physics-class-11-unit-02-chapter-04-accuracy-and-precision-of-measuring-instruments-lecture-4-4-2ldkopn9kbi

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Error Analysis </primary>
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- In absence of precise measurement we take many readings of the same measurement.

- $a_{\text {mean }}=\frac{a_1+a_2 \cdots+a_n}{n}$

- Absolute error: $\left|\Delta a_1\right|=\left|a_1-a_{\text {mean }}\right|$

- Mean absolute error: $\Delta a_{\text {mean }}=\frac{\left(\left|\Delta a_1\right|+|\Delta Q| \cdots+\left|\Delta a_n\right|\right)}{n}$

- $a_{\text {mean }}-\Delta a_{\text {mean }} \leqslant a \leqslant a+\Delta a_{\text {mean }}$

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<footer style = "font-size: 18px"> $\rightarrow$ Accuracy and Precision of Measuring Instruments $\rightarrow$ <span style = "color:blue"> Error Analysis </span> $\rightarrow$ Error Analysis $\rightarrow$ Combination of Errors </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Combination of Errors </primary>
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- $Z \pm \Delta Z=(A \pm A A)+(B \pm \Delta B)$

- $ =A B(A+B) \pm(\Delta A)$

- $\Delta Z=\Delta A+\Delta B  \pm(A B)$

- <span style="color:blue">Subtraction:</span>

- $Z=A-B$

- $Z \pm \Delta Z  =(A \pm \Delta A)-(B \pm \Delta B)$

- $ =(A-B) \pm \Delta A \pm \Delta B$

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<footer style = "font-size: 18px">Error Analysis $\rightarrow$ Combination of Errors $\rightarrow$ <span style = "color:blue"> Combination of Errors </span> $\rightarrow$ Combination of Errors $\rightarrow$ Combination of Errors </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Combination of Errors </primary>
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- **Product or Quotient**

- $ Z = \frac{A}{B} \text { or } A \times B$

- $ Z=A B $

- $ Z \pm \Delta Z=(A \pm \Delta A)(B \pm \Delta B) $

- $ Z \pm \Delta Z=(A B) \pm \Delta A B \pm \Delta B A \pm \Delta A\Delta B $

- Now, divide above equation by Z.

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<footer style = "font-size: 18px">Combination of Errors $\rightarrow$ Combination of Errors $\rightarrow$ <span style = "color:blue"> Combination of Errors </span> $\rightarrow$ Combination of Errors $\rightarrow$ Combination of Errors </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Combination of Errors </primary>
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- $1 \pm \frac{\Delta Z}{Z}=1 \pm \frac{\Delta A}{A} \pm \frac{\Delta B}{B} \pm \frac{\Delta A \Delta B}{A B}$

- **a)** Convert $\pm$ to "+" for maximum error.

- **b)** $(\Delta A \Delta B) \rightarrow$ neglect product of 2 small quantity.

- $\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$

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### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Combination of Errors </primary>
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- $ Z =\frac{A}{B}$

- $ \frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$

- If $ Z=\frac{A^n}{B^m}$

- $ \frac{\Delta Z}{Z}= \pm\left(n \frac{\Delta A}{A}+m \frac{\Delta B}{B}\right)$

- % error in Z = n(% error A)+ m(% error B)

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<footer style = "font-size: 18px">Combination of Errors $\rightarrow$ Combination of Errors $\rightarrow$ <span style = "color:blue"> Combination of Errors </span> $\rightarrow$ Problem $\rightarrow$ Significant Figures </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- If $H=I^2 R t$, where, $I$ is current, $R$ is resistance, $t$ is time, and $H$ Heat generate. Error in measurements of I, R and t are 2%, 3% and 1%. Find relative error in measurement of $H$ ?

- $\frac{\Delta H}{H}=\frac{2 \Delta I}{I}+\frac{\Delta R}{R}+\frac{\Delta t}{t}$

- $ \frac{\Delta H}{H} \times 100 $%$=2 \frac{\Delta I}{I} \times 100 $%$+\frac{\Delta R}{R} \times 100 $%

- $=2 \times 2 $%+3 %+1 %  = 8 %

- Error in $ \Delta H(\text { in })$ % $= \pm 8 $%

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<footer style = "font-size: 18px">Combination of Errors $\rightarrow$ Combination of Errors $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Significant Figures $\rightarrow$ Significant Figures </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Significant Figures </primary>
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- Whenever any measurement is reported, the uncertainty in the measurement is in the last digit.

- **For example:** Time period $=1.62 {s}$.

- 1 and 6 are reliable digits.

- $2 \rightarrow$ uncertainty.

- $1.62{s} \rightarrow$ correct upto 3 significant digits.

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<footer style = "font-size: 18px">Combination of Errors $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Significant Figures </span> $\rightarrow$ Significant Figures $\rightarrow$ Significant Figures </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Significant Figures </primary>
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- $1.62 \rightarrow 3$ significant digits.

- $287.5 \mathrm{cm} \rightarrow 4$ significant digits.

- Significant digits indicate the precision of an instrument which defend on the lead count.

- Choice of different units should nut affect the number of significant digits.

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<footer style = "font-size: 18px">Problem $\rightarrow$ Significant Figures $\rightarrow$ <span style = "color:blue"> Significant Figures </span> $\rightarrow$ Significant Figures $\rightarrow$ Significant Figures </footer>

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### <primary style="font-size:24px"> Significant Figures </primary>
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- $2.308 \mathrm{cm} \rightarrow 4$ significant.

- $2.308 \mathrm{cm}$ = $23.08 \mathrm{mm}$ = $0.02308 \mathrm{m}$.

- <span style="color:blue">**Rules:**</span>

- **1)** All non zero digits are significant.

- **2)** Zeros between 2 non zero digits ore significant irrespective of decimal point.

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<footer style = "font-size: 18px">Significant Figures $\rightarrow$ Significant Figures $\rightarrow$ <span style = "color:blue"> Significant Figures </span> $\rightarrow$ Significant Figures $\rightarrow$ Significant Figures </footer>

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### <primary style="font-size:24px"> Significant Figures </primary>
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- **Order of magnitude:**

- $a \times 10^b$

- If $ 1 \leqslant a \leqslant 5$

- round it off to 1

- $\approx 10^b \quad$ b is called **order of magnitude**.

- $5 \leqslant a \leqslant 10 \rightarrow$ round of to the next digit

- $ 10^{(b+1)} \rightarrow(b+1)$ is called the **order of magnitude.**
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### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Significant Figures </primary>
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- **For example:** $1.28 \times 10^7 \mathrm{m}$

- Order of magnitude $ \sim 10^7 \mathrm{m}$

- Diameter of Hydrogen atom: $1.06 \times 10^{-10} \mathrm{m}$

- Order of magnitude $\sim 10^{-10} \mathrm{m}$

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### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Significant Figures </primary>
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- Exact numbers can be considered to have an infinite number of significant figures.

- **For example :**

- Diameter $={2} \times$ radius

- 2 is exact number.

- So it has $\infty$ significant digits.

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<footer style = "font-size: 18px">Significant Figures $\rightarrow$ Significant Figures $\rightarrow$ <span style = "color:blue"> Significant Figures </span> $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rules for Arithmetic Operations with Significant Figures </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Rules for Arithmetic Operations with Significant Figures </primary>
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- Final result cannot be more accurate than the original measured values.

- In general, the final result should not have more significant figures than the original data from which it was obtained.

- **Example:**

- $ \text {  Density }=\frac{\text { mass }}{\text { volume }} =\frac{4.237 \mathrm{gms}}{2.51 \mathrm{cm}^3} $

- $ \text { Density } =1.68804780876 \frac{\mathrm{g}}{\mathrm{\textrm {cm } ^ { 3 }}}$

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<footer style = "font-size: 18px">Significant Figures $\rightarrow$ Significant Figures $\rightarrow$ <span style = "color:blue"> Rules for Arithmetic Operations with Significant Figures </span> $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rules for Arithmetic Operations with Significant Figures </footer>

- In multiplication or division, the final result should retain as many significant
- figures as are there in the original number with the least significant figures.

- $ m = 4.2379$, 4 significant digits.

- $V = 2.51 \mathrm{cm}^3$, 3 significant digits.

- Density should have three significant figures.

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<footer style = "font-size: 18px">Significant Figures $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ <span style = "color:blue"> Rules for Arithmetic Operations with Significant Figures </span> $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rules for Arithmetic Operations with Significant Figures </footer>

- $ \text {  Density }=\frac{\text { mass }}{\text { volume }}$

- $\quad$$\quad$$\quad$$=\frac{4.237 \mathrm{gms}}{2.51 \mathrm{~cm}^3} $

- $ \text { density } =1.68804780876 \frac{\mathrm{g}}{\mathrm{cm}^3}$

- $ =1.69 \mathrm{gm} / \mathrm{cm}^3$

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<footer style = "font-size: 18px">Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ <span style = "color:blue"> Rules for Arithmetic Operations with Significant Figures </span> $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rounding off the Uncertain Digits </footer>

- In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

- **For example**, the sum of the numbers 436.32 g, 227.2 g and 0.301 g by mere arithmetic
- addition, is 663.821 g. But the least precise measurement (227.2 g) is correct to only one decimal place. The final result should, therefore, be rounded off to 663.8 g.

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<footer style = "font-size: 18px">Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ <span style = "color:blue"> Rules for Arithmetic Operations with Significant Figures </span> $\rightarrow$ Rounding off the Uncertain Digits $\rightarrow$ Rounding off the Uncertain Digits </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Rounding off the Uncertain Digits </primary>
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- A number 2.746 rounded off to three significant figures is 2.75, while the number 2.743 would be 2.74.

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<footer style = "font-size: 18px">Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rules for Arithmetic Operations with Significant Figures $\rightarrow$ <span style = "color:blue"> Rounding off the Uncertain Digits </span> $\rightarrow$ Rounding off the Uncertain Digits $\rightarrow$ Rounding off the Uncertain Digits </footer>

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### <primary style="font-size:24px"> Rounding off the Uncertain Digits </primary>
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- Preceding digit is raised by 1 if the insignificant digit to be dropped (the
- underlined digit in this case) is more than 5,

- Is left unchanged if the latter is less than 5.

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<footer style = "font-size: 18px">Rules for Arithmetic Operations with Significant Figures $\rightarrow$ Rounding off the Uncertain Digits $\rightarrow$ <span style = "color:blue"> Rounding off the Uncertain Digits </span> $\rightarrow$ Rounding off the Uncertain Digits $\rightarrow$ Rounding off the Uncertain Digits </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Rounding off the Uncertain Digits </primary>
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- 2.745 rounded off to three significant figures becomes 1.74. 
 
- The number 2.735 rounded off to three significant figures becomes 1.74 since
- the preceding digit is odd.

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<footer style = "font-size: 18px">Rounding off the Uncertain Digits $\rightarrow$ Rounding off the Uncertain Digits $\rightarrow$ <span style = "color:blue"> Rounding off the Uncertain Digits </span> $\rightarrow$ Rounding off the Uncertain Digits $\rightarrow$ Rules for Determining the Uncertainty </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Rules for Determining the Uncertainty </primary>
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- $ L=16.2 \mathrm{cm} $, and $ B=10.1 \mathrm{cm}$

- Area = $ 163.62 \mathrm{cm}^2 $.

- $ L=16.2 \pm 0.1 \mathrm{cm} $, $ \Delta L=0.1 \mathrm{cm} $

- $ B=10.1 \pm 0.1 \mathrm{~cm} $

- $ A=(16.2 \pm 0.1)(10.1 \pm 0.1) \mathrm{cm}^2 $

- $\frac{\Delta L}{L}=\frac{0.1}{16.2} \times 100 $% $=0.6$ %

- $ \frac{\Delta B}{B}=\frac{0.1}{10.1} \times 100 $%=1 %

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<footer style = "font-size: 18px">Rounding off the Uncertain Digits $\rightarrow$ Rounding off the Uncertain Digits $\rightarrow$ <span style = "color:blue"> Rules for Determining the Uncertainty </span> $\rightarrow$ Rules for Determining the Uncertainty $\rightarrow$ Rules for Determining the Uncertainty </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Rules for Determining the Uncertainty </primary>
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- $ A={L B} ; \frac{\Delta A}{A}=\frac{\Delta L}{L}+\frac{\Delta B}{B}$

- $(\frac{\Delta A}{A}) $% = 0.6 %+1 %  = 1.6 %

- $\Delta A=\frac{1.6}{100} \times A $

- $ A=16.2 \times 10.1$

- $A=163.62 \mathrm{cm}^2 \pm 1.6 $%$ \leftarrow \text{2.6cm}$

- $ =164 \mathrm{cm}^2 \pm 3 \mathrm{cm}^2$

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<footer style = "font-size: 18px">Rounding off the Uncertain Digits $\rightarrow$ Rules for Determining the Uncertainty $\rightarrow$ <span style = "color:blue"> Rules for Determining the Uncertainty </span> $\rightarrow$ Rules for Determining the Uncertainty $\rightarrow$ Rules for Determining the Uncertainty </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Rules for Determining the Uncertainty </primary>
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- The relative error of a value of number specified to significant figures depends not
- only on n but also on the number itself.

- **For example**, the accuracy is $1.029 \pm 0.019$.

- Relative error $=\frac{0.01}{1.02} \times 100=1$%

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<footer style = "font-size: 18px">Rules for Determining the Uncertainty $\rightarrow$ Rules for Determining the Uncertainty $\rightarrow$ <span style = "color:blue"> Rules for Determining the Uncertainty </span> $\rightarrow$ Rules for Determining the Uncertainty $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Accuracy And Precision Of Measuring Instruments L-4 </Success>
### <primary style="font-size:24px"> Rules for Determining the Uncertainty </primary>
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- Similarly, $9.89 \mathrm{~g} \pm 0.01 \mathrm{g}$

- Relative error $=\frac{0.01}{9.89} \times 100 \% $ =0.1 %
 
- Intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.

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