Errors

Every measurement involves errors. It is important to understand their effects on final results and to evaluate the level of accuracy. Main Types of Errors:

1. Systematic Errors
2. Random Errors

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Systematic Errors

These are consistent and repeatable errors caused by imperfections in the measurement system. Causes:

1. Incorrect design or setup of the instrument (e.g., poor calibration).
2. Wrong reading or interpretation by the observer.
3. Limitation of the measurement method.
4. Low accuracy of the formula used.

Minimization of Systematic Errors:

1. Use well-designed and calibrated instruments.
2. Check for zero errors before measurement.
3. Follow standardized procedures.
4. Use corrections based on known errors.

Note: Systematic errors cannot be reduced by taking multiple readings.

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Random Errors

These are unpredictable and vary in magnitude and direction. They occur due to slight changes in the environment or the observer's skill. Causes:

1. Fluctuations in surrounding conditions (e.g., temperature or pressure).
2. Human limitations like parallax error.
3. Instrument insensitivity or imperfections.

Minimization of Random Error

Random errors occur due to unpredictable fluctuations. To minimize them:

1. Repeat the measurement several times.
2. Use highly sensitive and well-calibrated instruments.
3. Calculate the mean of all measurements: $\bar{x} = \frac{x_1 + x_2 + x_3 + \dots + x_n}{n}$

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Mistake

A mistake is an error due to incorrect operation, e.g., reading without accounting for zero error or misusing the instrument.

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Precision of Error

Precision shows how closely multiple measurements agree, regardless of whether they are correct. For example, if three readings of a wire diameter are: $d_1 = 1.23\, \text{mm},\quad d_2 = 1.24\, \text{mm},\quad d_3 = 1.23\, \text{mm},$ they are precise even if the actual diameter is 1.30 mm.

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Blunder

A blunder is a repeated mistake due to carelessness, not a random or systematic error.

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General precautions:

1. Read instructions before performing experiments.
2. Check instruments for zero errors and calibration.
3. Use proper setup and handle instruments correctly.

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Instrumental Errors

• Maintain and store instruments properly.
• Clean and calibrate instruments regularly.

Observation Errors

• Read scales at eye level to avoid parallax.
• Record actual readings, not guessed ones.

Adjustment Errors

• Adjust instruments to remove zero error.
• Ensure instruments are positioned correctly.

Random Errors

• Repeat measurements at different positions.
• Find the average to reduce random fluctuations: $\text{Average} = \frac{\text{Sum of readings}}{\text{Number of readings}}$

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a. The Meter Rule

Measures length with an accuracy of 1 mm or 0.1 cm. If it has 1000 divisions in 1 meter: $\text{Least count} = \frac{1\, \text{m}}{1000} = 0.001\, \text{m} = 1\, \text{mm}$

b. Vernier Calipers

Measures up to 0.01 cm accuracy using main and vernier scales. Suppose: $10\, \text{vernier divisions} = 9\, \text{main scale divisions}$ then the least count (L.C) is: $\text{L.C} = 1\, \text{main division} - 1\, \text{vernier division} = 1\, \text{mm} - 0.9\, \text{mm} = 0.1\, \text{mm} = 0.01\, \text{cm}$

c. Micrometer Screw Gauge

Measures very small lengths with an accuracy of 0.01 mm. Let: $\text{Pitch} = \frac{\text{distance moved by spindle}}{\text{number of turns}} = 0.5\, \text{mm}$ and the thimble has 50 divisions: $\text{L.C} = \frac{\text{pitch}}{\text{number of divisions}} = \frac{0.5\, \text{mm}}{50} = 0.01\, \text{mm}$ Total Reading: $\text{Reading} = \text{sleeve reading} + \text{(thimble reading)} \times \text{L.C}$

Diagram of micrometer screw gauge

A micrometer screw gauge is used in measuring diameters of the wires thickness of the metal sheets, diameter of ball bearings and other tiny lengths. Before using it the gap between anvil and spindle has to be closed to check for zero error. An object to be measured is placed between anvil and spindle.

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Percentage Error

The percentage error is calculated by multiplying the relative error (in decimal form) by 100. This makes it more convenient to express errors in percentage rather than as fractions or decimals. Formula for Percentage Error:

$$\text{Percentage Error} = \left(\frac{\Delta x}{x}\right) \times 100$$

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Error Analysis in a Sum of Quantities

When performing operations like addition or subtraction on measured quantities, the errors in these quantities also affect the final result. Here's how the error analysis works for the sum of two quantities: Given: Let: $a$ and $b$ be the two measured quantities, $x = a + b$ be the result of their sum, $\Delta a$ and $\Delta b$ be the absolute errors in $a$ and $b$, respectively, $\Delta x$ be the absolute error in the result $x$. Equation for the sum:

$$x = a + b$$

However, due to errors in $a$ and $b$, the measured result may deviate. Thus, the measured result is:

$$x \pm \Delta x = (a \pm \Delta a) + (b \pm \Delta b)$$

Expanding this equation gives:

$$x \pm \Delta x = a + b \pm (\Delta a + \Delta b)$$

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Errors in a Difference

When subtracting two quantities $x = a - b$, the absolute errors $\Delta a$ and $\Delta b$ in $a$ and $b$ are used to calculate the error in the difference. The equation for the difference is:

$$x \pm \Delta x = (a \pm \Delta a) - (b \pm \Delta b)$$

After expanding, the four possible values for $\Delta x$ are:

$$\Delta a + \Delta b, \quad \Delta a - \Delta b, \quad -\Delta a - \Delta b, \quad -\Delta a + \Delta b$$

The maximum possible error in $x$ is:

$$\Delta x = \Delta a + \Delta b$$

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Errors in a Product

For the product of two quantities $x = ab$, the equation becomes:

$$x \pm \Delta x = (a \pm \Delta a)(b \pm \Delta b)$$

To find the fractional error in the product, divide both sides by $x = ab$:

$$\frac{\Delta x}{x} = \pm \frac{\Delta a}{a} \pm \frac{\Delta b}{b}$$

Since $\frac{\Delta a}{a} \times \frac{\Delta b}{b}$ is very small, it can be neglected. Therefore, the maximum fractional error is:

$$\frac{\Delta x}{x} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$

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Errors in Division

For the division of two quantities $x = \frac{a}{b}$, the equation becomes:

$$x \pm \Delta x = \frac{a \pm \Delta a}{b \pm \Delta b}$$

To find the fractional error, divide both sides by $x = \frac{a}{b}$:

$$\frac{\Delta x}{x} = \pm \frac{\Delta a}{a} \pm \frac{\Delta b}{b}$$

The maximum fractional error in the division is:

$$\frac{\Delta x}{x} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$

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Errors in Exponents

For a quantity $x = a^n b^m$, applying natural logarithms to both sides:

$$\ln(x) = n \ln(a) - m \ln(b)$$

Taking the derivative of both sides:

$$\frac{dx}{x} = n \frac{da}{a} - m \frac{db}{b}$$

This can be written as:

$$\frac{\Delta x}{x} = n \frac{\Delta a}{a} - m \frac{\Delta b}{b}$$

The maximum fractional error is:

$$\frac{\Delta x}{x} = \pm n \frac{\Delta a}{a} + m \frac{\Delta b}{b}$$

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Example: Percentage Error in the Determination of $g$

Consider the formula:

$$g = \frac{4\pi^2 l}{T^2}$$

where: $l$ is measured with a $\pm 2\%$ error, $T$ is measured with a $\pm 3\%$ error. The maximum percentage error in $g$ is given by:

$$\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2 \times \frac{\Delta T}{T}$$

Substituting the given errors:

$$\frac{\Delta g}{g} = 2\% + 2 \times 3\% = 2\% + 6\% = 8\%$$

Therefore, The maximum percentage error in the determination of $g$ is $8\%$.