Resultant forces

​​In a nutshell

The resultant force is the net force that acts on an object. The forces on an object can be represented using a free-body diagram. Forces can be resolved into vertical horizontal components using scale diagrams.

Free body diagrams

A free body diagram is a diagram of the forces acting on an object or system. The forces are represented by vectors (arrows) with a size and direction. All arrows start from the same point - the object.

Example	Free body diagram
A scuba diver	1. Buoyancy force 2. Weight 3. Water resistance 4. Forward thrust
A ball rolling down a ramp	1. Normal contact force 2. Frictional force 3. Weight
Person with a parachute	1. Air resistance 2. Weight

Vector addition

The resultant force is the net force that acts on an object. The resultant force can be found from a free body diagram by adding the vectors.

Vectors that point in opposite directions are subtracted from each other. Vectors that point in the same direction are added together. 

​​Adding vectors	Subtracting vectors

Equilibrium

An object is in equilibrium if the resultant force on an object is equal to zero. This means that the individual force vectors cancel each other out. 

Note: The motion of an object in equilibrium stays constant! A moving object stays moving at that velocity. A non-moving object stays still.

Example

1. Normal contact force 2. Weight due to gravity

The vertical force vectors are equal in opposite directions so cancel each other out. There are no horizontal forces. This means the resultant force is zero and the object is in equilibrium.

Resolving vectors

Vectors directed at an angle to the horizontal (or vertical) can be resolved into its vertical and horizontal components. These vertical and horizontal components add together to create the original vector.

Resolving vectors makes finding resultant forces much easier. This is because all the horizontal and vertical components can be added or subtracted separately.

To resolve vectors, a scale diagram has to be drawn. This means the vector has to be drawn to the correctly scaled size and at the correct angle.

The vector can then be split into its component parts. This is done by drawing lines to the $$x$$​ and $$y$$​-axis. 

Example

A vector of magnitude $$15\,N$$​ is at a direction of  $$30\degree$$ from the north. Draw this vector to scale and resolve the vectors into its vertical and horizontal component parts.

The horizontal component has a magnitude of  $$7.5\,N$$​ and the vertical component has a magnitude of  $$13\,N$$.

Tip: Sometimes a question may give an angle in terms of a bearing. A bearing is an angle measured clockwise from the north arrow. It is always written using three digits e.g. $$038\degree$$​. 

Resultant forces

Two vectors pointing in perpendicular directions (e.g. along the $$x$$​ and $$y$$​-axis) can add together to show the resultant force. This is the opposite process to resolving a vector.

Drawing a resultant force

Procedure

Note: If the question asks for your final answer as a bearing, make sure to convert the angle to a bearing!

Example

A force of $$4\,N$$​ acts on an object in the vertical direction. A force of $$3\,N$$​ acts on the same object in the horizontal direction. What is the resultant vector on this object?

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Note: The forces are not always towards the north or towards the east. Sometimes they are towards the south (negative $$y$$​-axis) or towards the west (negative $$x$$​-axis). The same procedure can be used, just remember the resultant vector won't be in the north-east direction!

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