Equilibrium of Forces

What is Equilibrium?

Equilibrium means a state where everything is balanced. When an object is in equilibrium, it means:

• It stays at rest if it was already at rest, OR
• It keeps moving at a constant speed in a straight line if it was already moving

Think of a book sitting still on a table, or a car travelling at a steady 50 km/h on a straight road. Both are in equilibrium because nothing is making them change what they're doing.

For an object to be in equilibrium, two conditions must be met:

Condition 1: No resultant force

• All the forces acting on the object must balance out
• The upward forces equal the downward forces
• The leftward forces equal the rightward forces
• In other words, the forces cancel each other out completely

Condition 2: No resultant torque (no turning effect)

• All the turning effects must balance out
• Clockwise turning effects equal anticlockwise turning effects
• This is explained by the principle of moments

The Principle of Moments

What is a Moment?

A moment (also called torque) is the turning effect of a force. When you push or pull something at a distance from a pivot point (the point it can rotate around), you create a turning effect.

For example:

• Opening a door by pushing it near the handle creates a moment about the hinges (the pivot)
• Using a spanner to turn a nut creates a moment about the center of the nut
• A see-saw turning about its central support point

The moment of a force is calculated using:

Moment = Force × Perpendicular distance from pivot

In symbols: τ = F × d

Where:

• τ (tau) = moment (measured in newton-metres, N m)
• F = force (measured in newtons, N)
• d = perpendicular distance from the pivot to the line of action of the force (measured in metres, m)

Important: The distance must be perpendicular (at 90°) to the direction of the force.

Applying the Principle of Moments

Let's look at a horizontal beam balanced on a pivot:

• Force F₁ acts downward at distance d₁ to the left of the pivot (creates an anticlockwise moment)
• Force F₂ acts downward at distance d₂ to the right of the pivot (creates a clockwise moment)
• Force F₃ acts downward at distance d₃ to the right of the pivot (creates an anticlockwise moment)

For the beam to be in equilibrium:

Clockwise moment = Anticlockwise moments

F₂ × d₂ = (F₁ × d₁) + (F₃ × d₃)

Step-by-Step Example

Question: A uniform beam of weight 40 N is 5 m long and is supported by a pivot 2 m from one end. When a load of weight W is hung from that end, the beam is in equilibrium. What is the value of W?

Solution:

Step 1: List the known information

• Weight of beam (WB) = 40 N
• Length of beam = 5 m
• Distance from end to pivot = 2 m
• Unknown weight = W

Step 2: Find where the beam's weight acts

• The beam is uniform, so its weight acts from the center
• Center of beam = 5 ÷ 2 = 2.5 m from the end
• Distance from pivot to center = 2.5 - 2 = 0.5 m

Step 3: Identify the moments (taking moments about the pivot)

• Clockwise moment = weight of beam × its distance from pivot = 40 × 0.5 = 20 N m
• Anticlockwise moment = W × 2 (the hanging weight is 2 m from the pivot)

Step 4: Apply the principle of moments

• Clockwise moment = Anticlockwise moment
• 20 = W × 2
• W = 20 ÷ 2
• W = 10 N

Important Tips

1. Choose your pivot wisely: You can take moments about any point, but choosing the point where an unknown force acts can make calculations easier (because that force won't appear in your equation since its distance is zero).

2. Keep distances in the same units: Make sure all distances are in metres before calculating.

3. Identify directions correctly: Draw a diagram showing which forces create clockwise moments and which create anticlockwise moments.

4. Forces through the pivot: Any force that acts directly through the pivot creates NO moment (because the perpendicular distance is zero).