Cambridge: Mechanics

Mechanics is the branch of mathematics that deals with the motion of objects and the forces that cause that motion. This chapter introduces the fundamental principles for modelling physical systems. You will learn to describe motion using concepts like displacement, velocity, and acceleration (kinematics), and then explain why that motion occurs using forces and Newton's Laws (dynamics). The main skill is translating a real-world scenario, like a car braking or a box sliding down a ramp, into a set of mathematical equations that can be solved.

The chapter covers several key areas. It begins with kinematics in a straight line, for both constant and variable acceleration. You will then study forces, including weight, friction, tension in strings, and the normal reaction force. A major part of the topic involves applying Newton's Laws of Motion to single particles and to systems of connected particles, such as those linked by a string passing over a pulley. Drawing clear force diagrams and resolving forces into components are essential skills you will develop.

The difficulty often lies in correctly identifying all the forces acting on an object and setting up the initial equations of motion, especially for inclined planes or systems with multiple bodies. A strong foundation in algebra, particularly solving simultaneous equations, is necessary. Basic trigonometry is used constantly for resolving forces, and knowledge of differentiation and integration from Pure Mathematics is required for problems involving variable acceleration.

Formulas for Cambridge: Mechanics

Equation of motion (velocity-time): $v = u + at$ For constant acceleration. u, v are initial and final velocities.
Equation of motion (displacement-time): $s = ut + \frac{1}{2}at^2$ For constant acceleration. s is displacement.
Equation of motion (velocity-displacement): $v^2 = u^2 + 2as$ For constant acceleration.
Velocity from displacement: $v = \frac{ds}{dt}$ For variable acceleration, where v is velocity and s is displacement.
Acceleration from velocity: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ For variable acceleration, where a is acceleration and v is velocity.
Newton's Second Law: $F_{net} = ma$ The net force on an object equals its mass times its acceleration.
Weight: $W = mg$ W is weight, m is mass, and g is the acceleration due to gravity.
Limiting Friction: $F_{max} = \mu R$ F_max is the maximum frictional force, μ is the coefficient of friction, R is the normal reaction force.