Triangle of forces

In a nutshell

A triangle of forces is a method of drawing out a problem in a way that makes it easier to understand and solve. The normal contact force is the force on an object acting perpendicular from the surface it sits on, tension is a force that pulls on an object due to a string, upthrust is a force exerted on an object due to the surrounding fluid and friction is a force that opposes the direction of motion.

Definitions

Keyword	Definition
Triangle of forces	A method of drawing out a problem involving three forces acting on an object by drawing the forces pointing from each other in a triangle.
Normal contact force	The force on an object from the surface it sits on, acting perpendicular to the surface.
Tension	The pulling force acting on an object due to a string or similar object attached to it.
Upthrust	The upwards pushing force acting on an object due to it being submerged in fluid.
Friction	The force that opposes the direction of motion.
Coplanar forces	Forces that act in the same plane as each other.

Forces at equilibrium

When an object is at rest, it means that all the forces that act on it are balanced out, and their sum adds up to zero. Some examples of forces that act on an object are tension, upthrust, normal contact force and friction. Tension is a pulling force that results from objects such as a rope, while upthrust is a result of an object being submerged in a fluid and being pushed upwards by it.

Example

An object is suspended by two ropes. The free-body diagram looks like this.

Physics; Motion; KS5 Year 12; Triangle of forces

The tension forces counteract the weight force, preventing the object from falling.

Triangle of forces

A triangle of forces rearranges the forces in a system with three coplanar forces in a way that allows problems to be solved more easily. In order to draw a triangle of forces, each force is drawn tip to tail, leading on from each other to form a closed triangle. A triangle of forces can only be drawn when the forces are in equilibrium.

Example

The diagram above shows an object being suspended by two ropes. The triangle of forces for this set-up looks like this.

Physics; Motion; KS5 Year 12; Triangle of forces

Applying to problems

Drawing a triangle of forces allows for the use of Pythagoras' theorem and trigonometry to work out unknown angles and forces in a way that may have been hard to analyse in a regular free-body diagram.

Example

The following diagram shows an object at rest with three coplanar forces acting on it. Draw a triangle of forces and calculate the object's weight.

Physics; Motion; KS5 Year 12; Triangle of forces

The triangle of forces looks like this.

Physics; Motion; KS5 Year 12; Triangle of forces

Using Pythagoras' theorem, it is possible to directly found out the object's weight. First, write down the equation.

$a^2 = b^2 + c^2$ 

Next, rearrange the equation and substitute in the values.

$b = \sqrt{a^2-c^2}$

$W = \sqrt{15^2-10^2}$

$W = 11.2\,N$

The weight force is $\underline{11.2\,N}$ .