Fundamental Mechanical Quantities and Measures

Angles are defined as a dimensionless ratio of the arc length ($a$) to the radius ($r$) of the circle: $\theta_{\text{rad}} = \frac{a}{r}$

There are three common ways to express these measures:

1. Radians (rad): The standard scientific unit, derived from the ratio mentioned above.
2. Degrees ($^\circ$): A common unit where a full circle is divided into 360 parts.
3. Revolutions (rev): Used to describe complete turns or rotations.

The relationship between these units is fixed: $1 \text{ circle} = 360^\circ = 2\pi \text{ rad} = 1 \text{ rev}$

Exam Tip: Remember that an angle is a dimensionless ratio. When calculating angular displacement, ensure your units (rad vs. degrees) match the requirements of the formula you are using.

Time ($t$)

Time is treated as a monotonically increasing, uni-directional quantity. In physics, this is often referred to as "entropy's arrow," implying that time moves forward and cannot be reversed. In the SI system, the standard unit for time is the second (s). Time is the denominator in almost every rate-based calculation, from velocity to power.

Matter: Mass, Volume, and Density

Matter refers to any physical body or substance that occupies space—whether in solid, liquid, gas, or plasma form—and can be perceived. To quantify matter, we use three interconnected concepts: mass, volume, and density.

Mass ($m$)

Mass is a measure of how much matter is contained within a body. It is also the physical property that provides inertia, which is the resistance of an object to acceleration. The SI unit for mass is the kilogram (kg).

Volume ($V$)

Volume is the amount of three-dimensional space a material body occupies. The SI unit is the cubic metre ($\text{m}^3$). In practical applications, we often use the litre (L), where: $1 \text{ L} = 0.001 \text{ m}^3$ This is equivalent to a cube measuring $10\text{ cm} \times 10\text{ cm} \times 10\text{ cm}$.

Density ($\rho$)

Density is the ratio of an object's mass to the volume it occupies. It describes how "tightly packed" the matter is within a space. The formula for density is: $\rho = \frac{m}{V}$ The standard SI unit is $\text{kg/m}^3$.

Worked Example: Calculating Density If you have a block of material with a mass of $2,000\text{ kg}$ and it occupies a volume of $2\text{ m}^3$, what is its density?

Using the formula: $\rho = \frac{2,000\text{ kg}}{2\text{ m}^3}$ $\rho = 1,000\text{ kg/m}^3$

Note that this is exactly the density of water at standard temperature and pressure.

Key Concept: Mass and Weight are not the same. Mass is the amount of matter (measured in kg), while weight is the force exerted by gravity on that mass (measured in Newtons).

Summary Table of Fundamental Quantities