Quantities and Units
Mechanics is the branch of physics and mathematics concerned with the motion of objects and the forces that cause or change that motion. Before solving any mechanics problem, you need a precise language for describing the physical world — this language is built from quantities and units. A physical quantity is anything that can be measured, such as mass, time, velocity, or force. Every quantity must be expressed with both a numerical value and a unit; saying a car travels '60' is meaningless without knowing whether that is kilometres per hour, metres per second, or miles per hour. The international standard for scientific measurement is the Système International d'Unités, universally abbreviated SI. The SI system defines seven base units from which all other units in mechanics — and indeed all of science — can be derived. In mechanics, the three most important base units are the metre (m) for length, the kilogram (kg) for mass, and the second (s) for time. Every other mechanical quantity, such as speed, acceleration, force, and energy, can be expressed in terms of these three fundamental building blocks. A crucial distinction in mechanics separates scalar quantities from vector quantities. A scalar has only a magnitude — a single number with a unit. A vector has both a magnitude and a direction in space. Getting this distinction right is not merely a formality; mixing scalars and vectors, or ignoring direction, leads to physically wrong answers. Speed is a scalar, but velocity is a vector. Understanding quantities and units rigorously is the essential first step before tackling kinematics, dynamics, or any other area of mechanics.
- The SI Base Units in Mechanics
- Derived Quantities and Their Units
- Dimensional Analysis
- Scalars and Vectors
- Vector Representation and Components
The SI Base Units in Mechanics
The SI system has seven base units, but mechanics uses three most frequently. The metre (m) measures length or distance. The kilogram (kg) measures mass — note that mass is not the same as weight; mass is the amount of matter in an object, while weight is a force. The second (s) measures time. All three are defined with extreme precision by international standards bodies: the metre is defined via the speed of light, the kilogram via Planck's constant, and the second via atomic vibrations of caesium-133.
Derived Quantities and Their Units
Derived quantities are formed by combining base units through multiplication or division. For example, speed = distance ÷ time, so its SI unit is m/s (metres per second), written m s⁻¹. Acceleration = change in velocity ÷ time, giving units of m s⁻². Force is defined by Newton's second law, F = ma, so its unit is kg m s⁻², which is given the special name newton (N): 1 N = 1 kg m s⁻². Energy and work have units of N m = kg m² s⁻², called the joule (J). Pressure is force per unit area: N m⁻² = kg m⁻¹ s⁻², called the pascal (Pa). This dimensional analysis — tracing units through formulas — is a powerful tool for checking whether equations are physically consistent.
Dimensional Analysis
Every valid physical equation must be dimensionally homogeneous: the dimensions on both sides must match. Dimensions are written using square brackets, so [length] = L, [mass] = M, [time] = T. For example, checking kinetic energy KE = ½mv²: the right side has dimensions [M][L T⁻¹]² = M L² T⁻², and the left side is energy = force × distance = [M L T⁻²][L] = M L² T⁻². They match, confirming the formula is dimensionally consistent. Dimensional analysis can also be used to derive the form of equations when the exact formula is unknown — a technique called the method of dimensions.
Scalars and Vectors
A scalar quantity is completely described by a magnitude alone. Examples in mechanics include: distance (e.g., 15 m), speed (e.g., 20 m s⁻¹), mass (e.g., 5 kg), time (e.g., 3 s), and energy (e.g., 100 J). A vector quantity requires both a magnitude and a direction. Examples include: displacement (e.g., 15 m due North), velocity (e.g., 20 m s⁻¹ at 30° above horizontal), acceleration (e.g., 9.8 m s⁻² downward), force (e.g., 50 N to the right), and momentum. Vectors are typically written in bold type (v) or with an arrow above the symbol (v⃗). The magnitude of a vector v is written |v| or simply v in italics.
Vector Representation and Components
A vector in two dimensions can be resolved into perpendicular components using trigonometry. If a velocity vector v has magnitude v and makes angle θ with the positive x-axis, its components are vₓ = v cos θ (horizontal) and vᵧ = v sin θ (vertical). Conversely, given components vₓ and vᵧ, the magnitude is v = √(vₓ² + vᵧ²) and the direction is θ = arctan(vᵧ/vₓ). This component method is essential for adding vectors: to add two vectors, add their x-components separately and their y-components separately, then reconstruct the resultant.
The Key Equation
- F — Net force (newton (N = kg m s⁻²)): The vector sum of all forces acting on the object; a vector quantity pointing in the direction of acceleration.
- m — Mass (kilogram (kg)): A scalar measure of the amount of matter in the object; also measures the object's resistance to changes in motion (inertia).
- a — Acceleration (metre per second squared (m s⁻²)): The rate of change of velocity with respect to time; a vector quantity whose direction matches that of the net force.
Worked Example
Problem. A force of 24 N acts on an object of mass 6 kg. (a) Calculate the acceleration produced. (b) Verify that the units are consistent by showing the dimensional analysis. (c) If the force acts at 30° above the horizontal, find the horizontal and vertical components of the force.
Answer. Acceleration = 4 m s⁻²; Horizontal component of force ≈ 20.8 N; Vertical component = 12 N.
Origins · 1 minGiovanni Giorgi and how this was discovered
Giorgi was the electrical engineer and physicist who proposed the foundations of what became the modern SI unit system. In 1901 he proposed a coherent four-unit system (metre, kilogram, second, and a fourth electromagnetic unit) that resolved the inconsistencies between mechanical and electrical units. His proposal was adopted internationally and eventually evolved into the full SI system formally established in 1960. Before Giorgi's proposal, scientists routinely used two incompatible systems — the CGS (centimetre-gram-second) system and practical electrical units — leading to constant conversion headaches. Giorgi's elegant solution unified them all.
Related Concepts
- Displacement and Velocity — Displacement and velocity are the first vector quantities encountered in mechanics, directly applying the scalar/vector distinction introduced here.
- Newton's Laws of Motion — Newton's laws define force and acceleration, the key derived quantities introduced in this concept, and require correct use of SI units throughout.
- Dimensional Analysis — Dimensional analysis extends the unit ideas here into a powerful technique for checking equations and deriving physical relationships.
- Vector Addition and Resolution — The component method introduced here for resolving a force at an angle is developed more fully in vector operations, covering general 2D and 3D cases.