Applied Maths: Newton's Laws and Connected Particles (Strand 4) — Higher Level Notes

1. The three laws, stated usefully

• First law. A body stays at rest or moves at constant velocity unless a resultant force acts. So constant speed means the forces balance — a parcel sliding along the floor of a van at a steady 15 km/h has a resultant force of exactly zero on it.
• Second law. F = ma, where F is the resultant force.
• Third law. Every action has an equal and opposite reaction — acting on a different body.

2. Weight, tension and normal reaction

Weight is W = mg, always vertically downwards. Tension acts along a string, away from the body. The normal reaction acts perpendicular to the surface — which on a slope is not vertical.

On a plane inclined at angle A, the reaction balances only the perpendicular part of the weight:

R = mg cos A

So R is smaller than the weight, and it shrinks further as the slope steepens while the component pulling the body down the slope, mg sin A, grows. That is precisely why a block eventually slips.

3. The laws of friction

• Friction acts along the surfaces, opposing relative motion — never perpendicular to them.
• Friction is independent of the area of contact.
• Friction never exceeds μR, where μ is the coefficient of friction.

The subtle one: below slipping, friction is not automatically μR. It takes whatever value equilibrium requires, up to that limit. Only when the body is on the point of moving does F = μR.

Note also that μ is a ratio of two forces and so carries no units. Multiplying it by a mass instead of a weight is a frequent slip: for a 10 kg block with μ = 2/5, the limiting friction is (2/5)(10)(9.8) = 39.2 N, not 4 N.

4. Connected particles

Two masses joined by a light inextensible string over a smooth pulley share one acceleration, and the tension is the same throughout the string. The method is fixed:

1. Draw each particle separately with every force on it.
2. Write F = ma for each, taking the direction of motion as positive.
3. Add the equations to eliminate T, or subtract to find it.

For a hanging particle, the equation of motion is T - mg = ma. Getting that sign wrong is the difference between an accelerating and a decelerating system.

5. Particles on slopes

Resolve along and perpendicular to the plane, never horizontally and vertically. Along the slope you have mg sin A driving the motion and friction μR opposing it; perpendicular you have R = mg cos A.

If the body is on the point of sliding down an unaided slope, mg sin A = μmg cos A, so μ = tan A — a result worth recognising instantly.