Revision: 11th Std >> Measurements MAH-MHT CET (PCM/PCB) Measurements

The quantity of matter contained in a body is known as its mass.

Mass is the measure of the amount of matter in an object.

It is defined as 1/24 the part of the mean solar day.

The average of the varying solar days, when the earth completes one revolution around the sun, is called mean solar day.

The standard metre is defined in terms of the speed of light, according to which one metre is the distance travelled by light in `1/(299,792,458)` of a second in the air (or vacuum).

It is defined as the 1/1440 part of the mean solar day.

One year is defined as the time in which earth completes one complete revolution around the sun.

Measurement is the process of comparison of the given physical quantity with the known standard quantity of the same nature.

A standard metreis equal to 1650763.73 wavelengths in vacuum, of the radiation from krypton isotope of mass 86.

The physical quantities like mass, length and time which do not depend on each other are known as fundamental quantities.

The time taken by the earth to complete one rotation about its own axis is called solar day.

“A second is defined as 1/86400 the part of a mean solar day.”
OR
Second may also be defined “as to be equal to the duration of9,192,631,770 vibrations corresponding to the transition between two hyperfine levels of cesium – 133 atoms in the ground state.”

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

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)

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

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.

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.

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

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

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.

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.

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

The smallest value that can be measured by the measuring instrument is called its least count.

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.

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.

Accuracy is about how close your measured value is to the true, actual value of that quantity.

or

The quality or state cate of being accurate or the ability to work or perform without making mistakes.

Accuracy = Mean value - True Value

Precision is about getting reproducible results. If you measure the same thing multiple times and get nearly identical answers, your measurements are precise.

or

The quality, condition, or fact of being exact and accurate or the closeness of the set of values obtained from identical measurements of quantity.

Precision = Individual Value - Arithmetic Mean Value

In real experiments, it is very difficult to get exactly the same answer every single time. This difference or possibility of error is called uncertainty.

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|

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.

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.

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

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

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.

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:

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

The relative error as a percent: 

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

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

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

• 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

\[°F=\frac{9}{5}°C+32\]