Class 11 Units and Measurement Note

Class 11 Units and Measurement Note

  March 27, 2023    Schoolix

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 UNITS AND MEASUREMENTS 

# Physical Quantities

The quantity which can be measured by an instrument and which describes the laws of physical world.

= numerical value* unit(a comparison)

# Types of Physical Quantities

a) Fundamental Quantities: which do not depend upon any other quantities.

• Length (metre)
• Mass (kilogram)
• Time (second)
• Electric current (ampere)
• Thermodynamic temperature (kelvin)
• Amount of substance (mole)
• Luminous intensity (candela)

b) Derived Quantities: which are made up of fundamental quantities.

Example: Area = L*L = m^2

# Dimension

Dimensions are the powers to which fundamental units are raised in order to express the derived unit of a quantity.

• Length (metre)                                                                  = [L]
• Mass (kilogram)                                                                = [M]
• Time (second)                                                                   = [T]
• Electric current (ampere)                                                  = [A]
• Thermodynamic temperature (kelvin)                              = [K]
• Amount of substance (mole)                                            = [mol]
• Luminous intensity (candela)                                           = [cd]

# Principle of Homogeneity 

The dimensions of an equation must be same.

# Application of Dimension Analysis 

1) Checking the correctness of an equation: For a correct equation dimensions of each term of                                                                                       L.H.S  = dimensions of terms of R.H.S.

    Example: F = ma

                    [MLT^-2] = [M][LT^-2] 

                    Hence Proved

2) Conversion of unit: n₁u₁ = n₂u₂

     Example: Convert 1N to Dyne

                     the dimensional formula of Force F=[M¹L¹T⁻²]

                        So a = 1, b = 1, c = −2

                         Now because Newton is in m.k.s system, we have

                        M1​=1kg                                  M2​=1g

                        L1​=1m                                     L2​=1cm

                        T1​=1s                                        T2​=1s

                        n1​=1Newton                    n2​=?                       

                         Using the formula

                         n2​=n1​​[M1/M2]^a[L1/L2]^b[T1/T2]^c

                         n2​=1[1kg/1g]¹[1m/1cm]¹[1s/1s]⁻²

                         n2​=1[1000g/1g]¹[100cm/1cm]^1×1

                        n2​=1000×100=100000=10⁵ dynes

                     

3) Deriving relation between physical quantities:

    Example: Force on a particle depend upon mass of the particle and acceleration of the particle. Find                         the equation for F.

                     F ∝ m^x, F ∝ a^y

                        F = k m^x a^y

                        [MLT^-2] = [M]^x[LT^-2] ^y

                        [MLT^-2] = [M^xL^yT^-2y]

                        Therefore x = 1, y = 1

                         F = [MLT^-2] = km¹a¹

^{# Limitations of Dimension Analysis}

^{1) Can't find value of constant.}

^{2) Can't apply in expression having trignometric, exponential, logarithmic functions.}

^{3) Can't apply in equation having more than one term.}

^{# Rules of Significant digits}

Significant figures tells us that the number of digits in a measured value, we are confident of.

1) All non-zero digits are significant.

2) Zero between two non-zero and significant digits are significant.

3) Initial zero's are never significant.

4) Trailing zero's are significant if they appear after decimal.

5) Order of magnitude is never significant.

6) Pure number or constant have infinite significant figures.

NOTE: While changing units number of significant digits remain same.

# Calculation of significant figures

1) Addition and Subtraction: The result of this function is rounded off to the same number of decimal places as present in the value with least decimal places.

2) Multiplication and Division: The result is rounded off to same number of significant figures as present in the value with least significant figures.

# Error Analysis

Difference between the measured value and the actual value is error.

1) Absolute error: The magnitude of difference between the true value and the measured value of the quantity.

ΔX= X-X1

2) Mean absolute error: 

△X_mean  = |△X₁| + |△X₂|+ ……. |△X_n|/n

3) Relative/Fractional error: 

= △X/X

4) Percentage error:

=  △X/X*100

# Propagation of error

1) Addition and Subtraction: Error are always added and subtracted in addition and subtraction.

 △x =  △a ±  △b

2) Multiplication and Division: 

 △x/x =  △a/a +  △b/b

# Quantities raised to some power

x = a^pb^q/c^r

△x/x = p(△a/a) + q(△b/b) + r(△c/c)