System of Forces in A Level Mechanics

System of Forces in A Level Mechanics (with planes + friction, the friendly way)

Systems of forces come up in almost every A Level Mechanics paper — equilibrium, slopes, connected particles, or awkwardly angled pulls. The maths is rarely hard, but the setup catches people out.

Here’s how to think about it like a teacher would:
🔁 Draw → Resolve → Equations → Check.

You’ll meet the same process across AQA, Edexcel, and OCR. Once you lock the routine, the rest is just tidy algebra.

🔙 Previous topic:

“Return to variable acceleration before moving continuing with system of forces.”

📏 1) Core idea — what “system of forces” really means

A system means you’ve got more than one force acting on an object (or system of objects). For equilibrium, everything balances out:

Sum of horizontal forces = 0
Sum of vertical forces = 0
And if rotation’s involved, sum of moments = 0

✅ Exam marker tip (OCR/AQA): Always write “Take right and up as positive.” It’s a small line that earns method credit and saves a sign error later.

🔁 2) The 4-step method you can rely on

1️⃣ Draw a neat free-body diagram — include weight (mg), normal reaction (R), friction (F), tensions, pushes, and pulls.
2️⃣ Resolve in two perpendicular directions — on slopes, that means along and perpendicular to the plane.
3️⃣ Equations: for equilibrium, set each resolved sum = 0. For motion, use F = ma.
4️⃣ Check: are directions correct? Is friction at its limit (F = μR) or just below (F < μR)?

🧠 In my lessons, I always say: “Axes first, arrows second, algebra third.” Most mistakes come from skipping step one.

📚 3) Worked example — three forces and a mystery

The system below is in equilibrium. Find the missing force P and the angle θ.

(Picture it: forces of 6 N at 80°, 4 N at 70°, and the unknown P at angle θ.)

Take right and up as positive.

Horizontal:
6 cos80 + P sinθ − 4 cos70 = 0
⟹ P sinθ = 0.326

Vertical:
6 sin80 + 4 sin70 − 5 − P cosθ = 0
⟹ P cosθ = 4.668

Now divide:
tanθ = (P sinθ) / (P cosθ) = 0.326 / 4.668
⟹ θ ≈ 4.0°

Back-substitute: P ≈ 4.68 N

✅ Mark scheme tip (Edexcel): The cleanest full-mark setup shows both resolves and the tan step. Avoid skipping straight to a calculator line.

📏 4) Resolving on a plane (or slope)

When forces act on an incline of angle θ:

Component of weight down the slope = mg sinθ
Component perpendicular to slope = mg cosθ
Normal reaction: usually R = mg cosθ (unless there’s an extra vertical force)

❗ Common slip-up: Using R = mg on a slope. Nope — it’s mg cosθ.

Example

A 10 kg mass rests on a plane inclined at 20°. Find the components of its weight parallel and perpendicular to the plane.

Weight = 10 × 9.8 = 98 N

Parallel (down the slope): 98 sin20 ≈ 33.5 N
Perpendicular (into the slope): 98 cos20 ≈ 92.1 N

🧠 Quick memory aid: “Sin goes with the slope, cos goes with the crush.”

✅ 5) Choosing between sine and cosine

Here’s the rule of thumb teachers love to repeat:
If your force lies along your chosen axis, multiply by cos of the included angle.
If it’s angled away, use sin.

Or — just use the complementary angle and cosine anyway. That’s why 10 sin30 = 10 cos60.

The key thing? Label your angle on the diagram before deciding. That’s where most people slip up.

❗ 6) Friction — the force everyone mislabels

Friction always acts opposite to intended motion, not necessarily opposite to movement (if it hasn’t moved yet).

Three key cases:

Static friction: F ≤ μR (adjusts to whatever’s needed to prevent motion)
Limiting equilibrium: F = μR (about to move)
Kinetic friction: F = μₖR (sliding)

✅ Exam board habit: AQA says “rough plane” once and never reminds you again. If it’s rough, draw that friction arrow from the start — saves you from a missing-mark moment later.

Mini Example (Horizontal surface)

A 10 kg block is pushed along a rough floor. The force needed to overcome friction is 10 N. Find μ.

Vertical: R = 10g = 98 N
Horizontal: F = μR ⇒ 10 = μ × 98
⟹ μ = 10/98 ≈ 0.10

That’s all there is — just keep your directions consistent.

🧗 7) Friction on a plane (limiting case)

A 6 kg mass rests in limiting equilibrium on a rough plane inclined at 30° to the horizontal. Find μ.

“Limiting” means it’s just about to move — so F = μR.

Perpendicular to plane: R = 6g cos30
Parallel to plane: weight component = 6g sin30
At limit: F = μR ⇒ 6g sin30 = μ × 6g cos30
⟹ μ = tan30 = 1/√3 ≈ 0.577

✅ Exam tip (OCR/Edexcel): Always include “At limiting equilibrium, friction = μR.” It’s a mark on its own.

📏 8) Quick reference — resolving templates

Here’s your mini “cheat sheet” for the big three scenarios 👇

Situation	Along Plane / Horiz.	Perpendicular
Smooth slope	T − mg sinθ = 0	R = mg cosθ
Rough slope (limiting)	T − μR − mg sinθ = 0	R = mg cosθ
Horizontal rough surface	F = μR	R = mg

🧠 Remember: if acceleration is involved, replace the zeros with ma in the direction of motion.

🧠 Teacher aside — draw before you solve

Actually—hang on—this one’s huge. In mechanics, I tell students:

“Don’t touch the calculator until your arrows make sense.”

If your arrows are wrong, everything else collapses. A neat free-body diagram beats algebra every time.

❗ 9) Classic exam traps (and how to dodge them)

🚫 Forgetting to show the friction arrow.
✅ Fix: draw it even if you’re not sure. If it’s the wrong way, the negative sign will tell you.

🚫 Assuming R = mg on an incline.
✅ Fix: use R = mg cosθ. Always.

🚫 Treating μR as fixed when not limiting.
✅ Fix: write “F ≤ μR” unless the question says “about to move.”

🚫 Mixing sin and cos on slopes.
✅ Fix: sin for slope (parallel), cos for crush (perpendicular).

🚫 Forgetting direction assumptions.
✅ Fix: start every problem with “Take right/up as positive.” It anchors your algebra.

🧠 Reflection — what students usually get wrong

I once had a student — brilliant algebra, neat handwriting — who lost four marks on an Edexcel mechanics paper because they wrote “R = mg” on an incline. When we reviewed it, they said, “Oh, I forgot the slope was there.”

It’s never the equations; it’s the awareness.
So slow down. Mark your axes. Check which way’s up the plane. Write μR only when it’s limiting. Those tiny checks are what turn 7/10 into 10/10.

🚀 Next steps

Start your revision for A Level Maths today with our A Level Maths revision classes, where we break down mechanics, statistics, and pure maths step by step — using real AQA, Edexcel, and OCR examples.

It’s designed to make tricky topics click before the exam — and to stop those “sign error” heartbreaks once and for all.

About the Author

S. Mahandru is the Head of Mathematics at Exam.tips, specialising in A Level and GCSE Mathematics education. With over a decade of classroom and online teaching experience, he has helped thousands of students achieve top results through clear explanations, practical examples, and applied learning strategies.

Updated: November 2025

🧭 Next topic:

“Next, explore projectile motion — one of mechanics’ signature ideas.”

System of Forces in A Level Mechanics