Mechanics
Quantities & Units in Mechanics
Year 12 · Year 13
- ✓By the end of this lesson students will be able to recall and use the SI base units for mass, length, and time.
- ✓By the end of this lesson students will be able to understand and use derived SI units for common mechanical quantities.
- ✓By the end of this lesson students will be able to check the consistency of units in equations using dimensional analysis.
- ✓By the end of this lesson students will be able to understand the concept of a mathematical model in mechanics.
- ✓By the end of this lesson students will be able to identify and state appropriate modelling assumptions for given physical situations.
Key concepts
The International System of Units (SI) provides a coherent system of units for scientific and engineering measurements. In mechanics, the fundamental base units are for mass, length, and time. All other mechanical units can be derived from these three.
Derived units are combinations of base units. For example, velocity is a derived unit (length per time), and force is another (mass times acceleration). Understanding how derived units are formed from base units is crucial for dimensional analysis and ensuring consistency in calculations.
Dimensional analysis is the process of checking the consistency of units in an equation. Both sides of a valid physical equation must have the same base units. This technique can help identify errors in formulae or unit conversions.
A mathematical model is a simplified representation of a real-world situation, designed to make it amenable to mathematical analysis. Real-world problems are often too complex to solve directly, so assumptions are made to create a manageable model. The accuracy of the model depends on the appropriateness of the assumptions.
When creating a mathematical model in mechanics, specific assumptions are often made to simplify the problem. These assumptions are vital for solving the problem and must be clearly stated. Common assumptions include:
An object whose mass is concentrated at a single point. Its dimensions are negligible. This assumption is used when the size, shape, or rotational effects of an object are not relevant to the problem (e.g., a ball thrown through the air, where air resistance is considered but its spin is not).
An object whose shape and size do not change under the action of applied forces. This assumption is used when the dimensions of an object are important (e.g., for moments or stability) but it does not deform (e.g., a ladder leaning against a wall).
A string or rod whose mass is negligible compared to the masses it connects or supports. This implies that the tension in a light string is uniform throughout its length (assuming no external forces acting along the string itself).
A string whose length does not change under tension. This implies that objects connected by an inextensible string will have the same magnitude of acceleration along the line of the string.
A surface or pulley where there is no friction. For a smooth surface, the reaction force is perpendicular to the surface. For a smooth pulley, the tension in the string passing over it is the same on both sides.
A surface where friction is present. The frictional force opposes motion or the tendency of motion and its maximum value is given by F_max = μR, where μ is the coefficient of friction and R is the normal reaction force.
A rod or lamina (flat plate) where the mass is evenly distributed throughout its length or area. This means its centre of mass is at its geometric centre.
The resistive force exerted by the air on a moving object is ignored. This simplifies calculations significantly, as air resistance is often a complex function of velocity and shape.
The acceleration due to gravity (g) is assumed to be constant and uniform, typically taken as 9.8 m s⁻² (or sometimes 9.81 m s⁻² or 10 m s⁻² if specified) and acting vertically downwards.
Key facts to remember
- 1The three SI base units in mechanics are the metre (m) for length, the kilogram (kg) for mass, and the second (s) for time.
- 2Derived units are combinations of base units, e.g., velocity (m s⁻¹), acceleration (m s⁻²).
- 3The Newton (N) is the SI unit of force, equivalent to kg m s⁻².
- 4The Joule (J) is the SI unit of energy/work, equivalent to N m or kg m² s⁻².
- 5The Watt (W) is the SI unit of power, equivalent to J s⁻¹ or kg m² s⁻³.
- 6A 'particle' is a model where an object's mass is concentrated at a single point, ignoring its dimensions.
- 7A 'light' string or rod has negligible mass.
- 8An 'inextensible' string has a constant length, meaning connected objects have the same acceleration.
- 9A 'smooth' surface or pulley implies no friction.
Worked examples
Example 1
Show that the equation for kinetic energy, E_k = 1/2 mv², is dimensionally consistent, where m is mass and v is velocity.
Answer
The equation E_k = 1/2 mv² is dimensionally consistent as both sides have the base units kg m² s⁻².
Dimensional analysis only checks for consistency of units, not the correctness of numerical constants.
Example 2
A student throws a tennis ball vertically upwards. List three suitable modelling assumptions that could be made to analyse its motion.
Answer
Suitable modelling assumptions are: 1. The tennis ball is modelled as a particle. 2. Air resistance is negligible. 3. The acceleration due to gravity is constant (g = 9.8 m s⁻²).
Other valid assumptions could include 'the Earth is a fixed frame of reference' or 'no wind'.
Example 3
Determine the SI base units of pressure, given that Pressure = Force / Area.
Answer
The SI base units of pressure are kg m⁻¹ s⁻² (which is equivalent to the Pascal, Pa).
Pressure is often given in Pascals (Pa), where 1 Pa = 1 N m⁻². This example shows how to break down a derived unit into its fundamental base units.
Common mistakes
- ✗Using non-SI units (e.g., grams, centimetres, km/h) in calculations without converting them to kilograms, metres, and metres per second, respectively.
- ✗Confusing the modelling assumptions 'light' (negligible mass) with 'smooth' (no friction).
- ✗Forgetting to explicitly state all modelling assumptions made at the start of a problem, especially when asked to do so.
- ✗Incorrectly deriving the base units for complex quantities, often making errors with negative exponents or powers.
- ✗Assuming air resistance is always negligible without the problem stating it or without explicitly making and stating the assumption.
Exam tips
- ★Always work in SI units (m, kg, s) unless the question specifically instructs otherwise. Convert all given values to SI units before starting calculations.
- ★When asked to state modelling assumptions, be precise with your terminology (e.g., 'particle', 'light', 'inextensible', 'smooth', 'air resistance negligible').
- ★Use dimensional analysis to check the consistency of your equations. If the units don't match on both sides, your equation is incorrect.
- ★Practise deriving the base units for various physical quantities to become proficient in dimensional analysis.
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