Resolving Forces, Inclined Planes & Friction

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Chapter Overview

Resolving Forces

Split any force into horizontal (F cosθ) and vertical (F sinθ) components using SOHCAHTOA.

Inclined Planes

Resolve weight parallel (mg sinθ) and perpendicular (mg cosθ) to the slope. Apply F = ma along the plane.

Coefficient of Friction

Friction satisfies F ≤ μR. At its max: F_MAX = μR. Object moves only when the applied force exceeds F_MAX.

Friction & F = ma

Four steps: resolve → find R → find F_MAX → apply Newton's Second Law.

Connected Particles

Friction only on particles touching a rough surface. Compare mg sinθ values to decide which way the system moves.

Limiting Equilibrium

"On the point of moving" means F = μR exactly. For stationary objects, only assume F = μR when told the object is about to move.

Resolving Forces

Topic 1 — Resolving Forces & Finding Resultants

Imagine pushing a shopping trolley at an angle — part of your push moves the trolley forward, part lifts it slightly off the ground. That's exactly what resolving forces does: it separates the useful part of a force from the wasteful part.

Splitting a Force into Components

When a force F acts at angle θ above the horizontal, draw a right-angled triangle with F as the hypotenuse. Using SOHCAHTOA:

The Resolution Formula

Horizontal = F cosθ Vertical = F sinθ

cosine → the direction you measure θ from | sine → the direction 90° away

F N (hypotenuse) ↗ /| / | ← F sinθ (vertical component — pushes object up/down) / | /θ | ─────┘ F cosθ (horizontal component — moves object along)

Key insight: The two components together have the exact same effect as the original force. You're not adding anything or losing anything — you're just changing perspective.

Which is cos, which is sin? Cosine always goes with the component adjacent to θ. Sine always goes with the component opposite to θ. Draw a tiny triangle every time if unsure.

Finding the Resultant of Two Forces

When two forces act at different angles, they combine into one resultant force. You can either:

Resolve both into horizontal and vertical, then add. Use Pythagoras + tan for final magnitude and direction.
Use the triangle law — arrange the two vectors tip-to-tail, then use the cosine rule for the missing side and the sine rule for the missing angle.

Cosine Rule (Resultant)

R² = a² + b² − 2ab cos(C)

Use this when you have two sides and the angle between them (the interior angle in the force triangle, not the original angle between forces).

Angle trick: If two forces make an angle α between them, the interior angle inside the force triangle is 180° − α (co-interior angles on a straight line). Always redraw and check your interior angle before applying the cosine rule.

Practice Question 1

A force of 20 N acts at 60° above the horizontal. Find its horizontal and vertical components. Leave answers in surd form.

Practice Question 2

Two forces of 6 N and 10 N act on a particle. The angle between them is 60°. Find the magnitude and direction of the resultant.

Inclined Planes

Topic 2 — F = ma on Inclined Planes

parallel to the slope perpendicular to the slope

Resolving Weight on a Slope

mg straight down

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Resolving Forces, Inclined Planes & Friction

Chapter Overview

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