State that, when there is no resultant force and no resultant moment, an object is in equilibrium

1.5.2 Turning Effect of Forces

Objective

State that, when there is no resultant force and no resultant moment, an object is in equilibrium.

Key Concepts

• Resultant Force (\$\sum \mathbf{F}\$): The single force that has the same effect as all the forces acting on the object.
• Resultant Moment (\$\sum M\$): The single turning effect (torque) that has the same effect as all the moments acting about a chosen pivot.
• Equilibrium: The state in which an object does not accelerate linearly or rotate. This occurs when both the resultant force and the resultant moment are zero.

Conditions for Equilibrium

For an object to be in equilibrium the following two conditions must be satisfied simultaneously:

1. \$\sum \mathbf{F}=0\$ (no net translational force)
2. \$\sum M=0\$ (no net turning effect about any axis)

Understanding Moments

The moment (or torque) of a force about a point O is given by:

\$M = F \times d\$

where \$F\$ is the magnitude of the force and \$d\$ is the perpendicular distance from the line of action of the force to point O. The direction of the moment is indicated by the sense of rotation (clockwise or anticlockwise).

Table: Summary of Equilibrium Conditions

Worked Example

Problem: A uniform beam 4 m long is supported at its ends. A weight of 200 N is hung 1 m from the left support. Determine the forces exerted by the supports.

Solution:

1. Take moments about the left support (point A). Let \$R_B\$ be the reaction at the right support.
2. \$\sum MA = 0 \;\Rightarrow\; RB \times 4\,\text{m} - 200\,\text{N} \times 1\,\text{m} = 0\$
3. \$R_B = \frac{200 \times 1}{4} = 50\;\text{N}\$
4. Apply the condition \$\sum F_y = 0\$:
5. \$RA + RB - 200 = 0 \;\Rightarrow\; R_A = 200 - 50 = 150\;\text{N}\$

Both support forces are vertical, and the sum of moments about any point is zero, confirming equilibrium.

Key Take‑away

When an object is in equilibrium, the vector sum of all forces acting on it is zero and the algebraic sum of all moments about any point is also zero. This principle is fundamental for solving static problems in physics and engineering.