ERROR AND UNCERTAINTY ANALYSIS : EXPLANATION, SOLVED EXAMPLES AND PRACTICE PROBLEMS OF ERROR AND UNCERTAINTY


Error is a very important concept of Physics and Mathematics. It is generally said that nothing is Perfect, this statement is also true in the context of scientific calculations or measurements. Some content of Error always exists in any scientific results. So it is not possible to find the exact value of measurements. The study of error can help us to reduce the content of error in calculations. This topic is also an inseparable part of many exams like CSIR-NET/JRF, GATE, JEST, IIT-JAM, NEET, JEE, etc.

What to be discussed -

  1. What is Error 
  2. Formulae to calculate Error
  3. Solved Problem of Error
  4. What is RMS Error
  5. Solved Problem of RMS Error
  6. What is Uncertainty
  7. How to measure uncertainty
  8. Solved Problem
  9. Practice Problems of Error and Uncertainty Analysis

What is ErrorThe deviation of the measured value from true value is known as error. When we measured any quantity, it has two things, the first is the true value of that quantity which we cannot measure exactly and the other is the uncertainty in that measurement.
  Error = Measured Value-True Value
  •  Absolute Error  ΔX = XM-XT
  •  Relative Error  = ΔX/X
  •  Percentage Error = (ΔX/X) *100
Propagation of Error - Error is propagating through the Arithmetic operations. You should also remember that total Error is always the sum of errors in all quantities, whether it is addition or substruction. While we make mistake or error in measuring the two quantities and when we apply an arithmetic operation even subtraction then the total error will be the sum of each error. The following are the ways to find the total absolute error while we applying arithmetic operations on two or more quantities. 

1. Error in Addition of two quantities
Z = X+Y
ΔZ = Δ X+ Δ Y 

2. Error in Subtraction of two quantities
Z = X-Y
ΔZ = Δ X+ Δ Y 

3. Error in Product of two quantities
Z = X.Y
lnZ = lnX + lnY
ΔZ/Z = ΔX/X + ΔY/Y

4. Error in Division of two quantities
 Z = X/Y 
lnZ = lnX - lnY 
ΔZ/Z = ΔX/X + ΔY/Y

5. Error in Power of a quantity
Z=X^n
ΔZ/Z = n(ΔX/X)

Solved Problem -1   

A Student measures the displacement x from the equilibrium of a stretched spring and reports it be with a 1% error. The spring constant k is known to be 10N/m with 0.5% error. The percentage error in the estimate of the potential energy is 

(1) 0.8%
(2) 2.5%
(3) 1.5%
(4) 3.0%

Solution - The given Potential energy is 
                                                                  

For Relative Error, we have- 
                                                   
For Percentage Error - 


Root Mean Square Error  - It is generally defined as the square root of the mean square. Mathematically, RMS Error for the previous problem can be written as - 


Solved Problem - 2 
The Viscosityof a liquid is given by poiseuille's formula. Assume that l and V can be measured very accurately. but the pressure P has an RMS error of 1% and the radius has an independent RMS error of 3%. The RMS error of the viscosity is closest to 
(1) 2%
(2) 4%
(3) 12%
(4) 13%

Solution - 
                                          

where l and V measured accurately, So we can take these as constants for error calculation 

RMS Error in is given by 


In Percentage - 
 



Uncertainty The margin of error of measurement.

                True Value ± Uncertainty            

For example  X = 10 ± 0.5
How to calculate Uncertainty  

 

Solved Problem-3 
In a measurement of the viscous drag force experienced by spherical particles in a liquid, the force is found to be proportional towhere V is the measured volume of each particle. If V is measured to be, with an uncertainty of, the resulting relative percentage uncertainty in the measured force is 
(1) 2.08
(2) 0.09
(3) 6
(4) 3

Solution - Given:  
                                                       
                                                             , Where K is constant

Practice Problem-1
The decay constantsof the heavy pseudo-scalar mesons, in the heavy quark limit, are related to their massesby the relation
where a is an empirical parameter to be determined. The valuesand correspond to uncorrelated measurements of a meson. The error on the estimate of a is 
(1) 
(2) 
(3) 
(4) 


Practice Problem-2-
The experimentally measured values of the variables x and y areand respectively. What is the error in the calculated value offrom the measurements?
(1) 0.12
(2) 0.05
(3) 0.03
(4) 0.07


Comment your answers of the Practice problem below.
Thank You

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