Mathematics · Chapter 8
Study notes aligned to the official NEB syllabus.
Mechanics is the branch of applied mathematics that studies forces and motion. This chapter has two parts. Statics deals with forces in balance: how two or more forces acting at a point combine into a single resultant, how a single force is split into components, and the conditions for a particle to stay in equilibrium (the parallelogram law and Lami's theorem). Dynamics deals with motion: here we study motion in a straight line under a constant acceleration and the three equations of motion that link displacement, velocity, acceleration and time. A force is a vector, having both magnitude and direction, so combining forces uses vector rules, not ordinary addition.
A single force that has the same effect as two or more given forces is called their resultant. For two forces acting at a point the resultant is found by the parallelogram law of forces:
If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point.
If the two forces are $P$ and $Q$ with angle $\theta$ between them, the magnitude of the resultant $R$ and the angle $\alpha$ it makes with $P$ are
$$ \begin{aligned} R &= \sqrt{P^2 + Q^2 + 2PQ\cos\theta}, \ \tan\alpha &= \frac{Q\sin\theta}{P + Q\cos\theta}. \end{aligned} $$
Three special cases are worth remembering:
Worked example. Two forces of $5,\text{N}$ and $3,\text{N}$ act at a point with $60^{\circ}$ between them. Then
$$ \begin{aligned} R &= \sqrt{5^2 + 3^2 + 2\cdot 5\cdot 3\cos 60^{\circ}} \ &= \sqrt{25 + 9 + 30\cdot\tfrac{1}{2}} \ &= \sqrt{49} = 7,\text{N}, \end{aligned} $$
and the direction is given by
$$ \begin{aligned} \tan\alpha &= \frac{3\sin 60^{\circ}}{5 + 3\cos 60^{\circ}} \ &= \frac{3\cdot 0.866}{5 + 1.5} \ &= \frac{2.598}{6.5} \ &= 0.400 \end{aligned} $$
so $\alpha \approx 21.8^{\circ}$ measured from the $5,\text{N}$ force.
Mechanics is the branch of applied mathematics that studies forces and motion. This chapter has two parts. Statics deals with forces in balance: how two or more forces acting at a point combine into a single resultant, how a single force is split into components, and the conditions for a particle to stay in equilibrium (the parallelogram law and Lami's theorem). Dynamics deals with motion: here we study motion in a straight line under a constant acceleration and the three equations of motion that link displacement, velocity, acceleration and time. A force is a vector, having both magnitude and direction, so combining forces uses vector rules, not ordinary addition.
A single force that has the same effect as two or more given forces is called their resultant. For two forces acting at a point the resultant is found by the parallelogram law of forces:
If two forces acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point.
If the two forces are and with angle between them, the magnitude of the resultant and the angle it makes with are
Three special cases are worth remembering:
Worked example. Two forces of and act at a point with between them. Then
and the direction is given by
so measured from the force.