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

A force is a push or pull on an object, and it is a vector: it has a magnitude and a direction. It is measured in newtons (N). Forces you meet at AS include weight (acting vertically downwards), the normal reaction (perpendicular to a surface), tension (along a string or rope) and resistance (opposing motion). The resultant force is the single force with the same effect as all the forces acting on a particle together. A force can be given by its magnitude and direction, or by components such as $(4\mathbf{i}+7\mathbf{j})$ N, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

Add forces as vectors. With components, add the $\mathbf{i}$ parts and the $\mathbf{j}$ parts separately. For two perpendicular forces of $12$ N and $5$ N, the magnitude is $\sqrt{12^2+5^2}=13$ N and the angle to the $12$ N force is $\tan^{-1}\frac{5}{12}=22.6^\circ$. For forces given as $x\mathbf{i}+y\mathbf{j}$, the magnitude is $\sqrt{x^2+y^2}$ and the angle to $\mathbf{i}$ is $\tan^{-1}\frac{|y|}{|x|}$, taking care over the quadrant.

Newton's first law: a particle remains at rest, or continues to move with constant velocity (constant speed in a straight line), unless acted on by a resultant force. So if the resultant force is zero, the velocity does not change, and if the velocity is constant, the resultant force is zero. A particle at rest, or moving with constant velocity, is in equilibrium. A non-zero resultant means the velocity changes, which is covered by Newton's second law. A car driving at constant speed along a straight level road has driving force equal to the resistance. A crate pulled at constant velocity by a $150$ N tension has a $150$ N resistance.

To use the first law in two dimensions, resolve each force into perpendicular components and set the sum in each direction to zero. A force $F$ at angle $\theta$ to a direction has component $F\cos\theta$ along it and $F\sin\theta$ perpendicular to it. On a plane inclined at $\alpha$ to the horizontal, the weight $W$ has component $W\sin\alpha$ down the plane and $W\cos\alpha$ into the plane. A particle held at rest on a smooth plane by a string parallel to the plane has $T=W\sin\alpha$ and $R=W\cos\alpha$.

1. List every force on the particle and give its direction.
2. Choose two perpendicular directions and resolve all forces.
3. Write "resultant $=0$" in each direction as an equation.
4. Solve, and state magnitudes with units (N). Example: $60$ N particle on a smooth $30^\circ$ plane, held by a horizontal string $S$. Vertically, $R\cos30^\circ=60$ so $R=69.3$ N. Horizontally, $S=R\sin30^\circ=34.6$ N. For vector forces, equilibrium means $\mathbf{F}_1+\mathbf{F}_2+\mathbf{F}_3=\mathbf{0}$. With $\mathbf{F}_1=(4\mathbf{i}+7\mathbf{j})$ N and $\mathbf{F}_2=(-9\mathbf{i}+2\mathbf{j})$ N, $\mathbf{F}_3=(5\mathbf{i}-9\mathbf{j})$ N.