NEET Physics Units And Measurements Notes

Units And Measurements

A physical quantity can be completely represented by its magnitude and unit.

Physical quantity= magnitude (n) x unit (u)

The magnitude of a physical quantity and units are inversely proportional to each other.

The larger the unit, the smaller will be its magnitude.

i.e., nu = constant, or = n₁u₁ = n₂u₂ = constant

SI System

Supplementary units

• Radian (rad), is used to measure plane angle.
• Steradian (Sr), used to measure solid angle.

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Measurement of Large Distances

Larger distances such as the distance of a planet or a star from the Earth can be calculated using the parallax method

⇒ \(\mathrm{D}=\frac{\mathrm{b}}{\theta}\)

Where, b is the basis and 0 is a parallax angle.

The angular size of the planet can be calculated using the formula.

⇒ \(\alpha=\frac{d}{D}\)

Where ‘d’ is the diameter of the planet and ‘D’ is the distance between the planet and Earth.

Absolute error, Relative error, and Percentage error

Suppose a₁, a₂, a₃,….a_n, are the different-measured values of a physical quantity, then mean value or true value is given by,

⇒ \(a_{\text {mean }}=\frac{\left(a_1+a_2+\ldots .+a_n\right)}{n}\)

The magnitude of the difference between the true value and the individual measurement value is called the absolute error of the measurement. Absolute error |Δa| is denoted by meaning

⇒ \(\left|\Delta a_1\right|=a_{\text {mean }}-a_1\)

⇒ \(\left|\Delta a_2\right|=a_{\text {mean }}-a_2\)

⇒ \(\left|\Delta a_n\right|=a_{\text {mean }}-a_n\)

Absolute error |Δa | is always positive.

Units And Dimensions

The arithmetic mean of all absolute error is called the mean absolute error.

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⇒ \(\text { i.e., } \Delta a_{\text {mean }}=\frac{\left|\Delta a_1\right|+\left|\Delta a_2\right|+\ldots \ldots+\left|\Delta a_n\right|}{n}\)

Relative Error

Relative error is the ratio of the mean absolute error (Aa^^ )to the mean value (amean) of the quantity.

Relative error \(=\frac{\Delta \mathrm{a}_{\text {mean }}}{\mathrm{a}_{\text {mean }}}\)

Percentage error = Relative error x 100

⇒ \(\frac{\Delta \mathrm{a}_{\text {mean }}}{\mathrm{a}_{\text {mean }}} \times 100\)

Error of a sum or difference:

Let the physical quantity Z is given by,

⇒ Z = A + B

⇒ Then, ± ΔZ = ± ΔA ± ΔB

When two physical quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Error of a product or quotient:

Let the physical quantity Z is given by,

Z = AB Then the maximum relative error is given by,

⇒ \(\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=\frac{\Delta \mathrm{A}}{\mathrm{A}}+\frac{\Delta \mathrm{B}}{\mathrm{B}}\)

When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

Error in case of a measured quantity raised to a power:

Let the physical quantity Z is given by,

⇒ Z= Aⁿ

Then maximum relative error is given by

⇒ \(\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=2 \frac{\Delta \mathrm{A}}{\mathrm{A}}\)

In general if Z = \(\frac{A^x B^Y}{C^Z}\) then,

⇒ \(\frac{\Delta \mathrm{Z}}{\mathrm{Z}}=2 \frac{\Delta \mathrm{A}}{\mathrm{A}}\)

The relative error in a physical quantity raised to the power ‘n’ is n times the relative error in the individual quantity.

Units And Measurement Formula

Significant Figures

• All non-zero digits are significant.
• All the zeros between two non-zero digits are significant, no matter where the decimal point is.
• If the number is less than 1, the zeros on the right of decimal point but to the left of the first non-zero digit are not significant.
• The trailing zeros in a number without a decimal point are not significant.
• The trailing zeros in a number with a decimal point are significant.

Arithmetic operations with significant figures

• In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
• In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

Rounding off the digits

The preceding digit is raised by 1 if the digit to be dropped is more than 5, and is left unchanged if it is less than 5.

Note:

If the digit to be dropped is 5, and if the preceding digit is even, the digit is simply dropped, and if it is odd the preceding digit is raised by 1.

The nature of a physical quantity is represented by its dimensions:

• The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.
• An equation obtained by equating a physical quantity with its dimensional formula is called dimensional equation.

Unit And Dimensions Notes Pdf

Applications Of Dimensional Analysis

• Checking the dimensional consistency of equations.
  • (If an equation is dimensionally wrong, then it is wrong, but it may not be right also. Thus a dimensionally correct equation need not be an exact equation)
• Deducing relation among physical quantities.