A physical quantity is any property of a material or system that can be quantified by measurement using a numerical value and a unit.
Measurement is essentially a process of comparison of an unknown quantity with a standard reference of the same kind.
The numerical value (n) and the unit (u) of a physical quantity are inversely proportional to each other such that n × u = Constant.
If the unit of measurement is made smaller, the numerical value representing the quantity becomes larger to maintain the same total magnitude.
Physical quantities are categorized into two main groups known as fundamental quantities and derived quantities based on their independence.
Fundamental quantities are those that cannot be expressed in terms of other quantities, such as mass, length, and time.
Derived quantities are formed by combining fundamental quantities through multiplication or division, such as speed, area, or force.
Example: Speed is a derived quantity calculated as Distance ÷ Time, where both distance (length) and time are fundamental.
Systems of Units and the SI System
A complete set of units, including both fundamental and derived units, is known as a system of units.
The CGS system uses the centimeter, gram, and second as fundamental units for length, mass, and time respectively.
The FPS system is the British engineering system using the foot, pound, and second as its base measurement units.
The MKS system utilizes the meter, kilogram, and second, providing the foundation for the modern metric system.
International System of Units (SI) was adopted in 1971 to provide a globally uniform language for science and technology.
There are seven base units in the SI system: Meter (m), Kilogram (kg), Second (s), Ampere (A), Kelvin (K), Mole (mol), and Candela (cd).
Supplementary units in the SI system include the Radian (rad) for plane angles and the Steradian (sr) for solid angles.
Radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
Standard Definitions of Base Units
The Meter is currently defined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second.
One Kilogram is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015 × 10⁻³⁴ when expressed in J s.
The Second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of Cesium-133.
Ampere is the constant current which, if maintained in two straight parallel conductors of infinite length, produces a specific force between them.
The Kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
One Mole contains exactly 6.02214076 × 10²³ elementary entities, a number known as the Avogadro constant.
Candela measures luminous intensity in a given direction of a source that emits monochromatic radiation of frequency 540 × 10¹² Hz.
Standardization of these units ensures that scientific results can be replicated with high precision across different laboratories worldwide.
Measurement of Large and Small Distances
The Parallax Method is used to measure large distances such as those to planets or stars by observing the shift in position from two different locations.
Distance (D) in the parallax method is calculated as D = b / θ, where b is the basis (distance between observation points) and θ is the parallax angle.
One Astronomical Unit (AU) is the average distance between the Earth and the Sun, approximately equal to 1.496 × 10¹¹ meters.
A Light Year (ly) is the distance light travels in a vacuum in one year, calculated as 9.46 × 10¹⁵ meters.
Parsec (pc) is the distance at which an arc of 1 AU length subtends an angle of 1 second of arc, equal to 3.08 × 10¹⁶ meters.
Fermi (f) is a small unit of length used in nuclear physics, where 1 Fermi = 10⁻¹⁵ meters.
Angstrom (Å) is commonly used to measure the wavelength of light and atomic sizes, where 1 Å = 10⁻¹⁰ meters.
Microscopic distances like the size of a molecule can be estimated using the Oleic Acid layer method on a water surface.
Dimensional Analysis and Formulas
Dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity.
The dimensional formula is written using square brackets [Mᵃ Lᵇ Tᶜ], where M is mass, L is length, and T is time.
Speed has the dimensional formula [M⁰ L¹ T⁻¹] because it is length divided by time.
Force is defined as Mass × Acceleration, resulting in the dimensional formula [M¹ L¹ T⁻²].
Work and Energy share the same dimensional formula, which is [M¹ L² T⁻²], illustrating their physical relationship.
Pressure is Force per unit Area, giving it the dimensional formula [M¹ L⁻¹ T⁻²].
Dimensional constants are quantities that have fixed values and dimensions, such as the Universal Gravitational Constant (G).
Dimensionless quantities have no units or dimensions, such as refractive index, strain, and plane angle.
Applications of Dimensional Analysis
Principle of Homogeneity states that a physical equation is dimensionally correct if the dimensions of all terms on both sides are the same.
Dimensional analysis can be used to convert the units of a physical quantity from one system to another using n₁u₁ = n₂u₂.
Conversion formula: n₂ = n₁ × [M₁/M₂]ᵃ [L₁/L₂]ᵇ [T₁/T₂]ᶜ, where a, b, c are the dimensions of the quantity.
Example: To convert 1 Newton to Dynes, we use the dimensions of force [M¹ L¹ T⁻²] to find that 1 N = 10⁵ Dynes.
Deriving relationships between physical quantities is possible if we know the factors on which the quantity depends.
Limitations include the inability to determine dimensionless constants or handle functions like trigonometric, logarithmic, or exponential terms.
Dimensional analysis cannot distinguish between physical quantities having the same dimensions, such as Work and Torque.
Equations involving more than three fundamental quantities cannot be solved completely using this method if they are in product form.
Significant Figures and Rounding Off
Significant figures are the digits in a measured quantity that are known reliably plus one digit that is uncertain.
All non-zero digits in a measurement are always considered significant.
Zeros between two non-zero digits are significant, regardless of the position of the decimal point.
Trailing zeros in a number without a decimal point are generally not significant unless they come from a specific measurement.
Example: In the value 0.0075, there are only 2 significant figures, as leading zeros are placeholders.
Rounding rule: If the digit to be dropped is 5 followed by zeros, the preceding digit is increased by 1 if it is odd, and left unchanged if it is even.
In addition and subtraction, the final result should retain as many decimal places as are present in the term with the least decimal places.
In multiplication and division, the result must contain only as many significant figures as the original number with the least significant figures.
Errors in Measurement
Error is the difference between the true value of a physical quantity and the value obtained through measurement.
Systematic errors tend to be in one direction (positive or negative) and include instrumental errors and personal bias.
Random errors occur irregularly due to unpredictable fluctuations in experimental conditions or observer limitations.
Least count error is the error associated with the resolution of the measuring instrument.
Absolute error is the magnitude of the difference between the individual measurement and the true value of the quantity.
Mean absolute error is the arithmetic mean of the magnitudes of absolute errors in all measurements.
Relative error is the ratio of the mean absolute error to the mean value of the quantity being measured.
Percentage error is the relative error expressed as a percentage, calculated as (Mean Absolute Error / Mean Value) × 100%.
Combination and Propagation of Errors
When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors of the individual quantities.
Formula for Sum: If Z = A + B, then ΔZ = ΔA + ΔB.
When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors of the multipliers.
Formula for Product: If Z = A × B, then ΔZ/Z = (ΔA/A) + (ΔB/B).
Error in power of a quantity: If Z = Aⁿ, then the relative error ΔZ/Z = n × (ΔA/A).
General Error Formula: If Z = (Aᵃ Bᵇ) / Cᶜ, then ΔZ/Z = a(ΔA/A) + b(ΔB/B) + c(ΔC/C).
Example: If the error in radius is 2%, the error in the volume of a sphere (V = 4/3 π r³) is 3 × 2% = 6%.
Errors always add up in the worst-case scenario calculation to ensure the reliability of the scientific conclusion.
Accuracy, Precision, and Least Count
Accuracy refers to how close a measured value is to the true or accepted value of the quantity.
Precision describes the degree of exactness or refinement of a measurement, often limited by the instrument's least count.
A measurement can be precise but inaccurate if a systematic error, like a zero error in a vernier caliper, is present.
Least count is the smallest value that can be measured accurately by a specific measuring instrument.
Vernier Caliper least count is typically 0.01 cm, calculated as 1 Main Scale Division − 1 Vernier Scale Division.
Screw Gauge least count is usually 0.001 cm, calculated as (Pitch of the screw) ÷ (Total number of divisions on circular scale).
Zero error occurs when the zero mark of the measuring scale does not coincide with the reference mark when the jaws are closed.
Positive zero error must be subtracted from the observed reading, while negative zero error must be added to get the true reading.
Common Mistakes and Traps
Mistake: Adding quantities with different dimensions. Trap: Only quantities with identical dimensions can be added or subtracted (e.g., you cannot add 5 kg and 2 m).
Mistake: Confusing units with dimensions. Trap: Units like Joules and Erg are different, but their dimension [M¹ L² T⁻²] is the same.
Mistake: Forgetting that plane angles have units (radians) but are dimensionless. Trap: Many students assume dimensionless means unitless.
Mistake: Calculating absolute error in multiplication. Trap: Relative errors must be added for multiplication and division, not absolute errors.
Mistake: Incorrectly identifying significant figures in 100 vs 100.0. Trap: 100 has 1 significant figure, but 100.0 has 4 because the decimal indicates precision.
Mistake: Using wrong supplementary units. Trap: Using degrees instead of radians in formulas like Arc = Radius × Angle.
Mistake: Assuming all zeros are significant. Trap: Leading zeros (0.002) are never significant; they only locate the decimal point.
Mistake: Neglecting the power sign in error propagation. Trap: Even if a term is in the denominator (C⁻¹), its error is added, never subtracted.
Shortcuts and Tricks for Exams
To find the dimensional formula of a constant, rearrange the standard formula for that constant first (e.g., G = F r² / m₁ m₂).
If an equation is y = A sin(Bt + C), then (Bt + C) must be dimensionless, so the dimension of B is [T⁻¹].
In error percentage questions, identify the quantity with the highest power as it contributes most to the total error.
To convert units quickly, remember prefixes: Mega (10⁶), Kilo (10³), Centi (10⁻²), Milli (10⁻³), Micro (10⁻⁶), Nano (10⁻⁹).
Vedic Math trick for squares of numbers ending in 5 helps in area calculations (e.g., 2.5²: 2×3=6, then append 25 to get 6.25).
Use the method of elimination in MCQs by checking the dimensions of the options against the required physical quantity.
Remember the ratio: 1 Radian ≈ 57.3° for quick mental conversions between angular systems.
The number of significant figures in the result of a calculation cannot be more than the least number of sig-figs in the inputs.
Quick Reference / Formula Summary
SI Base Units: Meter (m), Kilogram (kg), Second (s), Ampere (A), Kelvin (K), Mole (mol), Candela (cd).
Parallax distance: D = b / θ (θ must be in radians).
Important Lengths: 1 AU = 1.5 × 10¹¹ m; 1 ly = 9.46 × 10¹⁵ m; 1 pc = 3.1 × 10¹⁶ m; 1 Å = 10⁻¹⁰ m.
Dimensional Formulas: Force [M L T⁻²], Work [M L² T⁻²], Power [M L² T⁻³], Pressure [M L⁻¹ T⁻²].
Error in Z = (Aᵃ Bᵇ) / Cᶜ: ΔZ/Z = a(ΔA/A) + b(ΔB/B) + c(ΔC/C).
Significant Figures: Non-zeros are always sig; trailing zeros after decimal are sig; leading zeros are never sig.
Least Count (LC): LC = Pitch / Number of divisions (for screw gauge) or 1 MSD − 1 VSD (for calipers).