Measurement and Vectors

Measurement and Vectors

Physics is an experimental science that relies on accurate measurements and a standard system of units. In engineering, precise measurement and vector analysis are crucial for designing structures and analyzing forces.

Units and Dimensions (SI System)

The International System of Units (SI) is the standard system used in physics and engineering. The seven base units are:

| Quantity | Unit Name | Symbol | |----------|-----------|--------| | Length | meter | m | | Mass | kilogram | kg | | Time | second | s | | Electric Current | ampere | A | | Thermodynamic Temperature | kelvin | K | | Amount of Substance | mole | mol | | Luminous Intensity | candela | cd |

Dimensional Analysis is a method used to check the consistency of equations. For example, velocity has dimensions of Length/Time $[L T^{-1}]$, and force has dimensions of Mass $\times$ Acceleration $[M L T^{-2}]$.

Scalar and Vector Quantities

Physical quantities are categorized as either scalars or vectors:

• Scalars: Quantities that have only magnitude (e.g., mass, time, temperature, distance, speed).
• Vectors: Quantities that have both magnitude and direction (e.g., displacement, velocity, acceleration, force).

Vectors are often represented graphically by arrows, where the length represents the magnitude and the arrowhead points in the direction.

Vector Addition and Components

Vectors can be added using geometric methods (triangle or parallelogram rule) or algebraic methods (component method).

Component Method

A vector $\mathbf{A}$ in 2D can be resolved into x and y components: $A_x = A \cos \theta$ $A_y = A \sin \theta$ where $A = |\mathbf{A}|$ is the magnitude and $\theta$ is the angle with the positive x-axis.

The magnitude and direction can be found from components: $A = \sqrt{A_x^2 + A_y^2}$ $\theta = \tan^{-1} \left( \frac{A_y}{A_x} \right)$

Vector Products

1. Dot Product (Scalar Product): The result is a scalar. $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \phi = A_x B_x + A_y B_y + A_z B_z$ The dot product is useful for calculating work done by a force: $W = \mathbf{F} \cdot \mathbf{d}$.

2. Cross Product (Vector Product): The result is a vector perpendicular to both operands. $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin \phi$ The direction is given by the right-hand rule. In components: $\mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ The cross product is essential for calculating torque: $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$.