Forces and Newton's first lawEdexcel A-Level Maths: Revision notes

A force is a push or a pull that can change the velocity of a body or deform it. Force is a vector: it has a magnitude and a direction. Its unit is the newton (N). Forces acting on a body combine to give a resultant force, found by vector addition, taking direction into account. Forces can be given in components, as $(a\mathbf{i}+b\mathbf{j})$ N, or as column vectors. The resultant is the vector sum, and its magnitude is $\sqrt{a^2+b^2}$. Example: $3$ N east and $4$ N north combine to a resultant of $\sqrt{3^2+4^2}=5$ N.

• Weight $W$: the gravitational force on a body, acting vertically downwards (towards the centre of the Earth).
• Normal reaction $R$: the force exerted by a surface on a body in contact with it, acting perpendicular to the surface.
• Tension $T$: the pulling force in a string, rope or cable, acting along it and away from the body it pulls.
• Thrust or compression: the pushing force in a rod or strut, along its length.
• Resistance: a force that opposes motion, such as friction between surfaces or air resistance. Also useful is the driving force, the force produced by an engine. Draw a force diagram for the body, showing every force with an arrow in its correct direction, before writing any equation.

Newton's first law: a body remains at rest, or continues to move with constant velocity (constant speed in a straight line), unless acted on by a resultant force. Equivalently:

• Resultant force zero $\Rightarrow$ the body's velocity does not change.
• Velocity changing $\Rightarrow$ a non-zero resultant force acts. A force is needed to change motion, not to keep it going. A car travelling at constant speed has a driving force exactly equal to the resistance, giving a zero resultant.

If a body is at rest or moving with constant velocity, the resultant force is zero. In vector form the forces sum to $\mathbf{0}$; in scalar form the forces balance in each of two perpendicular directions. Example: a box of weight $200$ N at rest on a floor with a horizontal push of $35$ N. Vertically $R=200$ N; horizontally friction $F=35$ N, opposing the push. Example in vectors: $\mathbf{F}_1=(3\mathbf{i}+5\mathbf{j})$, $\mathbf{F}_2=(-7\mathbf{i}+2\mathbf{j})$ and $\mathbf{F}_3=(p\mathbf{i}+q\mathbf{j})$ with constant velocity. Then $3-7+p=0$ and $5+2+q=0$, so $\mathbf{F}_3=(4\mathbf{i}-7\mathbf{j})$ N, with magnitude $\sqrt{65}\approx8.06$ N. The angle of a force with $\mathbf{i}$ is found from $\tan\theta=\frac{q}{p}$, checking the quadrant.

Questions often ask you to explain a situation using Newton's first law. Structure the answer: (1) name the forces; (2) say whether the velocity is constant; (3) state that the resultant force is zero (or not); (4) conclude. Example (skydiver): falling at constant speed, air resistance equals weight, so the resultant is zero. Before the parachute opens, the resistance is smaller than the weight, so there is a downward resultant and the skydiver accelerates until the resistance grows to equal the weight. Modelling words: a particle has no size; a light string has no mass; a smooth surface has no friction; inextensible means the string does not stretch.