Revision: Class 11 >> Units and Measurement NEET (UG) Units 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.

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

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

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

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

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

The error arising due to improper design or calibration, least count of the instrument, or zero error of the instrument is called instrumental error.

The deviation of a measured value from the true value of a quantity arising due to human error, instrument limitations, or environmental conditions is called error.

Mathematically: Error = Measured value − True value

Random errors are unpredictable fluctuations in measurements that vary in both magnitude and direction.

OR

The error that occurs irregularly with respect to sign and size, being unpredictable and varying in magnitude and direction — which can be minimised by taking a large number of observations — is called random error.

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

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

When we measure any physical quantity (length, mass, time, temperature, etc.), the value we obtain is usually not exactly equal to its true value. The difference between the measured value and the true value is called measurement error.

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"`

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"` \[\sum_{i=1}^n\triangle a_i\]

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 ratio of the mean absolute error to the mean value of the quantity measured is called relative error or fractional error.

The magnitude of the difference between the true value and the measured value of a quantity is called absolute error.

The arithmetic mean of the magnitudes of absolute errors in all the measurements of a quantity is called the mean absolute error.

When the relative/fractional error is expressed in percentage, it is called percentage error.

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

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

How far each reading is from the mean:

\[\Delta a_i=
\begin{vmatrix}
a-a_i
\end{vmatrix}\]

Average error over all readings:

\[\Delta a_{\mathrm{mean}}=\frac{\sum_{i=1}^n\Delta a_i}{n}\]

How big the error is, compared to the mean value (no units):

Relative Error: \[\frac {Δa_{mean}}{a}\]

The relative error as a percent: 

Percemtage Error: \[\frac{\Delta a_{\mathrm{mean}}}{a}\times100\%\]

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.