Revision: Physics and Measurement JEE Main Physics and Measurement

Mass is the amount of matter contained in a body. Its unit is a kilogram (kg).

The density of a substance is defined as the mass of a unit volume of that substance.

`"Density" = "Mass"/"Volume"`

One metre is defined as the distance travelled by light in the air in  `1/(299,792,458)` of a second.

The S. I. unit of length is meter.

Multiple of metre = Kilometre (km).

Submultiple of metre = Centimetre (cm)

A value, quantity, or magnitude in terms of which other values, quantities, or magnitudes are expressed is called a unit.

The smallest value up to which an instrument can measure is called the least count.

Units that are neither fundamental nor derived but are accepted in the SI system (e.g., radian for plane angle, steradian for solid angle) is called supplementary units.

Units that are derived from fundamental units — such as force, which is mass × acceleration — and are expressed algebraically using base units is called derived units.

The basic physical quantities that cannot be derived from other quantities and serve as the foundation for all measurements is called fundamental quantities.

The quantities that are derived from fundamental quantities through mathematical relationships is called derived quantities.

A quantity that can be measured by an instrument and through which we describe the laws of the physical world is called a physical quantity.

A set of particular physical quantities from which different other units can be obtained, which are neither derived from one another nor resolved into any other units is called fundamental units.

Unit is the quantity of a constant magnitude which is used to measure the magnitudes of other quantities of the same nature.

Mass is the measure of the amount of matter in an object. It is a fundamental property of matter and does not change with location or the object’s state.

One kilogram (kg) is the S.I. unit of mass. It is defined as the mass of the international prototype of the kilogram, which is a platinum-iridium alloy cylinder stored at the International Bureau of Weights and Measures in France.

For a given set of measurements of the same quantity, the arithmetic mean of all the absolute errors is called mean absolute error in the measurement of that physical quantity.

`triangle "a"_"mean" = (triangle"a"_1 + triangle"a"_2 + ......+ triangle"a"_"n")/"n" = 1/"n"` \[\sum_{i=1}^n\triangle a_i\]

The ratio of the mean absolute error in the measurement of a physical quantity to its arithmetic mean value is called relative error.

Relative error = `(triangle "a"_"mean")/"a"_"mean"`

When a physical quantity is measured incorrectly, it can result in an error.

Systematic errors are consistent deviations from the true value caused by flaws in the measurement system.

OR

The type of error that consistently occurs in the same direction (either positive or negative), arising from imperfect design or calibration of measuring instruments, imperfection in experimental technique, or carelessness of an individual is called systematic error.

When relative error is represented as percentage it is called the percentage error.

Percentage error = `(triangle"a"_"mean")/("a"_"mean") xx 100`

1. For a given set of measurements of a quantity, the magnitude of the difference between mean value (Most probable value) and each individual value is called absolute error (Δa) in the measurement of that quantity.
2. absolute error = |mean value - measured value|
   Δa₁ = |a_mean - a₁|
   Similarly,
   Δa₂ = |a_mean - a₂|,
             `\vdots           \vdots             \vdots`
   Δa_n = |a_mean - a_n|

The measured value of a physical quantity denoting the number of digits in which we have confidence — where a larger number indicates greater accuracy of measurement — is called significant figures.

The powers to which the fundamental quantities are raised to express the derived unit of a physical quantity is called dimensions.

An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.

The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity.

A constant quantity having no dimensions (e.g., numbers 1, 2, 3, π) is called a dimensionless constant.

The study of the relationship between physical quantities with the help of dimensions and units of measurement is called dimensional analysis.

A quantity that is variable but has no dimensions (e.g., angle, specific gravity, strain, efficiency of a machine) is called a dimensionless variable.

A physical quantity having a fixed value with certain dimensions (e.g., velocity of light in vacuum, gravitational constant) is called a dimensional constant.

Least count = \[\frac {\text {Smallest reading on main scale}}{\text {No. of divisions on main scale}}\]

Instrument Least Count = \[\frac {\text {Main scale least count}}{\text {Divisions on secondary scale}}\]

• Fundamental (base) units — independent units for fundamental quantities (length, mass, time, etc.)
• Derived units — combinations of base units (e.g., m/s for speed, kg/m³ for density, Pa = kg m⁻¹ s⁻² for pressure)

The SI system has 7 base units:

Temperature Conversions:

K = ^°C + 273.15

\[°F=\frac{9}{5}°C+32\]

Three main applications are:

1. Checking the correctness of the given physical relation
2. To derive the relationship between various physical quantities
3. Conversion of one system of units into the other

Limitations of Dimensional Analysis:

• No information about dimensionless variables and constants.
• Applicable only for quantities of mass (M), length (L), and time (T).
• Cannot establish relations containing addition or subtraction like Y = A + B − C.
• Not applicable for trigonometric, exponential, and logarithmic functions.