Equilibrium of Forces

Equilibrium is a state in which the resultant of all forces acting on a body is zero, and the sum of moments about any point is also zero. This study guide covers the mechanics of forces acting on particles and rigid bodies.

1. Equilibrium of Particles

A particle is in equilibrium if the vector sum of all forces acting on it is zero. For coplanar forces (forces acting in the same plane), equilibrium is often solved using three methods:

• Triangle of Forces: If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the sides of a triangle taken in order.
• Polygon of Forces: For more than three forces, equilibrium is achieved if the forces can be represented as sides of a closed polygon taken in order.
• Lami's Theorem: If three forces P, Q, and R act at a point and are in equilibrium, each force is proportional to the sine of the angle between the other two forces. Mathematically: P/sin α = Q/sin β = R/sin γ.

2. Principles of Moments

The moment of a force is the turning effect of the force about a pivot. It is the product of the force and the perpendicular distance from the line of action of the force to the pivot (M = F × d). The unit is Newton-meter (Nm).

The Principle of Moments

For a body in equilibrium, the sum of clockwise moments about any point must equal the sum of anticlockwise moments about that same point.

Moment of a Couple (Torque)

A couple consists of two equal and opposite parallel forces whose lines of action do not coincide. The torque (T) of a couple is calculated as: Torque = one force × perpendicular distance between the forces.

3. Equilibrium of Rigid Bodies

For a rigid body to be in total equilibrium, two conditions must be met:

• First Condition: The vector sum of all external forces must be zero (ΣFx = 0 and ΣFy = 0). This is the condition for translational equilibrium.
• Second Condition: The algebraic sum of moments about any point must be zero (ΣM = 0). This is the condition for rotational equilibrium.

Resolution and Composition of Forces

Forces can be resolved into two perpendicular components: Fx = F cos θ and Fy = F sin θ. The resultant is the single force that represents the combined effect of several forces, while the equilibrant is the single force that balances the others (equal in magnitude to the resultant but opposite in direction).

4. Centre of Gravity and Stability

The Centre of Gravity (C.G) is the point through which the entire weight of a body appears to act. Stability refers to the ability of a body to return to its original position after being slightly displaced.

• Stable Equilibrium: A small displacement causes the C.G to rise. The body returns to its original position (e.g., a ball at the bottom of a bowl).
• Unstable Equilibrium: A small displacement causes the C.G to fall. The body moves further from its original position (e.g., a cone balanced on its apex).
• Neutral Equilibrium: A displacement does not change the height of the C.G. The body remains in its new position (e.g., a ball on a flat table).