Resultant Forces and Free Body Diagrams

Most objects have several forces acting on them at once. Rather than track each one separately, physics combines them into a single equivalent force.

A resultant force is the single force that has the same effect on an object as all the individual forces acting on it added together.

Think of a stationary car being pushed by two people from behind. If each pushes with 200 N in the same direction, the car behaves exactly as if one person pushed with 400 N. That 400 N is the resultant force.

The resultant matters because it decides what happens to an object's motion:

• A non-zero resultant force changes an object's motion (it speeds up, slows down or changes direction).
• A zero resultant force means the forces are balanced, and the object stays at rest or keeps moving at constant velocity.

Because force is a vector, combining forces means taking direction into account, not just adding sizes.

When forces act along the same straight line, finding the resultant is straightforward: add forces pointing the same way and subtract forces pointing the opposite way.

Worked example — a sledge is pulled forwards with a force of 50 N while friction of 30 N acts backwards. Find the resultant force.

The forces are in opposite directions, so subtract:

$50\text{N}-30\text{N}=20\text{N forwards}$

The resultant force is 20 N in the forwards direction, so the sledge accelerates forwards.

Worked example — two dogs pull a sledge forwards, one with 120 N and one with 90 N, both in the same direction. A friction force of 40 N acts backwards. Find the resultant force.

Add the forces in the same direction, then subtract the opposing force:

$(120+90)-40=210-40=170\text{N forwards}$

The resultant force is 170 N forwards. Always state the direction, because the resultant is a vector.

A special and very common case is when the resultant force is exactly zero.

When the resultant force on an object is zero, the forces are balanced and the object stays at rest or continues at a constant velocity.

Consider a book resting on a table. Its weight (say 8 N) pulls it down, and the normal contact force from the table (8 N) pushes it up. These are equal and opposite:

$8\text{N down}-8\text{N up}=0\text{N}$

The resultant is zero, so the book stays still. The same idea applies to a car cruising at a steady 30 m/s: the driving force from the engine exactly balances the resistive forces, so there is no resultant force and the velocity stays constant. This links directly to Newton's First Law.

(Higher Tier only) The rest of this lesson — free body diagrams and resolving forces — is required for Higher Tier papers only.

A free body diagram is a clear way to show all the forces on one object so you can find the resultant.

A free body diagram shows a single isolated object with every force acting on it drawn as a labelled arrow. The arrow's length shows the size of the force and its direction shows the way the force acts.

The object is usually drawn as a simple box or dot, and every force is drawn as an arrow starting from it. The diagram below shows a car driving along a level road at steady speed.

Here weight balances the normal contact force (vertical), and the driving force balances the resistive forces (horizontal). Every pair is equal and opposite, so the resultant is zero and the car travels at constant velocity.

(Higher Tier only) Resolving a single force into two perpendicular components.

A single force acting at an angle can be replaced by two forces at right angles to each other that together have the same effect. These two forces are its components, usually a horizontal and a vertical one.

This is the reverse of finding a resultant. Instead of combining forces, you split one force into two perpendicular parts. It is useful when a force acts at an angle, for example a rope pulling a sledge at 30° to the ground: part of the pull drags the sledge forwards (the horizontal component) and part lifts it (the vertical component).

You can find the components using a scale (vector) diagram:

1. Choose a scale, for example 1 cm to represent 10 N.
2. Draw the original force as an arrow at the correct angle and correct length.
3. Draw a rectangle with the force as its diagonal, with sides horizontal and vertical.
4. The two sides of the rectangle are the horizontal and vertical components; measure their lengths and convert back using the scale.

The same scale-diagram method works in reverse to combine two perpendicular forces into a single resultant, reading off its magnitude and direction.

(Higher Tier only) Using a vector (scale) diagram to find a resultant's magnitude and direction.

When two forces act at an angle to each other, you cannot simply add or subtract them. A scale diagram finds the resultant graphically.

Worked example — a force of 30 N acts to the east and a force of 40 N acts to the north on the same object. Find the resultant.

Draw the two forces tip to tail to scale (for example 1 cm = 10 N), then draw the resultant from the start of the first arrow to the tip of the last. Because these two forces are perpendicular, the resultant is the diagonal of a right-angled triangle, so its size can be checked:

$\text{resultant}=\sqrt{30^2+40^2}=\sqrt{900+1600}=\sqrt{2500}=50\text{N}$

The resultant force is 50 N, pointing north-east between the two original forces. In the exam you find both the magnitude and the direction by measuring your scale diagram with a ruler and protractor.

An object is in equilibrium when the forces on it form a closed shape on a scale diagram, meaning the resultant is zero.

1. Adding forces without considering direction

Forces are vectors. Two 10 N forces give a 20 N resultant only if they point the same way. Pointing in opposite directions they give 0 N; at right angles they give about 14 N. Direction always matters.

2. Forgetting to state the direction of the resultant

A resultant force answer needs a direction as well as a size. "20 N" is incomplete; "20 N forwards" or "50 N north-east" is complete.

3. Thinking a moving object must have a resultant force

An object moving at constant velocity has a resultant force of zero. A resultant force is needed to change motion, not to keep it going.

4. Drawing forces on a free body diagram to the wrong scale

On a free body diagram the arrow lengths should reflect the force sizes. Balanced forces must be drawn the same length; a larger force must have a longer arrow.

5. Trying to add perpendicular forces arithmetically (Higher Tier)

Forces at right angles cannot simply be added. Use a scale diagram or Pythagoras: 30 N east and 40 N north give a 50 N resultant, not 70 N.