Q. What is Uncertainty in measurement? Explain its Types and Rules with examples: 


Answer:

Uncertainty:

The estimate of the possible range of an error in any measurement is called Uncertainty.

OR 

The magnitude of error in the measurement 0r doubt is called uncertainty.

⇒ Uncertainty estimates small or large the error is given by:

Measurement = Best Estimate ± Uncertainty


Types of uncertainty:

Absolute uncertainty: 

The least count of the measuring instrument is called Absolute Uncertainty. It's denoted by "Δ" and has the same unit as the quantity.


Fractional Or Relative Uncertainty:

Relative uncertainty = Absolute error / Measured Value

It is denoted by " ε " and has no unit 


Percentage uncertainty:

% uncertainty = fractional uncertainty x 100%

or

% uncertainty = (Absolute uncertainty / Measured value) x 100%


Rules for Uncertainty: 

(1) Addition and Subtraction: 

For the Assessment of total uncertainty in the final result of addition and subtraction, absolute Uncertainty is added. 

Example: Find displacement between points X₁ = 10.5  ± 0.1 cm & X₂ = 26.8 ± 0.1 cm

Given : 

X₁ = 10.5  ± 0.1 cm

X₂ = 26.8 ± 0.1 cm

Required:

displacement = X = ?

Solution:

X = X₂ - X₁ = (10.5  ± 0.1 cm) - (26.8 ± 0.1 cm) 

x = 16.3 ± 0.2 cm


(2) Multiplication and Division:

In the multiplication and division, the percentage uncertainty is added in the final result for the assessment of total uncertainty


Example:

if the potential difference of V= 5.2 ± 0.1V is applied across the ends of the conductor, and as the result the current I = 0.84 ± 0.05 passes through the conductor. Determine the resistance.


Given:

Potential difference V = 5.2 ± 0.1 V

Current I = 0.84 ± 0.05


Required:

Resistance R =?


Solution:

By Using Ohm's law

V = I R

or

R = `\frac {V}{I}`

by putting a best estimate

R = `\frac {5.2}{0.84}`

R = 6.2 Ω

Now

% uncertainty in V = `\frac {0.1}{5.2}` X `\frac {100}{100}`

% uncertainty in V = 2 %


% uncertainty in I = `\frac {0.05}{0.84}` X `\frac {100}{100}`

% uncertainty in I = 6 %


So therefore % uncertainty in R = 2% +6% = 8 %


So Resistance R = 6.2 Ω with 8% uncertainty 

R = 6.2 ± 8% Î©

R = 6.2 ± `\frac {8}{100}` Î©

R = 6.2 ± 0.08 Î©


(3) Power factor: 

In order to find out the total uncertainty in the measurement having an exponent, we multiply the percentage uncertainty by the power. 

Example: Find out the volume of a sphere whose radius 2.25 ± 0.01 cm?

Given:

Potential difference r = 2.25 ± 0.01 cm


Required:

Volume V = ?


Solution: 

The volume of the sphere 


V = `\frac {4}{3}`ã„«r³

V = `\frac {4}{3}`x (3.1416) (2.25)³

V = 47.7 cm³


Now % uncertainty in r = `\frac {0.01}{2.25}` X `\frac {100}{100}`

% uncertainty in r = 0.4 %

and 

% uncertainty in V = 3 x 0.4 % = 1.2 %

V = 47.7 cm³ with 1.2% uncertainty

V = 47.7 ± 1.2% cm³

V = 47.7 ± `\frac {1.2}{100}` cm³

V = 47.7 ± 0.6 cm³


(4) Uncertainty in the Average value of many measurements:

The uncertainty on the average value of many measurements is equal to the mean deviation.

Example:

The six measurements were taken of the diameter of the wire using a Screw gauge which is 1.20, 1.22, 1.23, 1.19, 1.22, 1.21. Determine the uncertainty in the final result?


Solution:

Measurement (in mm) = 1.20, 1.22, 1.23, 1.19, 1.22, 1.21


Average diameter of wire = `\frac {1.20+1.22+1.23+1.19+1.22+1.21}{6}`

The average diameter of wire = 1.21 mm


Deviation of each measurement from (Average diameter of wire = 1.21 mm ) is 

(0.01, 0.01, 0.02, 0.02, 0.1, 0) 

so

Mean deviation = `\frac {0.01+0.01+0.02+0.02+0.1+0}{6}` = 0.01


Thus 

Uncertainty in the mean value of deviation is = 0.01 mm

Hence diameter of wire is = 1.21 ± 0.01 mm 


(5) Uncertainty in Timing Experiment: 

The uncertainty in the time period of a body is found by dividing the least count of the timing devices by the number of Vibrations. 

Example: 

The Simple Pendulum completes 30 vibrations in 54.6 sec. The least count of Stopwatch is 0.1 sec. Find out the uncertainty in the time period of the Simple pendulum.


Given data: 

Time for 30 Vibrations = 54.6 sec


To find:

T = ?


Solution:

By unitary method

Time is taken for 30 Vibrations = 54.6 sec

Time taken for 1 vibration T = `\frac {54.6}{30}`

T =1.82 sec


Uncertainty = Least count / Total No of Vibration

Uncertainty = `\frac {0.1}{30}`

Uncertainty = 0.003 sec


Thus 

T = 1.82 ± 0.003 sec


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