Resultant Forces and Free Body Diagrams

The resultant force is the overall force acting on an object when all the individual forces are combined. It is a single force that would have the same effect as all the forces acting together.

$\text{Resultant force} = \text{sum of all forces (taking direction into account)}$

Forces in the Same Direction

When forces act in the same direction, add them together:

$F_{\text{resultant}} = F_1 + F_2$

Example: Two people push a car in the same direction with forces of 200 N and 150 N.

$F_{\text{resultant}} = 200 + 150 = 350 \text{ N (in the direction of pushing)}$

Forces in Opposite Directions

When forces act in opposite directions, subtract the smaller from the larger:

$F_{\text{resultant}} = F_{\text{larger}} - F_{\text{smaller}}$

The resultant acts in the direction of the larger force.

Example: A rocket engine provides 50,000 N of thrust upwards. The rocket's weight is 35,000 N downwards.

$F_{\text{resultant}} = 50{,}000 - 35{,}000 = 15{,}000 \text{ N upwards}$

Balanced Forces (Resultant = 0)

When the resultant force on an object is zero, the forces are balanced. The object will:

• Stay stationary if it was at rest, OR
• Continue moving at a constant speed in a straight line if it was already moving

This links directly to Newton's First Law.

Examples of balanced forces:

• A book sitting on a table (weight down = normal force up)
• A car cruising at constant speed (driving force = friction + air resistance)
• A skydiver at terminal velocity (weight = air resistance)

Unbalanced Forces (Resultant ≠ 0)

When the resultant force is not zero, the forces are unbalanced. The object will accelerate in the direction of the resultant force.

"Accelerate" means:

• Speed up
• Slow down
• Change direction

Examples of unbalanced forces:

• A ball being kicked (applied force > friction)
• A car braking (friction > driving force)
• A falling object before reaching terminal velocity (weight > air resistance)

A free body diagram is a simplified drawing that shows all the forces acting on a single object. The object is usually represented as a box or dot.

Rules for Drawing Free Body Diagrams

1. Draw the object as a simple shape (box, circle, or dot)
2. Draw each force as an arrow starting from the object
3. The length of each arrow represents the size of the force (longer = bigger)
4. The direction of each arrow shows the direction of the force
5. Label each force with its name and value (if known)
6. Only include forces acting on the object (not forces the object exerts on others)

Common Forces to Include

Step-by-Step Method

1. Draw a free body diagram
2. Choose a positive direction (e.g., right or up)
3. Add forces in the positive direction
4. Subtract forces in the negative direction
5. The answer gives the resultant force (magnitude and direction)

Example: Horizontal Forces

A car has a driving force of 3000 N forwards and a total resistance (friction + air resistance) of 2200 N backwards.

$F_{\text{resultant}} = 3000 - 2200 = 800 \text{ N forwards}$

The car accelerates forwards.

Example: Vertical Forces

A helicopter has a lift force of 45,000 N upwards and a weight of 40,000 N downwards.

$F_{\text{resultant}} = 45{,}000 - 40{,}000 = 5{,}000 \text{ N upwards}$

The helicopter accelerates upwards.

When forces don't act along the same line, we need to use scale diagrams or vector resolution to find the resultant.

Method 1: Scale Drawing (Tip-to-Tail)

1. Choose a suitable scale (e.g., 1 cm = 10 N)
2. Draw the first force as an arrow to scale
3. From the tip of the first arrow, draw the second force
4. Draw the resultant from the start of the first arrow to the tip of the last arrow
5. Measure the length and angle of the resultant

Method 2: Resolving into Components

Any force at an angle can be split into horizontal and vertical components:

$F_x = F \cos \theta$ $F_y = F \sin \theta$

Where $\theta$ is the angle from the horizontal.

Then find the resultant of all horizontal components and all vertical components separately, and combine using Pythagoras:

$F_{\text{resultant}} = \sqrt{F_x^2 + F_y^2}$

The direction is found using:

$\theta = \tan^{-1}\left(\frac{F_y}{F_x}\right)$

• The resultant force is the single force that replaces all forces acting on an object
• Same direction: add the forces
• Opposite directions: subtract (resultant in the direction of the larger force)
• Balanced (resultant = 0): object at rest or constant velocity
• Unbalanced (resultant ≠ 0): object accelerates
• Free body diagrams show all forces on one object with labelled arrows
• Higher: use Pythagoras and trigonometry for forces at angles