Quick Revision Notes

Class 11 Physics - Chapter 2: Units and Measurements

MP Board

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1. Introduction to Physical Quantities

• Physical Quantity: All quantities that can be measured directly or indirectly, and in terms of which the laws of physics are expressed, are called physical quantities (e.g., length, mass, time, force).
• Formula: Physical Quantity = Numerical Value (n) × Unit (u) (Note: As the unit changes, the numerical value changes such that n × u = constant)

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2. Fundamental and Derived Units

• Fundamental Quantities / Units: These are independent of other quantities and cannot be derived from any other physical quantity (e.g., Length, Mass, Time).
• Derived Quantities / Units: These are quantities derived from fundamental quantities (e.g., Speed = Distance / Time).

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3. Systems of Units

1. CGS System: Centimetre, Gram, Second
2. FPS System: Foot, Pound, Second
3. MKS System: Metre, Kilogram, Second
4. SI System (International System of Units): The universally accepted system containing 7 fundamental units and 2 supplementary units.

The 7 Fundamental SI Quantities:

Physical Quantity	SI Unit	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electric Current	Ampere	A
Thermodynamic Temperature	Kelvin	K
Luminous Intensity	Candela	cd
Amount of Substance	Mole	mol

Supplementary Quantities:

• Plane Angle: Radian (rad)
• Solid Angle: Steradian (sr)

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4. Dimensional Analysis

• Dimensions: The powers to which fundamental quantities are raised to represent a physical quantity.
• Dimensional Formula: An expression showing how and which of the fundamental quantities enter into the dimensions of a physical quantity. Represented as [M^a L^b T^c].

Common Dimensional Formulas:

Physical Quantity	Formula	Dimensional Formula
Area	Length × Breadth	[M^0 L^2 T^0]
Volume	Length × Breadth × Height	[M^0 L^3 T^0]
Density	Mass / Volume	[M^1 L^-3 T^0]
Velocity / Speed	Displacement / Time	[M^0 L^1 T^-1]
Acceleration	Velocity / Time	[M^0 L^1 T^-2]
Force	Mass × Acceleration	[M^1 L^1 T^-2]
Work / Energy	Force × Displacement	[M^1 L^2 T^-2]
Power	Work / Time	[M^1 L^2 T^-3]
Pressure / Stress	Force / Area	[M^1 L^-1 T^-2]
Momentum	Mass × Velocity	[M^1 L^1 T^-1]
Frequency	1 / Time Period	[M^0 L^0 T^-1]

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5. Principal of Homogeneity of Dimensions

• According to this principle, only quantities having the same dimensions can be added or subtracted from each other.
• Application: It is used to check the correctness of a physical equation and to derive relations between physical quantities.

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6. Significant Figures

• Definition: The reliable digits plus the first uncertain digit in a measured value are known as significant figures.

Rules for Counting Significant Figures:

1. All non-zero digits are significant (e.g., 432 has 3 sig figs).
2. All zeros between two non-zero digits are significant (e.g., 402 has 3 sig figs).
3. If the number is less than 1, the zero(s) on the right of the decimal point but to the left of the first non-zero digit are not significant (e.g., 0.0045 has 2 sig figs).
4. Trailing zeros in a number without a decimal point are not significant (e.g., 4300 has 2 sig figs).
5. Trailing zeros in a number with a decimal point are significant (e.g., 4.300 has 4 sig figs).

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7. Errors in Measurement

• Absolute Error ($\Delta a$): The difference between the true value ($a_{mean}$) and the measured value ($a_i$). \Delta a_i = a_{mean} - a_i
• Mean Absolute Error ($\Delta a_{mean}$): The arithmetic mean of all absolute errors.
• Relative Error: The ratio of the mean absolute error to the mean value. Relative Error = \Delta a_{mean} / a_{mean}
• Percentage Error: Relative error expressed in percentage. Percentage Error = (\Delta a_{mean} / a_{mean}) \times 100\%

Combination of Errors:

• For Addition/Subtraction ($Z = A + B$ or $Z = A - B$): \Delta Z = \Delta A + \Delta B
• For Multiplication/Division ($Z = A \times B$ or $Z = A / B$): (\Delta Z / Z) = (\Delta A / A) + (\Delta B / B)
• For Powers ($Z = A^n B^m / C^p$): (\Delta Z / Z) = n(\Delta A / A) + m(\Delta B / B) + p(\Delta C / C)

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8. Rounding Off Rules

1. If the digit to be dropped is less than 5, the preceding digit remains unchanged (e.g., 7.34 rounds to 7.3).
2. If the digit to be dropped is greater than 5, the preceding digit is raised by 1 (e.g., 7.36 rounds to 7.4).
3. If the digit to be dropped is 5 followed by non-zero digits, the preceding digit is raised by 1.
4. If the digit to be dropped is 5 alone or 5 followed by zeros, the preceding digit remains unchanged if it is even, and is raised by 1 if it is odd (Round to even rule).