CAIE A-Level · Mathematics 9709 · Newton's Laws of Motion

Newton's Laws of Motion: Linear Motion of a Particle (9709 Mechanics 4.4)

9 min readSyllabus 4.4PreviewBy Uzair Khan

Syllabus objective

Apply Newton's laws of motion to the linear motion of a particle of constant mass moving under the action of constant forces, which may include friction, tension in an inextensible string and thrust in a connecting rod. If any other forces resisting motion are to be considered (e.g. air resistance) this will be indicated in the question.

Introduction

Newton's Laws of Motion are the engine of classical mechanics. In 9709 Mechanics, the most frequently examined skill is applying Newton's Second Law (F=maF = ma) to find the acceleration of, or the forces acting on, a particle moving in a straight line. Questions routinely feature inclined planes, connected particles joined by strings, friction, and resisting forces. Mastery of this topic is essential: it underpins every dynamics question in the paper.


Core Concept

Newton's Three Laws (as applied in 9709)

  • First Law: A particle remains at rest or moves with constant velocity unless acted upon by a resultant force.
  • Second Law: The resultant force on a particle equals the product of its mass and acceleration: Fnet=maF_{\text{net}} = ma.
  • Third Law: If particle A exerts a force on particle B, then B exerts an equal and opposite force on A.

Setting Up an Equation of Motion

The strategy is always the same:

  1. Draw a force diagram — mark every force acting on the particle.
  2. Choose a positive direction (usually the direction of motion or of acceleration).
  3. Apply Fnet=maF_{\text{net}} = ma, taking forces in the positive direction as positive and those opposing it as negative.
  4. Solve for the unknown.

Key Force Types

ForceSymbolNotes
WeightW=mgW = mgAlways acts vertically downward
Normal reactionRR or NNPerpendicular to the surface; found from equilibrium perpendicular to motion
FrictionFrμRF_r \leq \mu ROpposes motion; at limiting friction Fr=μRF_r = \mu R
TensionTTActs along a string, away from the particle
ThrustPPActs along a connecting rod, towards the particle (compressive)
Air resistance / driving forceGiven in questionDirection stated or deducible

Normal Reaction on an Incline

For a particle on a plane inclined at angle α\alpha to the horizontal, resolving perpendicular to the plane (no acceleration in that direction):

R=mgcosαR = mg\cos\alpha

The component of weight along the plane (down the slope) is mgsinαmg\sin\alpha.


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