Revision: Physical World and Measurement >> Units and Measurements Physics Science (English Medium) Class 11 CBSE

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.

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

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

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.

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

Derived quantities are physical quantities that depend on and can be calculated using fundamental quantities.

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.