3.4.1.1MechanicsFoundation

Forces and Free-Body Diagrams

A force is a $vector$ push or pull measured in newtons (N). Before you can apply Newton’s laws you have to see the forces clearly — that means drawing a $free-body diagram$ , finding the single $resultant$ force, and resolving awkward forces into perpendicular components.

30 min Video by Science Shorts 3.4.1 Force, energy and momentum

Balanced Forces — AS/A-Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Name the common forces: weight, normal (reaction), tension, friction, drag and thrust
Draw a free-body diagram showing every force acting on one object
Combine forces to find the resultant, both graphically and by components
Resolve a single force into perpendicular components using sine and cosine
Recognise when forces are balanced so the object is in equilibrium

Forces as vectors

Every force has a size and a direction, so forces add like vectors, not like ordinary numbers. The most common forces at A-Level are $weight$ (gravity), the $normal contact force$ (perpendicular to a surface), $tension$ in strings, $friction$ and $drag$ (opposing motion), and driving forces such as $thrust$ .

Weight is the pull of gravity on a mass and always acts vertically downward, from the object’s centre of mass. On Earth $g \approx 9.81$ N/kg.

W = m g

Weight (N) = mass (kg) × gravitational field strength (N/kg).

Free-body diagrams

A free-body diagram isolates a single object and shows only the forces acting $on$ it, drawn as arrows starting at the object. The length of each arrow suggests the size of the force and the direction shows which way it acts.

The golden rule: never draw a force the object exerts on something else — only the forces the surroundings exert on your chosen object belong on its diagram.

Tip — Label each arrow with what causes it (e.g. “weight”, “normal from ramp”, “tension in rope”). If you can’t name the thing pushing or pulling, the force probably shouldn’t be there.

Resultant force and resolving

The $resultant$ is the single force that has the same effect as all the forces combined. For two forces at right angles you can find its magnitude with Pythagoras and its direction with trigonometry.

Going the other way, any single force $F$ at an angle $θ$ to a chosen axis can be split into two perpendicular $components$ — one along the axis and one at right angles to it. This is the key trick for ramps, tensions and projectiles.

F_{x} = F cos θ, F_{y} = F sin θ

Components of a force at angle θ to the x-axis.

F_{res} = F_{x}^{2} + F_{y}^{2}

Magnitude of the resultant of two perpendicular components.

1The forces are perpendicular, so use Pythagoras for the size.

F_{res} = 3. 0^{2} + 4. 0^{2} = 25 = 5.0

3Direction:

tan θ = 4.0/3.0

, so

θ = 5 3^{\circ}

north of east.

Answer

5.0

N at

5 3^{\circ}

north of east

Balanced forces and equilibrium

When the resultant force on an object is $zero$ , the forces are said to be balanced and the object is in $equilibrium$ . It either stays at rest or keeps moving at constant velocity — a direct preview of Newton’s first law.

For forces in two dimensions this means the components balance separately: the horizontal components must cancel and the vertical components must cancel.

Tip — To test for equilibrium, resolve everything into two perpendicular directions and check that each direction adds up to zero. If both do, the resultant is zero.

Equation recap

W = m g

Weight from mass and gravitational field strength.

F_{x} = F cos θ, F_{y} = F sin θ

Resolving a force into perpendicular components.

F_{res} = F_{x}^{2} + F_{y}^{2}

Resultant of two perpendicular forces.

Common mistakes to avoid

Adding forces as plain numbers when they point in different directions.

Forces are vectors. Resolve into components (or draw a scale diagram) before combining them.

Swapping sine and cosine when resolving a force.

The component next to (adjacent) the angle uses cosine; the component opposite the angle uses sine. Always mark the angle first.

Drawing forces the object exerts on other things on its free-body diagram.

A free-body diagram shows only forces acting ON the chosen object, each starting at the object.

Assuming the normal contact force always equals the weight.

The normal force only equals the weight on flat ground with no vertical acceleration. On a slope it equals mg cos θ.

Key takeaways

Forces are vectors measured in newtons; weight is W = mg acting downward from the centre of mass.
A free-body diagram shows only the forces acting on one chosen object.
Resolve forces into perpendicular components with F cos θ and F sin θ.
Balanced forces (zero resultant) mean the object is in equilibrium — at rest or at constant velocity.

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