Errors Of Measurements - Practice Questions & MCQ

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Error in sum and Error in difference of two physical quantities, Error in product and Error in division of two physical quantities are considered the most difficult concepts.
Errors of measurements, Error in quantity raised to some power are considered the most asked concepts.
29 Questions around this concept.

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The period of oscillation of a simple pendulum is given by

$T\ =2\pi\sqrt{\frac{l}{g}}$ , Where $l$ is about $100\ cm$ known with an accuracy of $1\ mm$ . This time is about $2\ seconds$ . The time up to $100$ oscillations is measured with a stopwatch of at least $0.1\ seconds.$ The percentage error in g is:

$0.1\ %$

$1\ %$

$0.2\ %$

$0.8\ %$

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Concepts Covered - 4

Errors of measurements

It is the magnitude of the difference between the true value and the measured value of the quantity.

It may be positive in certain cases and negative in certain other cases

If $a_{1}, a_{2},a_{3}.........a_{n}$ are a measured value

then $a_{m}= \frac{a_{1}+a_{2}+......a_{n}}{n}$

where a_m = true value

then

1)Absolute Error for nth reading = $\Delta a_n= a_{m}- a_{n}$ = true value - measured value

So $\Delta a_{1}$ = $a_{m}- a_{1}$

$\Delta a_{2}$ = $a_{m}- a_{2}$

2) Mean absolute error

$\dpi{100} \Delta\bar{ a}= \frac{\left | \Delta a_{1} \right |+\left | \Delta a_{2} \right |+....\left | \Delta a_{n} \right |}{n}$

3) Relative error or Fractional error

The ratio of mean absolute error to the mean value of the quantity measured.

$Relative \ \ error=\frac{\Delta\bar{ a}}{a_{m}}$

$\Delta\bar{a} - mean\: absolute\: error$

$a_{m}= mean \ \ value$

4) Percentage error

$Percentage \ \ error=\frac{\Delta\bar{ a}}{a_{m}}\times100$ %

Error in sum and Error in difference of two physical quantities

1)Error in sum (x=a+b)

Error in sum (x=a+b):-
absolute error in x= $\Delta x= \pm \left ( \Delta a+\Delta b \right )$

where

$\Delta a$ = absolute error in measurement of a

$\Delta b$ = absolute error in measurement of b

$\Delta x$ = absolute error in measurement of x

The percentage error in the value of x = $\frac{\Delta x}{x} \times 100= \frac{\left ( \Delta a+\Delta b \right )}{a+b}\times 100$

2) Error in difference (x=a-b)

Absolute error in x = $\Delta x= \pm \left ( \Delta a+\Delta b \right )$
Percentage error in the value of x= $\frac{\Delta x}{x} \times 100= \frac{\left ( \Delta a+\Delta b \right )}{a-b}\times 100$

Error in product and Error in division of two physical quantities

1) Error in product $x=ab$

maximum fractional error= $\frac{\Delta x}{x}= \pm \left ( \frac{\Delta a}{a} +\frac{\Delta b}{b} \right )$

where

$\Delta a$ = absolute error in measurement of a

$\Delta b$ = absolute error in measurement of b

$\Delta x$ = absolute error in measurement of x

The percentage error in the value of x= $\dpi{100} \frac{\Delta x}{x}*100= \pm \left ( \frac{\Delta a}{a}*100+ \frac{\Delta b}{b}*100 \right )$

=(% error in value of a + % error in value of b)

2) Error in division $x=\frac{a}{b}$

maximum fractional error in x = $\frac{\Delta x}{x}=\pm \left ( \frac{\Delta a}{a}+\frac{\Delta b}{b} \right )$
The percentage error in the value of x= $\dpi{100} \frac{\Delta x}{x}*100= \pm \left ( \frac{\Delta a}{a}*100+ \frac{\Delta b}{b}*100 \right )$

=(% error in value of a + % error in value of b)

Error in quantity raised to some power

when $(x = \frac{a^n}{b^m})$

The maximum fractional error in x is:- $\frac{\Delta x}{x}=\pm \left ( n\frac{\Delta a}{a}+m\frac{\Delta b}{b} \right )$
Percentage error in the value of x== $\frac{\Delta x}{x}*100=\pm \left ( n\frac{\Delta a}{a}*100+m\frac{\Delta b}{b} *100\right )$