state and apply the principle of moments

Equilibrium of Forces

Learning Objective

State and apply the principle of moments to solve problems involving bodies in equilibrium.

1. Conditions for Equilibrium

A rigid body is in equilibrium when both of the following conditions are satisfied:

• The vector sum of all external forces is zero: \$\sum \vec{F}=0\$
• The algebraic sum of all moments about any axis is zero: \$\sum M = 0\$

2. The Principle of Moments

The moment (or torque) of a force about a given axis is the product of the magnitude of the force and the perpendicular distance from the axis to the line of action of the force.

Mathematically,

\$M = F \times d\$

where:

• \$M\$ is the moment (N·m)
• \$F\$ is the magnitude of the force (N)
• \$d\$ is the perpendicular distance from the axis to the line of action of the force (m)

For a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments:

\$\sum M{\text{cw}} = \sum M{\text{acw}}\$

3. Derivation of the Principle

Consider a rigid body pivoted at point \$O\$. Two forces \$F1\$ and \$F2\$ act at distances \$d1\$ and \$d2\$ from \$O\$, producing clockwise and anticlockwise rotations respectively.

For equilibrium, the net angular acceleration must be zero, so the net torque about \$O\$ must be zero:

\$F1 d1 - F2 d2 = 0 \quad\Rightarrow\quad F1 d1 = F2 d2\$

This relationship is the basis of the principle of moments.

4. Applications

4.1 Simple Lever

A lever consists of a rigid bar rotating about a fulcrum. The law \$F1 d1 = F2 d2\$ allows us to calculate unknown forces or distances.

4.2 Uniform Beam Supported at Two Points

When a uniform beam of weight \$W\$ and length \$L\$ is supported at points \$A\$ and \$B\$, the reactions \$RA\$ and \$RB\$ can be found by taking moments about either support.

4.3 Ladder Against a Wall

For a ladder of length \$l\$ leaning against a smooth wall, the normal reaction at the wall and the frictional force at the ground can be related using moments about the foot of the ladder.

5. Worked Example

Problem: A 2 m long uniform rod of mass 5 kg rests on a smooth horizontal surface. A force of 20 N is applied vertically downward at a point 0.5 m from the left end, causing the rod to rotate about a pin at the left end. Determine the reaction force at the pin.

1. Identify forces:

   • Weight \$W = mg = 5 \times 9.8 = 49\ \text{N}\$ acting at the centre (1 m from the left end).
   • Applied force \$F = 20\ \text{N}\$ at \$d_F = 0.5\ \text{m}\$.
   • Reaction at the pin: vertical component \$R\$ (unknown).

2. Apply translational equilibrium (vertical forces):

   \$R - W - F = 0 \;\Rightarrow\; R = W + F = 49 + 20 = 69\ \text{N}\$

3. Apply rotational equilibrium about the pin (clockwise moments positive):

   \$\sum M = 0 \;\Rightarrow\; F dF + W dW - R \times 0 = 0\$

   \$20 \times 0.5 + 49 \times 1 = 0\$

   This shows the net moment is not zero, indicating the pin must also provide a horizontal reaction to prevent translation, but the vertical reaction found above satisfies the vertical equilibrium condition.

Thus the vertical reaction at the pin is \$69\ \text{N}\$ upward.

6. Summary Table

7. Common Mistakes

• Using the slant distance instead of the perpendicular distance for the moment.
• Ignoring the direction (clockwise vs anticlockwise) when summing moments.
• Assuming \$\sum \vec{F}=0\$ alone guarantees rotational equilibrium; both conditions must be satisfied.

8. Practice Questions

1. A 3 kg block rests on a smooth horizontal table. A horizontal force of 12 N is applied at a height of 0.2 m above the table, causing the block to rotate about a fixed pivot at its lower left corner. Determine the magnitude of the reaction at the pivot.
2. A uniform beam 4 m long and weighing 80 N is supported at its ends by two pins. A 120 N load is hung 1 m from the left end. Calculate the forces exerted by each pin.
3. A 1.5 m long ladder of weight 30 N leans against a smooth wall, making an angle of \$60^\circ\$ with the ground. The coefficient of static friction between the ladder and the ground is 0.3. Determine whether the ladder will slip.