physics-class-11-unit-02-chapter-03-measurements-and-introduction-to-error-analysis-lecture-3-4-aglkwyh-j5e

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- n divisions of vernier scale =(n-1) divisions of main scale.

- eg; 10 divisions on vernier scale(V) = 9 divisions on main sale.

- 1 division of $V=\frac{9}{10}$ division of $S$.

- s=1 mm, V=0.9 mm.

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<footer style = "font-size: 18px">Measurements and Introduction to Error Analysis $\rightarrow$ Vernier Callipers $\rightarrow$ <span style = "color:blue"> Vernier Callipers </span> $\rightarrow$ Vernier Callipers $\rightarrow$ Vernier Callipers </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- The bottom scale is the main scale.

- The top scale is a movable scale, is called the Vernier scale.

- The 0 of the Vernier scale is not matching with the 0 of the main scale.

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

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- AB = AC - BC

- AB = 6S - 6V

- Least count of Vernier Calliper =S-V

- $ n V=(n-1) s $

- $ V=\frac{n-1}{n} s $

- Least count of Vernier Calliper $=S-\frac{n-1}{n} S=\frac{S}{n} $.

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### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- $s=1 \mathrm{mm}$

- 0 of Vernier occurs after 2 readings of main scale.

- Main scale reading = 2 mm

- The 4th reading of V coincides with main scale.

- $x=4S-4 V=4 \times 1-4 \times 0.9= 0.4 mm$

- The total reading of that length which the Vernier caliper is showing to us,

- 2 + 0.4 = 2.4 mm

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### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- In a Vernier caliper there is a moving jaw which is adjusted.

- You find how many divisions of the main scale are and then, matching with how many divisions on the Vernier scale.

- The Vernier scale is at the bottom.

- The number of divisions on the Vernier scale are 50 and they match with 49 divisions of the main scale.

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### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- The main scale reading is 3.3 centimeters.

- The Vernier scale we have the 20th reading.

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### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- 2 Vernier Callipers in which on mainscale 1 cm $\rightarrow$ 10 divisions

- $ V_1: S_1=1 \mathrm{~mm}$

- $ V_2: S_2=1 \mathrm{~mm}$

- **Calliper 1** : 10 equal divisions of Vernier = 9 divisions of main scale.

- **Second Calliper** :10 divisions of V = 11 divisions of S

- V = 1.1 mm

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### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- Second case: $x_1$ has to be measured.

- $x_1=$ distance between main scale reading and zero of vernier.

- 10 readings of the Vernier were matching 11 readings of the main scale.

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### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Vernier Callipers </primary>
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- $x_1$ $=8 \mathrm{S}-7 \mathrm{V}$

- $=8 \mathrm{mm}-7 \times 1.1 \mathrm{mm}$

- $ =(8-7.7) \mathrm{mm}$

- $ =0.3 \mathrm{~mm}$

- $C_2$ $\text { reading }=2.8 \mathrm{cm}+0.3 \mathrm{mm}=2.83 \mathrm{cm}$

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<footer style = "font-size: 18px">Vernier Callipers $\rightarrow$ Vernier Callipers $\rightarrow$ <span style = "color:blue"> Vernier Callipers </span> $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ Screw Gauge and Spherometer </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Screw Gauge and Spherometer </primary>
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- Thread of a screw has a helical shape which implies, if we make rotation on the thread of screw, we also make an axial movement.

- The axial distance moved when we complete 1 revolution on screw = pitch

- **Pitch** - Linear dimension corresponds to $360^{\circ}$ rotation on a screw.

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<footer style = "font-size: 18px">Vernier Callipers $\rightarrow$ Vernier Callipers $\rightarrow$ <span style = "color:blue"> Screw Gauge and Spherometer </span> $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ Screw Gauge and Spherometer </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Screw Gauge and Spherometer </primary>
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- The linear scale can measure the axial distance which the screw moves.

- The partial rotations of the screw are measured by circular scale.

- When these two are added we get the total reading.

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<footer style = "font-size: 18px">Vernier Callipers $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ <span style = "color:blue"> Screw Gauge and Spherometer </span> $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ Screw Gauge and Spherometer </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Screw Gauge and Spherometer </primary>
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- Pitch of the screw corresponds to the smallest reading of main scale. We rotate the spindle by 1 revolution and observe the axial distance moved on the mainscale.

- <span style="color:green">Typically ~ 0.5 mm or 1 mm </span>

- Circular scale enables us to read fractional reading.

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<footer style = "font-size: 18px">Screw Gauge and Spherometer $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ <span style = "color:blue"> Screw Gauge and Spherometer </span> $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ Screw Gauge and Spherometer </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Screw Gauge and Spherometer </primary>
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- Let number of divisions on circular scale be n.

- Let pitch of the screw be p.

- Least Count of the screw = $\frac{p}{n}$

- If Circular scale reading = m.

- Extra distance / length to be added to main scale reading = ${m} \times\frac{p}{n}$.

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### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Screw Gauge and Spherometer </primary>
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- For example, pitch p = 0.5 mm

- Number of divisions on the circular scales n = 50

- Least Count = $\frac{0.5}{50}$ = 0.01 mm

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<footer style = "font-size: 18px">Screw Gauge and Spherometer $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ <span style = "color:blue"> Screw Gauge and Spherometer </span> $\rightarrow$ Micrometer Screw Gauge and Zero Error $\rightarrow$ Accuracy, Precision of Instruments </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Micrometer Screw Gauge and Zero Error </primary>
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- The pitch of this instrument is 0.5 millimeters.

- The circular scale has 50 divisions.

- The least count of this instrument is 0.01 mm.

- Zero error has to be substracted from the reading.

- Zero error can be the positive (+ve) or negative (-ve).

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<footer style = "font-size: 18px">Screw Gauge and Spherometer $\rightarrow$ Screw Gauge and Spherometer $\rightarrow$ <span style = "color:blue"> Micrometer Screw Gauge and Zero Error </span> $\rightarrow$ Accuracy, Precision of Instruments $\rightarrow$ Accuracy and Precision </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Accuracy, Precision of Instruments </primary>
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- <span style="colorblue">Errors associated in measurements</span>
- Any measurement taken is not 100% accurate and the uncertainity of deviation of
- measured value from actual value = <span style="color:green">error</span>.

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<footer style = "font-size: 18px">Screw Gauge and Spherometer $\rightarrow$ Micrometer Screw Gauge and Zero Error $\rightarrow$ <span style = "color:blue"> Accuracy, Precision of Instruments </span> $\rightarrow$ Accuracy and Precision $\rightarrow$ Accuracy and Precision </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Accuracy and Precision </primary>
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- **<span style="color:green">ACCURACY</span> :-** how close we are to the actual numerical value.

- **<span style="color:green">PRECISION</span> :-** resolution of the quantity measured.

- If we take measurements from a typical scale $\rightarrow$ precision 1 mm.

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<footer style = "font-size: 18px">Micrometer Screw Gauge and Zero Error $\rightarrow$ Accuracy, Precision of Instruments $\rightarrow$ <span style = "color:blue"> Accuracy and Precision </span> $\rightarrow$ Accuracy and Precision $\rightarrow$ Systematic Errors </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Accuracy and Precision </primary>
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- Accuracy depends on many factors include precision.

- Eg; True Length of a line = 3.678 cm.

- Measure with scale of Least Count 1 mm = 3.5 cm.

- Measure with an instrument of Least Count 0.1 mm = 3.38 cm.

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

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### <primary style="font-size:24px"> Systematic Errors </primary>
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- Errors tend to be in 1 direction, i.e., they are either the positive or negative.

- <span style="color:blue">**a) Instrumental Error :-**</span> Calibration of instrument is not perfect,  zero error.

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<footer style = "font-size: 18px">Accuracy and Precision $\rightarrow$ Accuracy and Precision $\rightarrow$ <span style = "color:blue"> Systematic Errors </span> $\rightarrow$ Systematic Errors $\rightarrow$ Random Error and Least Count Error </footer>

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### <primary style="font-size:24px"> Systematic Errors </primary>
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- <span style="color:blue">**b) Experimental Technique :-** </span>e.g. we measure temperature of the human body, we can keep the thermometer under tongue or under armpit. Different techniques of measurement give  different temperatures.

- <span style="color:blue">**c) Personal Errors :-**</span> Individual bias, setting, carelessness of observer.

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<footer style = "font-size: 18px">Accuracy and Precision $\rightarrow$ Systematic Errors $\rightarrow$ <span style = "color:blue"> Systematic Errors </span> $\rightarrow$ Random Error and Least Count Error $\rightarrow$ Measurement </footer>

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### <primary style="font-size:24px"> Random Error and Least Count Error </primary>
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- Occur irregularly, are random unpredictable fluctuations in instruments, (temperature, voltage, vibrations in set up)

- The smallest resolution of the instrument.

- The least count of a vernier caliper is 0.1 mm

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<footer style = "font-size: 18px">Systematic Errors $\rightarrow$ Systematic Errors $\rightarrow$ <span style = "color:blue"> Random Error and Least Count Error </span> $\rightarrow$ Measurement $\rightarrow$ Error </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Absolute Error </primary>
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- $a_1 \ldots a_n$

- true value not known, mean is taken as true value.

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

- $\Delta a_1=a_1-a_{\text {mean }}$

- $\Delta a_2=a_2-a_{\text {mean }}$

- $\therefore a_n=a_n-a_{\text {mean }}$

- **<span style="color:green">Take absolute values.</span>**

- $|\Delta a| \leftarrow$ always positive.

- Arithmetic mean of all absolute errors, called mean absolute error of $a$.

- $\Delta a_{\text {mean }}=\frac{\left(\left|\Delta a_1\right|+\left|\Delta a_2\right| \cdots+|| a_n \mid\right)}{n}$

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<footer style = "font-size: 18px">Absolute Error $\rightarrow$ Absolute Error $\rightarrow$ <span style = "color:blue"> Absolute Error </span> $\rightarrow$ Absolute Error $\rightarrow$ Relative or Percentage Error </footer>

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### <primary style="font-size:24px"> Absolute Error </primary>
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- Single measurement, we expect this to be in range.

- $a_{\text {mean }} \pm \Delta a_{\text {mean }}$

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

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<footer style = "font-size: 18px">Absolute Error $\rightarrow$ Absolute Error $\rightarrow$ <span style = "color:blue"> Absolute Error </span> $\rightarrow$ Relative or Percentage Error $\rightarrow$ Relative or Percentage Error </footer>

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### <primary style="font-size:24px"> Relative or Percentage Error </primary>
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- Relative error $=\frac{\Delta a_{\text {mean }}}{a_{\text {mean }}}$

- Relative error is always a fraction.f

- % error : $\delta a=\frac{\Delta a_{\text {mean }}}{a_{\text {mean }}} \times 100$%

- **Example:** Time period of oscillations
- 5 measurements

- $2.63 \mathrm{s}$, $2.56 \mathrm{s}$, $2.42 \mathrm{s}$, $2.71 \mathrm{s}$, and $2.80 \mathrm{s}$

- Total sum = $\sum 13.12 \mathrm{s}  $

- $\text { mean } =\frac{13.12}{5}$

- $=2.624 \mathrm{s}=2.62 \mathrm{s}$

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<footer style = "font-size: 18px">Absolute Error $\rightarrow$ Absolute Error $\rightarrow$ <span style = "color:blue"> Relative or Percentage Error </span> $\rightarrow$ Relative or Percentage Error $\rightarrow$ Relative or Percentage Error </footer>

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### <primary style="font-size:24px"> Relative or Percentage Error </primary>
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- **Example:** Time period of oscillations

- <span style="color:green">Absolute errors :</span>

- $\Delta a_1=2.63-2.62=0.01 \mathrm{s}$

- $\Delta a_2=2.56-2.62=-0.06 \mathrm{s}$

- $\Delta a_3=2.43-2.62=-0.20 \mathrm{s}$

- $\Delta a_4=2.71-2.62=0.09 \mathrm{s}$

- $\Delta a_3=2.80-2.62=0.18 \mathrm{s}$

- absolute error $=\frac{(0.01+0.06+0.20+0.09+0.18)}{ 5}$

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<footer style = "font-size: 18px">Absolute Error $\rightarrow$ Relative or Percentage Error $\rightarrow$ <span style = "color:blue"> Relative or Percentage Error </span> $\rightarrow$ Relative or Percentage Error $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Measurements And Introduction To Error Analysis L-3 </Success>
### <primary style="font-size:24px"> Relative or Percentage Error </primary>
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- <span style="color:blue">Absolute error</span> = $\frac{0.54}{5}=0.11 \mathrm{~s}$

- $T=2.62 \pm 0.11 \mathrm{~s}$

- $\text { error }>0.1 \mathrm{~s}$

- $T=2.6 \pm 0.1 \mathrm{~s}$

- relative error % $= \frac{0.1}{2.6} \times 100=4 $%

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<footer style = "font-size: 18px">Relative or Percentage Error $\rightarrow$ Relative or Percentage Error $\rightarrow$ <span style = "color:blue"> Relative or Percentage Error </span> $\rightarrow$ Thank You $\rightarrow$  </footer>