to zero.
- For rotational equilibrium, sum all torques (moments) and set them equal to zero.
- Write out the equations based on your coordinate system.

### Steps To Solve Rotational Equilibrium Problems

1. **Follow Steps 1-4 from the Equilibrium Problems:**

2. **Calculate Torque:**

- Identify the axis of rotation.
- Calculate the torque for each force by multiplying its magnitude by the perpendicular distance from the axis.

3. **Apply Rotational Equilibrium Condition:**

- Sum all torques and set them equal to zero.
- This equation ensures that the object is not rotating.

### Steps To Solve Translational Equilibrium Problems:

1. **Follow Steps 1-4 from the Equilibrium Problems:**

2. **Resolve Forces:**

- If forces are not purely along the x or y-axis, resolve them into their respective components.
- Ensure that the resolved forces align with your chosen coordinate system.

3. **Apply Translational Equilibrium Condition:**

- Sum all forces in the x and y directions and set them equal to zero.
- This equation ensures that the object is not accelerating in any direction.

### Additional Tips On Solving Equilibrium Problems

**Include Direction:**- Pay attention to the direction of the forces. Include signs in your equations to denote direction.

**Check Units:**- Ensure that all forces and distances are in consistent units.

**Use Algebra:**- Solve the equations algebraically for unknowns.

## Worked Examples

### Example 1

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.

**Click here to 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.

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

### Example 2

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.

**Click here to 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.)

### Example 3

Two people are carrying a 2 m long couch with a mass of M = 20 kg level with the horizontal by lifting it from it ends. The center of mass of the couch is 1 m from each end. While it is being carried, another person with a mass m of 60 kg sits on the couch 0.5 m from one end. How much force will each person carrying the couch have to exert upwards in order to continue carrying the couch levelly?

**Click here to show/hide answer**

The couch above is in equilibrium : Rotational equilibrium and translational equilibrium.

Taking moments about the left point,

$$\begin{aligned} \text{Anti-clockwise moments about left point} &= \text{Clockwise moments about right point} \\ F_{\text{By right person}} \times 2 &= F_{\text{Weight of person on couch}} \times 0.5 + F_{\text{Weight of couch}} \times 1 \\ F_{\text{By right person}} \times 2 &= (60)(9.8)(0.5) + (20)(9.8)(1) \\ F_{\text{By right person}} &= \frac{490}{2} \\ F_{\text{By right person}} &= 245 \, \text{N} \end{aligned} $$

Taking moments about the right point,

$$ \begin{aligned} \text{Clockwise moments about right point} &= \text{Anti-clockwise moments about left point} \\ F_{\text{By left person}} \times 2 &= F_{\text{Weight of person on couch}} \times 1.5 + F_{\text{Weight of couch}} \times 1 \\ F_{\text{By left person}} \times 2 &= (60)(9.8)(1.5) + (20)(9.8)(1) \\ F_{\text{By left person}} &= 539 \, \text{N} \end{aligned}$$

Answer: 245N and 539N