Measurement involves comparing a physical quantity to a standard unit of measurement. This comparison yields a numerical value that expresses the magnitude of the measured quantity in relation to the chosen standard unit. Measurements are essential in science, engineering, and various aspects of daily life, providing a means to quantify and describe different aspects of the physical world accurately. Standard units, such as meters, kilograms, seconds, and degrees Celsius, serve as reference points to maintain consistency and uniformity in measurements.
Physical Quantities
Physical quantities encompass all those properties or attributes that can be measured, either directly or indirectly. These quantities serve as the fundamental building blocks for describing the principles and laws of physics.
Units
Units are standardized quantities of physical quantities that are selected for the purpose of measuring other quantities of the same kind. These units are chosen to be easily reproducible and internationally accepted, ensuring consistency and uniformity in measurements across the globe.
Fundamental quantities are physical quantities that are independent of each other. The units used to measure these fundamental quantities are known as fundamental units. These fundamental units serve as the building blocks for constructing units for other derived physical quantities.
| Fundamental Quantity | Fundamental Unit | Unit Symbol | Definition |
| Length | Metre | m | The meter represents the distance light travels in a vacuum in 1/299,792,458th of a second (1983). |
| Mass | Kilogram | kg | The kilogram is defined as the mass of the International prototype, a platinum-iridium alloy cylinder, kept at the International Bureau of Weights and Measures in Sevres, France (1889). |
| Time | Second | s | The second is the time it takes for 9,192,631,770 cycles of radiation corresponding to a cesium-133 atom’s transition between two hyperfine levels of the ground state (1967). |
| Electric Current | Ampere | A | An ampere is the constant current that, when maintained in two infinitely long parallel conductors with negligible cross-section and placed 1 meter apart in a vacuum, produces a force of 2 x 10^(-7) newtons per meter (1948). |
| Thermodynamic Temperature | Kelvin | K | The kelvin is defined as 1/273.16 of the thermodynamic temperature of the triple point of water (1967). |
| Amount of Substance | Mole | mol | A mole represents the amount of substance containing the same number of elementary entities as there are atoms in 0.012 kg of carbon-12 (1971). |
| Luminous Intensity | Candela | cd | The candela is the luminous intensity emitted in a specific direction by a source that emits monochromatic radiation with a frequency of 540 x 10^12 Hz and has a radiant intensity of 1/683 watt per steradian (1979). |
| Supplementary Fundamental Quantity | Supplementary Unit | Unit Symbol | Definition |
| Plane Angle | Radian | rad | One radian is defined as the angle formed at the center of a circle by an arc with a length equal to the radius of the circle, i.e., θ = (arc length) / (radius). |
| Solid Angle | Steradian | sr | One steradian is defined as the solid angle subtended at the center of a sphere by a surface area on the sphere that is equal in size to the square of the sphere’s radius, i.e., Ω = (surface area) / (radius²). |
Derived Quantities
Derived quantities and their associated units refer to physical measurements that are calculated or derived from fundamental quantities. Examples of derived quantities include velocity, acceleration, force, and work. These derived units are obtained through mathematical combinations or relationships involving fundamental units.
Systems of units encompass a comprehensive collection of units, encompassing both fundamental and derived units, for various physical quantities. In the field of mechanics, several common systems of units are utilized:
CGS System: In this system, the unit of length is the centimeter, the unit of mass is the gram, and the unit of time is the second.
FPS System: In this system, the unit of length is the foot, the unit of mass is the pound, and the unit of time is the second.
MKS System: In this system, the unit of length is the meter, the unit of mass is the kilogram, and the unit of time is the second.
SI System: The International System of Units, abbreviated as SI, is the globally accepted system of units for measurement. This system comprises seven fundamental units and two supplementary fundamental units.
Relationship between Some Mechanical SI Units and Commonly Used Units
Length: (a) 1 micrometer = 10^-6 meters (b) 1 nanometer = 10^-9 meters (c) 1 angstrom = 10^-10 meters
Mass: (a) 1 metric ton = 10^3 kilograms (b) 1 pound = 0.4537 kilograms (c) 1 atomic mass unit (amu) = 1.66 x 10^-27 kilograms
Volume: 1 liter = 10^-3 cubic meters
Force: (a) 1 dyne = 10^-5 newtons (b) 1 kilogram-force (kgf) = 9.81 newtons
Pressure: (a) 1 kgf/m^2 = 9.81 N/m^2 (b) 1 millimeter of mercury (mm Hg) = 133 N/m^2 (c) 1 pascal = 1 N/m^2 (d) 1 atmosphere pressure = 76 cm of mercury = 1.01 x 10^5 pascals
Work and Energy: (a) 1 erg = 10^-7 joules (b) 1 kilogram-force meter (kgf-m) = 9.81 joules (c) 1 kilowatt-hour (kWh) = 3.6 x 10^6 joules (d) 1 electronvolt (eV) = 1.6 x 10^-19 joules
Power: (a) 1 kilogram-force meter per second (kgf-ms^-1) = 9.81 watts (b) 1 horsepower (hp) = 746 watts
| Physical Quantities | Dimensional Formula (with explanations) |
| Area | [L]^2 (The dimension of length squared) |
| Volume | [L]^3 (The dimension of length cubed) |
| Velocity | [LT]^-1 (The dimensions of length divided by time) |
| Acceleration | [LT]^-2 (The dimensions of length divided by time squared) |
| Force | [MLT]^-2 (The dimensions of mass times length divided by time squared) |
| Work or Energy | [MLT]^-2 (The dimensions of mass times length divided by time squared) |
| Power | [MLT]^-2 (The dimensions of mass times length divided by time squared) |
| Pressure or Stress | [MLT]^-1 (The dimensions of mass times length divided by time squared) |
| Linear Momentum or Impulse | [MLT]^-1 (The dimensions of mass times length divided by time) |
| Density | [ML]^-3 (The dimensions of mass divided by length cubed) |
| Strain | Dimensionless (No dimensions, unitless) |
| Modulus of Elasticity | [MLT]^-1 (The dimensions of mass times length divided by time squared) |
| Surface Tension | [MT]^-2 (The dimensions of mass times time divided by length squared) |
| Velocity Gradient | [T]^-1 (The dimensions of time^-1) |
| Coefficient of Viscosity | [MLT]^-1 (The dimensions of mass times length divided by time) |
| Gravitational Constant | [MLT]^-3 (The dimensions of mass times length divided by time cubed) |
| Moment of Inertia | [ML^2] (The dimensions of mass times length squared) |
| Angular Velocity | [T]^-1 (The dimensions of time^-1) |
| Angular Acceleration | [T]^-2 (The dimensions of time^-2) |
| Angular Momentum | [MLT]^-1 (The dimensions of mass times length divided by time) |
| Specific Heat | [L^2T]^-2 (The dimensions of length squared divided by time squared) |
| Latent Heat | [L^2T]^-2 (The dimensions of length squared divided by time squared) |
| Planck’s Constant | [ML^2T]^-1 (The dimensions of mass times length squared divided by time) |
| Universal Gas Constant | [ML^2T]^-2 (The dimensions of mass times length squared divided by time squared) |
Dimensions are a fundamental concept in physics and engineering, and they have several important applications, as you mentioned:
(i) To check the accuracy of physical equations:
- Dimensions are used to verify the consistency of equations. In a correct physical equation, the dimensions of the left-hand side (LHS) must be equal to the dimensions of the right-hand side (RHS). This is known as dimensional analysis. If the dimensions do not match, it indicates an error in the equation or calculation.
(ii) To change a physical quantity from one system of units to another system of units:
- Dimensions provide a way to convert a physical quantity from one system of units to another. By expressing a quantity in terms of its dimensions, you can easily convert it to different units by using conversion factors. For example, if you have a length in meters and want to express it in feet, you can use the conversion factor of 1 meter = 3.28084 feet.
(iii) To obtain a relation between different physical quantities:
- Dimensions help derive relationships between different physical quantities. By analyzing the dimensions of various physical parameters in an equation, you can determine how they are related to each other. This is particularly useful in deriving new equations or understanding the dependencies between variables in a physical system.
Significant figures
Significant figures, also known as significant digits, are the digits in a measured value of a physical quantity that convey meaningful information about the precision or accuracy of the measurement. They are the digits we are sure about, plus one more digit that represents uncertainty. Here’s how they work:
- All non-zero digits are always considered significant. For example, in the number 3456, all four digits (3, 4, 5, and 6) are significant.
- Any zeros between significant figures are also considered significant. For example, in the number 203, both the 2 and the 3 are significant.
- Leading zeros (zeros to the left of the first non-zero digit) are not considered significant. For example, in the number 0.00721, only 721 are significant figures.
- Trailing zeros (zeros to the right of the last non-zero digit) in a decimal number are considered significant. For example, in the number 12.500, all five digits (1, 2, 5, 0, and 0) are significant.
- In scientific notation (expressing a number as a coefficient multiplied by a power of 10), all digits in the coefficient are significant. For example, in 3.45 x 10^4, both 3 and 45 are significant.
- In exact numbers (numbers that have no uncertainty or infinite precision), all digits are considered significant. For example, in the number 12 (without any decimal point or uncertainty), both 1 and 2 are significant.
Error
The lack in accuracy in the measurement due to the limitation of accuracy of the measuring instruments or due to any other factor is called an error. The difference between the measured value and the true value of a quantity is known as the error in the measurement
Absolute Error
The difference between the true value and the measured value of a quantity is called absolute error. If a₁, a₂, a₃, …, aₙ are the measured values of any quantity ‘a’ in an experiment performed ‘n’ times, then the arithmetic mean of these values is called the true value (aₘ) of the quantity:
aₘ = (a₁ + a₂ + a₃ + … + aₙ) / n
The absolute error in the measured values is given by:
Δa₁ = aₘ – a₁
Δa₂ = aₘ – a₂
Δaₙ = aₘ – aₙ
Each Δaᵢ represents the absolute error associated with the corresponding measurement.
Mean Absolute Error
The mean absolute error (MAE) is defined as the arithmetic mean of the magnitudes of absolute errors in all the measurements. It represents the average magnitude of the absolute errors. For a set of ‘n’ measurements of a quantity ‘a,’ the MAE (Dₐ) is calculated as:
Dₐ = (|Δ₁| + |Δ₂| + |Δ₃| + … + |Δₙ|) / n
Where:
- Δ₁, Δ₂, Δ₃, …, Δₙ are the absolute errors associated with each of the ‘n’ measurements.
- |Δᵢ| represents the magnitude (absolute value) of each absolute error.
The MAE provides a measure of how close, on average, the measured values are to the true value or the expected value of the quantity ‘a.’ It is a useful metric for assessing the overall accuracy and precision of a set of measurements.
Relative Error
Relative error = Mean absolute error /True value =(Δa̅) / aₘ
Percentage Error
The percentage error is defined as the relative error expressed as a percentage of the true value. It quantifies the magnitude of the error relative to the true or expected value of a quantity ‘a.’ The percentage error (PE) is calculated as:
PE = (Δa̅/ aₘ) * 100%
Where:
- PE represents the percentage error.
- Δa̅ is the absolute error.
- aₘ is the true value or the expected value of the quantity ‘a.’
Combination of Errors
(i) Error in Addition or Subtraction:
- When adding or subtracting two quantities a and b with measured values (a ± Δa) and (b ± Δb), the maximum absolute error in their addition or subtraction is given by: Δx = ± (Δa + Δb)
(ii) Combination of Errors in Multiplication or Division:
- When multiplying or dividing two quantities a and b with measured values (a ± Δa) and (b ± Δb), the maximum relative error in the result is given by: Δx / x = Δa / a + Δb / b
(iii) Error in Case of a Measured Quantity Raised to a Power:
- When raising a quantity z to a power with measured values (a ± Δa), (b ± Δb), and (c ± Δc), the maximum error in the result is given by: Δz / z = p(Δa / a) + q(Δb / b) + r(Δc / c)