1  Units, Vectors, and Scalar Quantities

1.1 SI Units

The book adopts the Système International (SI) of units. The base units most relevant to mechanics are:

Quantity Unit Symbol
Mass kilogram kg
Length metre m
Time second s
Force newton N
Energy / Work joule J
Power watt W
Pressure / Stress pascal Pa

Derived units follow from these. For example:

\[1 \text{ N} = 1 \text{ kg·m/s}^2 \qquad 1 \text{ Pa} = 1 \text{ N/m}^2 \qquad 1 \text{ J} = 1 \text{ N·m}\]

1.2 Scalar and Vector Quantities

Scalar quantities have magnitude only and obey ordinary arithmetic:

  • Examples: mass, speed, energy, temperature, time

Vector quantities have both magnitude and direction and must be combined using vector algebra:

  • Examples: force, velocity, acceleration, displacement, momentum

A vector is represented graphically as an arrow — its length denotes magnitude, its orientation denotes direction.

1.3 Vector Addition

1.3.1 Triangle Rule

Two vectors \(\mathbf{A}\) and \(\mathbf{B}\) are added tip-to-tail; the resultant \(\mathbf{R}\) closes the triangle:

\[\mathbf{R} = \mathbf{A} + \mathbf{B}\]

1.3.2 Parallelogram Rule

Both vectors are drawn from the same point; the resultant is the diagonal of the parallelogram they form.

1.3.3 Polygon Rule

For more than two vectors, they are placed tip-to-tail in sequence; the resultant closes the polygon from the tail of the first to the tip of the last.

1.4 Vector Subtraction

Subtracting \(\mathbf{B}\) from \(\mathbf{A}\) is equivalent to adding the negative of \(\mathbf{B}\):

\[\mathbf{R} = \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})\]

1.5 Resolution of Vectors

Any vector \(\mathbf{F}\) can be resolved into two perpendicular components. For a vector at angle \(\theta\) to the horizontal:

\[F_x = F\cos\theta\] \[F_y = F\sin\theta\]

This is the most widely used technique in the book — resolving all forces into horizontal and vertical components before applying equilibrium conditions.

1.6 Resultant of Several Vectors

To find the resultant of a system of vectors analytically:

  1. Resolve each vector into \(x\) and \(y\) components
  2. Sum all components in each direction:

\[\Sigma F_x = F_{1x} + F_{2x} + \cdots \qquad \Sigma F_y = F_{1y} + F_{2y} + \cdots\]

  1. Find the magnitude of the resultant:

\[R = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2}\]

  1. Find the direction:

\[\theta = \arctan\left(\frac{\Sigma F_y}{\Sigma F_x}\right)\]

1.7 The Laws of Sines and Cosines

For cases involving triangles of vectors, these laws provide an analytical alternative to graphical methods.

Sine rule:

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

Cosine rule:

\[c^2 = a^2 + b^2 - 2ab\cos C\]

1.8 Summary of Equations

Concept Equation
Horizontal component \(F_x = F\cos\theta\)
Vertical component \(F_y = F\sin\theta\)
Resultant magnitude \(R = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2}\)
Resultant direction \(\theta = \arctan(\Sigma F_y / \Sigma F_x)\)
Cosine rule \(c^2 = a^2 + b^2 - 2ab\cos C\)

1.9 Summary

The resolution-and-recombination technique — breaking every vector into perpendicular components, summing each direction separately, then reconstructing the resultant — is the single most important skill introduced in this chapter. It underpins equilibrium analysis, dynamics, stress transformation, and virtually every other topic in the book.