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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.
48 Questions around this concept.
The magnitude of difference between the true value and measured value of quantity is called
The value of absolute error of first measurement in a measured value a1,a2..............am is equal to
[am is the true value]
The ratio of mean absolute error to the mean value of the quantity measured is called:
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The unit of percentage error is :
For a measurement of the radius of a ball following readings are taken:
3.26cm 3.28cm 3.31cm
absolute error for the first reading is :
For the measurement of the cylinder following readings are taken :
1.52cm 1.50cm 1.51cm 1.48cm
mean absolute error for the measurement is :
The value of the two resistor are $R_1=(6 \pm 0.3) K \Omega$ and $R_2=(10 \pm 0.2) K \Omega$.The maximum absolute error in equivalent resistance when they are connected in series will be:
Two roads of length $(3.161 \pm 0.3) \mathrm{cm}$ and $(1.121 \pm 0.1) \mathrm{cm}$. What is the percentage error in the measurement of their difference :
If the length of the stick $P$ is $(28.7 \pm 0.5) \mathrm{cm}$ and that of the stick $Q$ is $(19.6 \pm 0.3) \mathrm{cm}$. What will be the percentage error in $R$, if $R=P+Q$
If the mass of box $A$ is $(3.25 \pm 0.01) \mathrm{kg}$ and that of $B$ is $(4.19 \pm 0.01) \mathrm{kg}$, then box $B$ is heavier than $A$ by:
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 $a_1, a_2, a_3 \ldots \ldots \ldots a_n$ are a measured value then
$
a_m=\frac{a_1+a_2+\ldots \ldots a_n}{n}
$
where am = true value
then
1)Absolute Error for nth reading $=\Delta a_n=a_m-a_n=$ true value - measured value
$$
\begin{aligned}
S_0 \Delta a_1 & =a_m-a_1 \\
\Delta a_2 & =a_m-a_2
\end{aligned}
$$
2) Mean absolute error
$$
\Delta \bar{a}=\frac{\left|\Delta a_1\right|+\left|\Delta a_2\right|+\ldots .\left|\Delta a_n\right|}{n}
$$
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
$$
\text { Percentage error }=\frac{\Delta \bar{a}}{a_m} \times 100 \%
$$
1)Error in sum (x=a+b)
Error in sum (x=a+b):-
absolute error in $\mathrm{x}=\Delta x= \pm(\Delta a+\Delta b)$
where
$\Delta a=$ absolute error in measurement of a
$\Delta x=$ absolute error in measurement of $\times$
The percentage error in the value of x
= $\frac{\Delta x}{x}=\frac{(\Delta a+\Delta b)}{a+b} \times 100$
2) Error in difference (x=a-b)
absolute error in $\mathrm{x}=\Delta x= \pm(\Delta a+\Delta b)$
Percentage error in the value of $\mathrm{x}=\frac{\Delta x}{x}=\frac{(\Delta a+\Delta b)}{a-b} \times 100 \%$
1) Error in product x=a.b
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=
$=\frac{\Delta x}{x} * 100= \pm\left(\frac{\Delta a}{a} * 100+\frac{\Delta b}{b} * 100\right)$
=(% error in value of a + % error in value of b)
2) Error in division x = a/b
maximum fractional error in
$
=\frac{\Delta x}{x}= \pm\left(\frac{\Delta a}{a}+\frac{\Delta b}{b}\right)
$
The percentage error in the value of $x=$
$
=\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)
when $\left(x=\frac{a^n}{b^m}\right)$
- The maximum fractional error in $\times$ is:- $\frac{\Delta x}{x}= \pm\left(n \frac{\Delta a}{a}+m \frac{\Delta b}{b}\right)$
- Percentage error in the value of $\mathrm{x}==\frac{\Delta x}{x} * 100= \pm\left(n \frac{\Delta a}{a} * 100+m \frac{\Delta b}{b} * 100\right)$
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