Base Units and Derived Units: The Building Blocks of Physics

1. Introduction: Why Do We Need Base Units?

When we measure a physical quantity like 'length', we have a huge number of units to choose from. We could use meters, centimeters, kilometers, miles, or even light-years. While having options is great, in science it can lead to real confusion and inconsistency. To fix this and establish a universal standard that scientists and engineers can rely on, a set of fundamental units was established.

The key to understanding this system is a simple analogy: Think of base units as the Lego building blocks of all other units in physics. They are the fundamental pieces from which every other measurement is constructed.

2. The Six Fundamental Base Units

SI Base Units are the fundamental, mutually independent units upon which the entire International System of Units (SI) is built. This means they are the core building blocks and cannot be expressed in terms of each other.

For A-Level Physics, there are six fundamental quantities and their corresponding SI Base Units you need to know:

Base Quantity

SI Base Unit

1. 1. Mass     kilogram (kg)

   2. Length    meter (m)

   3. Time   second (s)

   4. Current    Ampere (A)

   5. Temperature    Kelvin (K)

   6. Amount of substance   mole (mol)

1. Special Note on Mass: The kilogram is the only base unit that has a prefix (kilo-). The fundamental unit of mass is the kilogram (kg), not the gram.

2. Special Note on Temperature: The base unit for temperature is the Kelvin (K), not degrees Celsius (°C). You will explore the Kelvin scale in more detail later in your physics course.

Now that we have our fundamental building blocks, let's see how we can combine them to create units for everything else.

3. Derived Units: Building with the Blocks

Derived units are units of measurement that are formed by combining the fundamental base units. If a unit is not one of the six fundamental base units, it must be a derived unit.

The single most important technique for finding any derived unit is this: The trick is to always think what the formula for that quantity is, and then derive the unit from there.

Let's break this down with a few examples.

1. Example: Speed

   • Formula: speed = distance / time (v = d/t)

   • Unit Substitution: The unit for distance is meters (m) and the unit for time is seconds (s). This gives us meters / seconds.

   • Final Derived Unit: m/s or, using negative powers, m s⁻¹. Remember from your GCSE maths, the rule of negative indices states that x⁻ⁿ = 1/xⁿ, so s⁻¹ is simply another way of writing 1/s.

2. Example: Momentum

   • Formula: momentum = mass × velocity (p = mv)

   • Unit Substitution: The base unit for mass is the kilogram (kg) and the unit for velocity is m s⁻¹. This gives us kilogram × (meters/second).

   • Final Derived Unit: kg m s⁻¹

3. Example: Density

   • Formula: density = mass / volume (ρ = m/V)

   • Unit Substitution: The base unit for mass is the kilogram (kg). Volume is length × width × height, so its unit is meters × meters × meters, or m³. This gives us kilogram / m³.

   • Final Derived Unit: kg/m³ or, using negative powers, kg m⁻³.

Briefing on the Principles of Experimental Uncertainties in Physics

1. Addition or Subtraction: The absolute uncertainties of the quantities are added together.

2. Multiplication or Division: The percentage uncertainties of the quantities are added together.

3. Raising to a Power: The percentage uncertainty of the quantity is multiplied by the power to which it is raised.

The source material provides a clear framework and worked examples for each rule, establishing a practical guide for calculating and expressing uncertainty in derived quantities. The tutorial explicitly covers common calculations but notes that determining uncertainty from a graph is a separate topic not addressed in the provided context.

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1. Defining and Calculating Uncertainty

1.1 Absolute Uncertainty

Absolute uncertainty quantifies the margin of error in a direct measurement. It is defined in two primary ways depending on the instrument type.

• For Analog Instruments: The absolute uncertainty is typically the smallest measurement that can be taken with the instrument.

  • Example: For a ruler marked in millimeters, the smallest measurement possible is one millimeter. Therefore, the absolute uncertainty is 1 mm.

  • An alternative interpretation, noted as being used by some exam boards, defines absolute uncertainty as half of the smallest measurement. The tutorial proceeds by using the former definition.

• For Digital Instruments: The absolute uncertainty is indicated by the last significant digit of the reading.

  • Example: A voltmeter reading of 5.46 V implies that the final digit is uncertain. The true value is understood to be "sandwiched" between 5.45 V and 5.47 V. The absolute uncertainty is therefore ±0.01 V.

1.2 Percentage Uncertainty

Percentage uncertainty expresses the absolute uncertainty as a fraction of the measured value, multiplied by 100. It provides a relative measure of the uncertainty.

• Formula: Percentage Uncertainty = (Absolute Uncertainty / Experimental Value) × 100

• Example Calculation:

  • Given a measurement of 5.46 V with an absolute uncertainty of ±0.01 V.

  • The percentage uncertainty is calculated as: (0.01 / 5.46) × 100 = 0.18%.

  • The full measurement can be expressed as 5.46 V ± 0.18%.

1.3 Converting Between Uncertainty Types

It is possible to convert from a given percentage uncertainty back to an absolute uncertainty.

• Example Calculation:

  • Given a measurement of 50 V with a percentage uncertainty of ±5%.

  • The absolute uncertainty is 5% of 50 V, which is 2.5 V.

  • The measurement can be written with its absolute uncertainty as 50 V ± 2.5 V.

2. Rules for Combining Uncertainties

2.1 Rule 1: Addition and Subtraction

Rule: When adding or subtracting quantities, add their absolute uncertainties.

• Scenario: Calculating the total voltage across two components measured separately.

  1. Measurement 1 (V₁): 5.0 ± 0.1 V

  2. Measurement 2 (V₂): 4.0 ± 0.2 V

• Calculation:

  1. Calculate the Total Value: The total voltage is V_total = 5.0 V + 4.0 V = 9.0 V.

  2. Combine Absolute Uncertainties: The combined absolute uncertainty is 0.1 V + 0.2 V = 0.3 V.

  3. Final Result: The total voltage is 9.0 V ± 0.3 V.

• Derived Percentage Uncertainty: The percentage uncertainty for this total voltage can then be calculated: (0.3 / 9.0) × 100 ≈ 3.3%.

2.2 Rule 2: Multiplication and Division

Rule: When multiplying or dividing quantities, add their percentage uncertainties.

• Scenario: Finding the uncertainty in resistance (R) calculated using Ohm's Law (R = V/I).

  1. Voltage (V): 5.0 ± 0.1 V

  2. Current (I): 1.1 ± 0.1 A

• Calculation:

  1. Calculate Individual Percentage Uncertainties:

     • % Uncertainty in V = (0.1 / 5.0) × 100 = 2.0%

     • % Uncertainty in I = (0.1 / 1.1) × 100 ≈ 9.09%

  2. Combine Percentage Uncertainties: The percentage uncertainty in R is the sum of the percentage uncertainties in V and I.

     • % Uncertainty in R = % Uncertainty in V + % Uncertainty in I

     • % Uncertainty in R = 2.0% + 9.09% ≈ 11.09%

  3. Final Result: The final percentage uncertainty in the resistance is approximately 11% (rounded to two significant figures).

2.3 Rule 3: Raising to a Power

Rule: When a quantity is raised to a power (n), its percentage uncertainty is multiplied by that power.

• Formula: % Uncertainty in (yⁿ) = n × (% Uncertainty in y)

• Scenario: Finding the percentage uncertainty in the volume (V) of a cube, calculated from its side length (L), where V = L³.

  1. Side Length (L): 4.0 ± 0.1 m

• Calculation:

  1. Calculate the Final Value: The volume is V = (4.0 m)³ = 64 m³.

  2. Calculate the Percentage Uncertainty of the Base Measurement:

     • % Uncertainty in L = (0.1 / 4.0) × 100 = 2.5%

  3. Apply the Power Rule: The volume is the length raised to the power of 3. Therefore, multiply the percentage uncertainty in the length by 3.

     • % Uncertainty in V = 3 × (% Uncertainty in L) = 3 × 2.5% = 7.5%

  4. Final Result: The calculated volume can be expressed as 64 m³ ± 7.5%.

The Distinction Between Scalar and Vector Quantities

The fundamental distinction between scalar and vector quantities in physics lies in the role of direction. A scalar quantity is defined solely by its magnitude, or size, with direction being irrelevant. In contrast, a vector quantity is defined by both its magnitude and its direction, with direction being an "absolutely key" component.

This concept is illustrated through an analogy of two journeys from a starting point A to an ending point B. One traveler follows a direct, straight-line path, representing the vector quantity of displacement. The rate of this travel is velocity. The other traveler takes a longer, meandering route, representing the scalar quantity of distance. The rate of this travel is speed. Although both travelers begin and end at the same locations, the physical quantities used to describe their journeys differ because one accounts for direction (vector) while the other does not (scalar). This core principle allows for the classification of nearly all physical quantities, such as mass and energy (scalars) versus force and momentum (vectors).

The Fundamental Difference: Magnitude vs. Magnitude and Direction

The classification of any physical quantity as either a scalar or a vector depends entirely on whether direction is a component of its definition.

• Scalar Quantity: A scalar possesses only magnitude, which is described as a "posh word for size." For these quantities, such as mass or energy, direction is not an important factor. For example, the source notes, "you don't have 10 kg to the left and 10 kg to the right."

• Vector Quantity: A vector possesses both magnitude and direction. The inclusion of direction is what differentiates it from a scalar and is considered "absolutely key." An example provided is momentum, where "momentum to the left is going to be very different to the momentum to the right."

Classification of Common Physical Quantities

The source states that "pretty much every single quantity that we know can be split into maybe a vector or a scaler." The following table organizes the examples provided into their respective categories.

Standard Form and SI units