define and apply the moment of a force

Turning Effects of Forces

Learning Objective

Define the moment (torque) of a force and apply the concept to solve quantitative problems involving turning effects.

Definition of Moment

The moment (or torque) of a force about a chosen axis is the tendency of the force to cause rotation about that axis.

Mathematically, the moment \$\boldsymbol{\tau}\$ of a force \$\mathbf{F}\$ applied at a point with position vector \$\mathbf{r}\$ (relative to the axis) is

\$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}\$

where "\$\times\$" denotes the vector cross‑product. The magnitude is

\$\tau = r\,F\,\sin\theta\$

with \$\theta\$ the angle between \$\mathbf{r}\$ and \$\mathbf{F}\$. The direction of \$\boldsymbol{\tau}\$ follows the right‑hand rule.

Key Points

• Moment is a vector quantity; its direction indicates the axis of rotation.
• SI unit: newton‑metre (N·m). In practice, the sign (+/–) is used to denote clockwise or anticlockwise rotation.
• Only the component of the force perpendicular to the lever arm contributes to the moment.
• A couple consists of two equal and opposite forces whose lines of action do not coincide; it produces a pure turning effect with no resultant force.

Sign Convention

For problems in a vertical plane:

• Counter‑clockwise moments are taken as positive (+).
• Clockwise moments are taken as negative (–).

Resultant Moment

The net turning effect about an axis is the algebraic sum of all individual moments:

\$\sum \tau = \tau1 + \tau2 + \dots + \tau_n\$

For rotational equilibrium (no angular acceleration), the condition is

\$\sum \tau = 0\$

Table of Symbols

Worked Example

Problem: A uniform beam 4.0 m long is hinged at its left end (point A) and supported by a rope at its right end (point B). The beam carries a load of 800 N at its centre. The rope makes an angle of \$30^{\circ}\$ above the horizontal. Determine the tension in the rope.

1. Choose the hinge A as the axis of rotation. This eliminates the unknown hinge reaction forces from the moment equation.
2. Calculate the perpendicular distances:

   • Weight \$W = 800\,\$N acts at \$2.0\,\$m from A.
   • The tension \$T\$ acts at \$4.0\,\$m from A, with a vertical component \$T\sin30^{\circ}\$ producing a clockwise moment.

3. Write the moment equilibrium about A (counter‑clockwise positive):

   \$\sum \tau_A = 0 = W(2.0) - T\sin30^{\circ}(4.0)\$

4. Solve for \$T\$:

   \$800 \times 2.0 = T \times \frac{1}{2} \times 4.0\$

   \$1600 = 2.0\,T\$

   \$T = 800\ \text{N}\$

The tension in the rope is \$800\,\$N.

Common Mistakes

• Using the full magnitude of the force instead of its component perpendicular to the lever arm.
• Neglecting the sign convention, leading to incorrect addition/subtraction of moments.
• Forgetting to include all forces that produce a moment about the chosen axis.

Suggested Diagram

Practice Questions

1. A force of \$120\,\$N is applied at the end of a \$0.5\,\$m long wrench, making an angle of \$45^{\circ}\$ with the wrench. Calculate the moment about the bolt.
2. A uniform rectangular plate \$0.8\,\$m by \$0.6\,\$m rests on a frictionless pivot at its left edge. A force of \$50\,\$N is applied at the top right corner vertically downwards. Determine the angular acceleration if the plate’s moment of inertia about the pivot is \$0.12\,\$kg·m².
3. Two forces of \$30\,\$N and \$40\,\$N act on a door at distances of \$0.6\,\$m and \$0.9\,\$m from the hinges, respectively, both perpendicular to the door. What is the net moment about the hinges? Is the door in rotational equilibrium?

Summary

The moment of a force quantifies its turning effect about an axis and is given by the cross‑product \$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}\$. Correctly identifying lever arms, perpendicular components, and applying the sign convention are essential for solving A‑Level physics problems involving rotational equilibrium and dynamics.