Mathematical modelling in mechanicsEdexcel International A Level Maths: Revision notes

Mathematical modelling means turning a real situation into a simplified mathematical problem so that it can be solved. The cycle is:

1. State the real problem and decide what to include.
2. Make assumptions that simplify it, and set up equations.
3. Solve and obtain predictions.
4. Compare the predictions with real data.
5. If they disagree, refine the model by relaxing an assumption and repeat. Every model is a simplification, so you must be able to say what each assumption means for the forces and motion, and in which direction it makes the prediction wrong.

• A particle has mass but no size, so all of its mass is at one point. Rotation, size and air resistance on its shape are ignored. Use it for a ball, a car or a person when their dimensions are small compared with the distances involved.
• A lamina is a flat object of negligible thickness (a thin sheet), with mass spread over its area.
• A rigid body is an object of finite size that does not bend or change shape. Assuming a particle means the weight acts at a single point and there are no turning effects.

• A rod is a long thin rigid body, treated as having length but no thickness.
• A light rod has negligible mass; a uniform rod has its mass evenly distributed, so its weight acts at the midpoint; a non-uniform rod has its mass unevenly distributed, so its weight acts at a point other than the midpoint.
• A light string has negligible mass, so the tension is the same throughout (while the string is straight and not passing over a rough surface).
• An inextensible string does not stretch, so connected particles have the same speed and acceleration along the string.
• A light smooth pulley has no mass and no friction, so the tension is the same on both sides of the pulley. Example: for masses 3 kg and 5 kg over such a pulley, 5g − T = 5a and T − 3g = 3a, giving a = 2.45 m s⁻².

• A smooth surface exerts no friction, only a normal reaction perpendicular to the surface.
• A rough surface can exert friction, which opposes motion or tendency to move along the surface.
• A bead is a particle with a hole through it, so it can slide along a wire or string.
• A wire is a thin rigid fixed support; a bead on a smooth wire experiences a reaction perpendicular to the wire.
• A peg is a fixed support of negligible size; a string or rod resting on a smooth peg experiences a reaction perpendicular to the string or rod, and tension in a string passing over a smooth peg is unchanged.

To evaluate a model, link each assumption to a physical effect.

• Ignoring air resistance means predicted speeds and heights are too large, because resistance opposes motion.
• Treating a floor as smooth means predicted accelerations are too large.
• Treating a pulley as light and smooth ignores the extra force needed to turn it and the friction, so the predicted acceleration is too large.
• Treating a string as light and inextensible is fair for a thin string, but poor for a stretchy rope or a heavy chain. Refine by adding a resistance force, friction, a pulley mass or an extensible string, and compare predictions with data again.