Resultant Forces and Equilibrium

This lesson covers how forces combine to produce a resultant force, and what happens when forces are balanced (equilibrium) — as required by the Edexcel GCSE Physics specification (1PH0), Topic 2: Motion and Forces. You need to be able to calculate resultant forces, draw and interpret free body diagrams, and understand equilibrium.

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What Is a Force?

A force is a push or a pull that acts on an object due to its interaction with another object. Forces are measured in newtons (N) and are vector quantities — they have both a magnitude (size) and a direction.

Common forces you need to know:

Force	Description	Direction
Weight	The gravitational pull on an object (W = mg)	Downward, towards the centre of the Earth
Normal contact force	The support force from a surface	Perpendicular to the surface
Friction	Opposes the motion of an object	Opposite to the direction of motion
Air resistance (drag)	Friction from air acting on a moving object	Opposite to the direction of motion
Tension	The pull in a stretched string, rope or cable	Along the string, away from the object
Upthrust	The upward force on an object submerged in a fluid	Upward
Applied force	A push or pull applied by a person or engine	In the direction of the push/pull

Exam Tip: Always name forces precisely. Do not just write "gravity" — write "weight" or "gravitational force." The Edexcel mark scheme requires correct terminology.

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Free Body Diagrams

A free body diagram shows all the forces acting on a single object, represented as arrows. Each arrow:

• Starts on the object (at the point where the force acts).
• Points in the direction of the force.
• Has a length proportional to the magnitude of the force.

Example: A Book on a Table

Force	Direction	Size
Weight (W)	Downward	e.g. 5 N
Normal contact force (N)	Upward	e.g. 5 N

The book is in equilibrium because the two forces are equal in size and opposite in direction — the resultant force is zero.

Example: A Skydiver Falling at Constant Speed

Force	Direction	Size
Weight (W)	Downward	e.g. 800 N
Air resistance (R)	Upward	e.g. 800 N

At constant speed (terminal velocity), air resistance equals weight — the resultant force is zero.

Exam Tip: In free body diagrams, if the object is in equilibrium (stationary or moving at constant velocity), the arrows representing opposing forces must be the same length. If the object is accelerating, the arrow in the direction of acceleration must be longer.

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Calculating Resultant Forces

The resultant force is the single force that has the same effect as all the individual forces acting on an object combined. It determines whether the object accelerates, decelerates, or remains at constant velocity.

Forces Along the Same Line

When forces act along the same line, you calculate the resultant by adding or subtracting:

• Same direction: Add the forces.
• Opposite directions: Subtract the smaller from the larger; the resultant acts in the direction of the larger force.

Worked Example 1: Two Forces in the Same Direction

A car engine provides a forward force of 3000 N. A passenger pushes from behind with 200 N.

Resultant force = 3000 + 200 = 3200 N forward

Worked Example 2: Two Forces in Opposite Directions

A car engine provides a forward force of 3000 N. Friction and air resistance provide a backward force of 1200 N.

Resultant force = 3000 − 1200 = 1800 N forward

Worked Example 3: Three Forces

A box is pushed to the right with 50 N. Friction acts to the left with 20 N. A second person pushes to the right with 15 N.

Total rightward force = 50 + 15 = 65 N Resultant = 65 − 20 = 45 N to the right

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Equilibrium

An object is in equilibrium when the resultant force acting on it is zero. This means:

• If the object is stationary, it remains stationary.
• If the object is moving, it continues to move at constant velocity (constant speed in a straight line).

This is a direct consequence of Newton's First Law.

Conditions for Equilibrium

For an object to be in equilibrium:

1. The resultant force must be zero — all forces balance.
2. The resultant moment must also be zero (covered in the next lesson).

Examples of Equilibrium

Situation	Forces	Why Equilibrium?
Book on a table	Weight down, normal force up	Forces are equal and opposite
Car at constant velocity	Engine force forward, drag backward	Forward force = backward force
Hanging object at rest	Weight down, tension up	Tension = weight
Floating boat	Weight down, upthrust up	Upthrust = weight

Exam Tip: A common misconception is that an object in equilibrium must be stationary. This is wrong. An object moving at constant velocity is also in equilibrium — there is no resultant force, so there is no acceleration, and the velocity does not change.

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Resolving Forces at Angles (Higher Tier)

When forces act at angles to each other, you may need to resolve them into horizontal and vertical components or use scale drawings.

Scale Drawing Method

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

Worked Example: Two Forces at Right Angles

A force of 30 N acts to the right and a force of 40 N acts upward on the same object.

Using Pythagoras' theorem:

Resultant = √(30² + 40²) = √(900 + 1600) = √2500 = 50 N

The angle θ from the horizontal:

tan θ = 40 ÷ 30 = 1.333

θ = tan⁻¹(1.333) ≈ 53.1° above the horizontal

Exam Tip: At Higher tier, you may be asked to use Pythagoras' theorem or trigonometry to find the resultant of two forces at right angles. Make sure you are comfortable with both methods.

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Forces on Objects on Slopes

When an object is on a slope, its weight acts vertically downward, but you can resolve it into two components:

1. A component parallel to the slope (pulling the object down the slope).
2. A component perpendicular to the slope (pushing the object into the surface).

graph TD
    A["Object on a Slope"] --> B["Weight acts<br/>vertically downward"]
    B --> C["Component parallel<br/>to slope<br/>(causes sliding)"]
    B --> D["Component perpendicular<br/>to slope<br/>(balanced by normal force)"]
    A --> E["Normal contact force<br/>perpendicular to surface"]
    A --> F["Friction force<br/>parallel to surface<br/>(opposes sliding)"]

    style A fill:#2c3e50,color:#fff
    style B fill:#c0392b,color:#fff
    style C fill:#e74c3c,color:#fff
    style D fill:#e67e22,color:#fff
    style E fill:#2980b9,color:#fff
    style F fill:#27ae60,color:#fff

Key Points for Slopes

• The steeper the slope, the larger the component of weight parallel to the slope.
• If friction is large enough to balance the parallel component, the object stays in equilibrium.
• If the parallel component exceeds friction, the object slides down the slope.

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Forces on Hanging Objects and Bridges

Hanging Objects

When an object hangs from a single string at rest:

• Weight acts downward.
• Tension in the string acts upward.
• Tension = weight (equilibrium).

When an object hangs from two strings at angles, the vertical components of the two tensions must add up to equal the weight, and the horizontal components must cancel out.

Bridges

A bridge supported at two ends carries a load. The support forces (reactions) at each end must add up to equal the total weight of the bridge plus any load on it. The distribution depends on where the load is placed (this links to moments — see next lesson).

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Summary

• A force is a push or pull measured in newtons (N); it has both magnitude and direction (vector).
• Free body diagrams show all forces acting on a single object as arrows.
• The resultant force is the overall force found by combining all individual forces.
• For forces along the same line: add if same direction, subtract if opposite.
• An object is in equilibrium when the resultant force is zero — it is stationary or moving at constant velocity.
• At Higher tier, forces at angles can be resolved using scale drawings, Pythagoras' theorem or trigonometry.
• On slopes, weight is resolved into components parallel and perpendicular to the surface.
• For hanging objects and bridges, the support forces must balance the weight for equilibrium.

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Extended Worked Examples

Worked Example 4: Tug of War (Three Forces on One Line)

Team A pulls a rope to the left with 450 N. Team B pulls the rope to the right with 620 N. A third person joins Team A and adds 180 N to the left. What is the resultant force?

• Leftward total = 450 + 180 = 630 N
• Rightward total = 620 N
• Resultant = 630 − 620 = 10 N to the left

The rope and both teams move slowly to the left. If the teams pull with equal force, the rope is in equilibrium and the resultant is zero.

Worked Example 5: Aircraft in Steady, Level Flight

An aircraft in cruise has four main forces acting on it:

Force	Direction	Size
Weight	Downward	75 000 N
Lift	Upward	75 000 N
Thrust	Forward	18 000 N
Drag	Backward	18 000 N

Vertical resultant = 75 000 − 75 000 = 0 N. Horizontal resultant = 18 000 − 18 000 = 0 N. The aircraft is in equilibrium — it flies at constant velocity.

Worked Example 6: Box on a Frictionless Slope (Higher Tier)

A 40 kg box sits on a slope angled at 30 degrees. Weight W = mg = 40 × 10 = 400 N.

• Component parallel to slope = W sin(30) = 400 × 0.5 = 200 N (down the slope)
• Component perpendicular to slope = W cos(30) = 400 × 0.866 = 346 N (into slope)

With no friction, the resultant along the slope is 200 N, so the box accelerates down the slope.

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Scalar vs Vector: A Key Distinction

Quantity type	Definition	Examples
Scalar	Magnitude only	Mass, speed, distance, energy, time, temperature
Vector	Magnitude and direction	Force, velocity, acceleration, displacement, momentum, weight

Common mistake: Students often treat weight as a scalar. Weight is a force measured in newtons and has a direction (toward the Earth's centre). Mass is a scalar measured in kilograms.

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Common Mistakes

Common mistake: Forgetting that a stationary object has a resultant force of zero. It is NOT true that "no movement means no forces" — a parked car has weight and a normal contact force acting on it, and they balance.

Common mistake: Using sine when you should use cosine, or vice versa, when resolving vectors. Rule of thumb: the component adjacent to the angle uses cos, the component opposite the angle uses sin.

Common mistake: Drawing free body arrows from random points rather than from the object. Arrows should originate on the object itself.

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graph TD
    A["Resultant force zero?"] --> B{Decide}
    B -->|Yes| C["Equilibrium<br/>Stationary or constant velocity"]
    B -->|No| D["Non-equilibrium<br/>Object accelerates<br/>in direction of resultant"]
    C --> E["Free body arrows<br/>balanced"]
    D --> F["Longer arrow shows<br/>acceleration direction"]

    style A fill:#2c3e50,color:#fff
    style B fill:#8e44ad,color:#fff
    style C fill:#27ae60,color:#fff
    style D fill:#c0392b,color:#fff
    style E fill:#16a085,color:#fff
    style F fill:#e67e22,color:#fff

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Answering at different grade levels

Grade band	Expected response to: "Explain why a skydiver at terminal velocity has no resultant force"
Grade 3–4	"The forces are balanced, so the skydiver falls at a steady speed."
Grade 5–6	"The weight pulling the skydiver down is equal in size to the air resistance pushing up. Because they are equal and opposite, the resultant force is zero, so the skydiver moves at constant velocity."
Grade 7–9	"Weight is a vector acting vertically downward; air resistance is a vector acting vertically upward. At terminal velocity these vectors have equal magnitude, so the vector sum (the resultant force) is zero. By Newton's First Law, no resultant force means no change in velocity — the skydiver continues at constant velocity. This is an example of dynamic equilibrium: the object moves, but the forces are balanced."

Use precise terms such as scalar vs vector, resultant force, dynamic equilibrium, and link to Newton's three laws where appropriate to access Grade 7–9 marks.

Edexcel alignment: This content is aligned with Edexcel GCSE Physics (1PH0) specification Topic 2 Motion and forces — specifically 2.1 Motion (vectors and scalars), 2.2 Newton's laws (equilibrium and resultant force), and 2.3 Forces and braking (force diagrams). Assessed on Paper 1.