Errors in Measurements and Dimensional Analysis

1. Errors in Measurements

In physics, measurements are fundamental. However, no measurement is perfectly accurate. There will always be some degree of uncertainty. This uncertainty is called an 'error'. Understanding and quantifying these errors is crucial for interpreting experimental results correctly.

1.1 Types of Errors

Errors can be broadly classified into two main categories:

1.1.1 Systematic Errors

Systematic errors are those that tend to occur in the same direction and by the same amount (or proportionally) during each measurement. They arise from a flaw in the experimental setup, the instrument, or the method used. These errors can often be identified and corrected if their cause is known.

• Instrumental Errors: These errors occur due to faulty calibration of the measuring instrument. For example, a weighing scale that always shows 10 grams more than the actual weight, or a voltmeter that reads 0.1V even when no current is flowing.
• Imperfections in Experimental Setup: Errors due to the surroundings or the way the experiment is set up. For instance, measuring the length of an object in direct sunlight might lead to expansion and an incorrect reading, or an experiment on heat transfer might be affected by drafts in the room.
• Personal Errors: These arise from the individual's lack of proper technique or bias. Examples include parallax error when reading a scale (looking at the scale from an angle instead of perpendicularly), or consistently reading a thermometer too high or too low due to poor eyesight.
• Theory Errors: Sometimes, the theoretical model used has limitations that introduce errors. For example, assuming air resistance is zero in projectile motion when it actually exists.

Systematic errors are often difficult to detect because they consistently shift the measured value away from the true value. However, if an instrument is consistently off by a certain amount, this can be detected by comparing it with a standard instrument or by performing a known measurement.

1.1.2 Random Errors

Random errors are unpredictable and fluctuate from one measurement to another. They occur due to unpredictable variations in experimental conditions or limitations in the precision of the measuring instrument. These errors can be positive or negative, meaning they can cause the measured value to be higher or lower than the true value.

• Fluctuations in Conditions: Unforeseen changes in temperature, pressure, or vibration during an experiment.
• Observer's Limitations: Even with careful technique, there's a limit to how precisely an observer can read a scale or estimate a value. This inherent uncertainty leads to random errors.
• Instrumental Limitations: The inherent precision of an instrument limits the accuracy of readings. For example, a meter scale marked only in millimeters will lead to random errors when trying to measure a length to the nearest tenth of a millimeter.

Random errors cannot be eliminated entirely, but their effect can be minimized by repeating the measurement multiple times and taking the average of the readings. The average value is more likely to be closer to the true value than any single measurement.

1.2 Representation of Errors

Errors are typically represented in different ways to quantify their magnitude and impact.

1.2.1 Absolute Error

The absolute error is the magnitude of the difference between the true value of a quantity and its measured value. If $a_{true}$ is the true value and $a_{measured}$ is the measured value, then the absolute error ($\Delta a$) is given by:

$\Delta a = |a_{measured} - a_{true}|$

In practice, the true value is often unknown. When multiple measurements are taken, the average value is often taken as the best estimate of the true value. If $a_1, a_2, ..., a_n$ are $n$ measurements of a quantity, the mean value ($\bar{a}$) is:

$\bar{a} = \frac{a_1 + a_2 + ... + a_n}{n}$

The absolute error in each measurement is then $|\bar{a} - a_i|$. The mean absolute error is the average of these individual absolute errors:

Mean Absolute Error $= \frac{|\bar{a} - a_1| + |\bar{a} - a_2| + ... + |\bar{a} - a_n|}{n}$

1.2.2 Relative Error

The relative error is the ratio of the absolute error to the true value (or the mean value as an estimate of the true value). It expresses the error as a fraction of the measured quantity and is often more informative than the absolute error, especially when comparing measurements of different magnitudes.

Relative Error $= \frac{\text{Absolute Error}}{\text{True Value}} = \frac{\Delta a}{a_{true}}$

Using the mean value as the estimate for the true value:

Relative Error $= \frac{\text{Mean Absolute Error}}{\bar{a}}$

1.2.3 Percentage Error

The percentage error is simply the relative error multiplied by 100%. It provides a clear understanding of the error in terms of percentage.

Percentage Error $= \text{Relative Error} \times 100\% = \frac{\Delta a}{a_{true}} \times 100\%$

1.3 Propagation of Errors

When a quantity is determined from several other quantities, each of which has its own error, the errors in the original quantities propagate to the final result. The rules for error propagation depend on the mathematical operations (addition, subtraction, multiplication, division, powers) involved.

1.3.1 Error in Sum or Difference

If $Z = A + B$ or $Z = A - B$, where A and B are measured quantities with absolute errors $\Delta A$ and $\Delta B$ respectively, then the absolute error in Z, $\Delta Z$, is the sum of the absolute errors of A and B:

$\Delta Z = \Delta A + \Delta B$

Example: Let the length of two rods be measured as $L_1 = 10.5 \pm 0.2$ cm and $L_2 = 8.3 \pm 0.1$ cm. If we want to find the difference in their lengths, $L = L_1 - L_2$.

The difference in length is $L = 10.5 - 8.3 = 2.2$ cm.

The absolute error in the difference is $\Delta L = \Delta L_1 + \Delta L_2 = 0.2 + 0.1 = 0.3$ cm.

So, the difference in lengths is $2.2 \pm 0.3$ cm.

1.3.2 Error in Product or Quotient

If $Z = A \times B$ or $Z = \frac{A}{B}$, where A and B are measured quantities with relative errors $\frac{\Delta A}{A}$ and $\frac{\Delta B}{B}$ respectively, then the relative error in Z is the sum of the relative errors of A and B:

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

Example: Let the length of a rectangle be $l = 10.0 \pm 0.1$ cm and the width be $w = 5.0 \pm 0.1$ cm. Calculate the area and its error.

Area $A = l \times w = 10.0 \times 5.0 = 50.0$ cm$^2$.

Relative error in length: $\frac{\Delta l}{l} = \frac{0.1}{10.0} = 0.01$.

Relative error in width: $\frac{\Delta w}{w} = \frac{0.1}{5.0} = 0.02$.

Relative error in area: $\frac{\Delta A}{A} = \frac{\Delta l}{l} + \frac{\Delta w}{w} = 0.01 + 0.02 = 0.03$.

Absolute error in area: $\Delta A = A \times 0.03 = 50.0 \times 0.03 = 1.5$ cm$^2$.

So, the area is $50.0 \pm 1.5$ cm$^2$.

1.3.3 Error in Power

If $Z = A^n$, where A is a measured quantity with relative error $\frac{\Delta A}{A}$ and n is a power (which can be an integer or a fraction), then the relative error in Z is n times the relative error in A:

$\frac{\Delta Z}{Z} = n \frac{\Delta A}{A}$

Example: If the radius of a sphere is measured as $r = 5.0 \pm 0.2$ cm, find the percentage error in its volume.

Volume $V = \frac{4}{3} \pi r^3$. Here, $n=3$.

Relative error in radius: $\frac{\Delta r}{r} = \frac{0.2}{5.0} = 0.04$.

Relative error in volume: $\frac{\Delta V}{V} = 3 \frac{\Delta r}{r} = 3 \times 0.04 = 0.12$.

Percentage error in volume: $\frac{\Delta V}{V} \times 100\% = 0.12 \times 100\% = 12\%$.

1.3.4 General Formula for Error Propagation

For a quantity $Z = f(A, B, C, ...)$, the relative error in Z is given by:

$\frac{\Delta Z}{Z} = \left| \frac{\partial f}{\partial A} \frac{A}{f} \Delta A \right| + \left| \frac{\partial f}{\partial B} \frac{B}{f} \Delta B \right| + ...$

This is a more general form, but for exam purposes, the specific rules for sum, difference, product, quotient, and powers are usually sufficient.

1.4 Significant Figures

Significant figures are the digits in a number that are known with some degree of certainty. They indicate the precision of a measurement. Understanding significant figures helps in reporting results accurately and avoiding the false impression of higher precision than actually exists.

1.4.1 Rules for Significant Figures

• All non-zero digits are significant. (e.g., 123 has 3 significant figures)
• Zeros between non-zero digits are significant. (e.g., 1007 has 4 significant figures)
• Leading zeros (zeros to the left of the first non-zero digit) are not significant. (e.g., 0.0052 has 2 significant figures: 5 and 2)
• Trailing zeros (zeros to the right of the last non-zero digit) are significant only if the number contains a decimal point. (e.g., 12.00 has 4 significant figures; 1200 has 2 significant figures, but 1200. has 4 significant figures). Scientific notation is best for clarity: $1.2 \times 10^3$ (2 sig figs), $1.200 \times 10^3$ (4 sig figs).

1.4.2 Operations with Significant Figures

• Addition and Subtraction: The result should have the same number of decimal places as the number with the fewest decimal places.
  Example: $12.345 + 0.12 = 12.465$. Rounded to two decimal places (from 0.12), the result is $12.47$.
• Multiplication and Division: The result should have the same number of significant figures as the number with the fewest significant figures.
  Example: $2.34 \times 5.6 = 13.104$. Since 5.6 has 2 significant figures, the result is rounded to 2 significant figures: $13$.

2. Dimensional Analysis

Dimensional analysis is a powerful technique used in physics to check the correctness of an equation, to derive relationships between physical quantities, and to convert units from one system to another. It is based on the principle that equations must be dimensionally homogeneous, meaning that the dimensions on both sides of an equation must be the same.

2.1 Fundamental and Derived Quantities

Physical quantities are classified into two types:

• Fundamental Quantities: These are quantities that are independent of each other and are chosen as the basic units. The seven fundamental quantities in the International System of Units (SI) are:
  • Length (L)
  • Mass (M)
  • Time (T)
  • Electric Current (A - Ampere)
  • Thermodynamic Temperature ($\Theta$ - Theta)
  • Amount of Substance (N)
  • Luminous Intensity (I or J)
• Derived Quantities: These quantities can be expressed in terms of fundamental quantities. Examples include velocity, acceleration, force, energy, etc.

2.2 Dimensional Formulas

The dimensional formula of a physical quantity expresses how it is related to the fundamental quantities of length, mass, and time. The dimensions are usually represented by square brackets [ ]. For example:

• Length: [L]
• Mass: [M]
• Time: [T]
• Area: Length $\times$ Length = [L] $\times$ [L] = [L$^2$]
• Volume: Length $\times$ Length $\times$ Length = [L$^3$]
• Velocity: $\frac{\text{Displacement}}{\text{Time}} = \frac{[\text{L}]}{[\text{T}]} = [\text{LT}^{-1}]$
• Acceleration: $\frac{\text{Velocity}}{\text{Time}} = \frac{[\text{LT}^{-1}]}{[\text{T}]} = [\text{LT}^{-2}]$
• Force: Mass $\times$ Acceleration = [M] $\times$ [LT$^{-2}$] = [MLT$^{-2}$]
• Work/Energy: Force $\times$ Distance = [MLT$^{-2}$] $\times$ [L] = [ML$^2$T$^{-2}$]
• Pressure: $\frac{\text{Force}}{\text{Area}} = \frac{[\text{MLT}^{-2}]}{[\text{L}^2]} = [\text{ML}^{-1}\text{T}^{-2}]$

Often, dimensions of other fundamental quantities like Electric Current ([A]), Temperature ([$\Theta$]), etc., are also included. For example, the dimensional formula for Electric Charge (Q) is [AT].

2.3 Principle of Homogeneity of Dimensions

This principle states that for an equation to be physically meaningful, the dimensions of all the terms on both sides of the equation must be the same. This means that quantities can only be added or subtracted if they have the same dimensions.

2.4 Applications of Dimensional Analysis

2.4.1 Checking the Dimensional Correctness of an Equation

We can use dimensional analysis to verify if a given physical equation is likely to be correct. If the dimensions on both sides of an equation do not match, the equation is definitely incorrect. However, if the dimensions match, it does not guarantee that the equation is correct, as it might be missing dimensionless constants or have incorrect relationships.

Example: Check the equation $v = u + at$, where $v$ is final velocity, $u$ is initial velocity, $a$ is acceleration, and $t$ is time.

• Dimension of $v$: [LT$^{-1}$]
• Dimension of $u$: [LT$^{-1}$]
• Dimension of $at$: [LT$^{-2}$] $\times$ [T] = [LT$^{-1}$]

Since the dimensions of all terms ($v$, $u$, and $at$) are [LT$^{-1}$], the equation is dimensionally correct.

2.4.2 Deriving Relationships Between Physical Quantities

Dimensional analysis can be used to find a possible formula for a physical quantity if we know the quantities on which it depends. This method is particularly useful when the exact relationship is unknown or complex.

Example: Find the time period (T) of a simple pendulum, assuming it depends on its length (L), the mass of the bob (m), and the acceleration due to gravity (g).

We assume the relation is of the form: $T = k L^a m^b g^c$, where $k$ is a dimensionless constant and $a, b, c$ are exponents to be determined.

Dimensions of each term:

• [T] = [L]$^a$ [M]$^b$ [LT$^{-2}$]$^c$
• [M$^0$L$^0$T$^1$] = [L$^{a+c}$ M$^b$ T$^{-2c}$]

Equating the exponents of M, L, and T on both sides:

• For M: $b = 0$
• For L: $a + c = 0$
• For T: $-2c = 1 \implies c = -\frac{1}{2}$

Substituting $c = -\frac{1}{2}$ into $a + c = 0$: $a - \frac{1}{2} = 0 \implies a = \frac{1}{2}$.

So, the relationship is $T = k L^{1/2} m^0 g^{-1/2} = k \sqrt{\frac{L}{g}}$.

The dimensional analysis gives $T \propto \sqrt{\frac{L}{g}}$. The actual experiment or theory shows that $k = 2\pi$. So, $T = 2\pi \sqrt{\frac{L}{g}}$.

2.4.3 Conversion of Units

Dimensional analysis can be used to convert a physical quantity from one system of units to another.

Example: Convert 1 Joule of energy into ergs. (1 J = ? erg)

We know that 1 Joule is the SI unit of energy, and 1 erg is the CGS unit of energy.

The dimensional formula for energy is [ML$^2$T$^{-2}$].

In the SI system (MKS):

• Mass $M_1 = 1$ kg
• Length $L_1 = 1$ m
• Time $T_1 = 1$ s
• So, 1 J = 1 kg $\cdot$ m$^2$ $\cdot$ s$^{-2}$

In the CGS system:

• Mass $M_2 = 1$ g
• Length $L_2 = 1$ cm
• Time $T_2 = 1$ s
• So, 1 erg = 1 g $\cdot$ cm$^2$ $\cdot$ s$^{-2}$

We need to find the value of $x$ such that 1 J = $x$ ergs.

1 kg $\cdot$ m$^2$ $\cdot$ s$^{-2}$ = $x$ g $\cdot$ cm$^2$ $\cdot$ s$^{-2}$

Substitute SI units in terms of CGS units:

• 1 kg = 1000 g
• 1 m = 100 cm

So, (1000 g) $\cdot$ (100 cm)$^2$ $\cdot$ s$^{-2}$ = $x$ g $\cdot$ cm$^2$ $\cdot$ s$^{-2}$

1000 g $\cdot$ 10000 cm$^2$ $\cdot$ s$^{-2}$ = $x$ g $\cdot$ cm$^2$ $\cdot$ s$^{-2}$

10$^3$ $\times$ 10$^4$ (g $\cdot$ cm$^2$ $\cdot$ s$^{-2}$) = $x$ (g $\cdot$ cm$^2$ $\cdot$ s$^{-2}$)

$10^7$ = $x$

Therefore, 1 Joule = $10^7$ ergs.

2.5 Limitations of Dimensional Analysis

• It cannot determine dimensionless constants (like $2\pi$, or the constant $k$ in the pendulum example).
• It cannot be used to derive formulas involving more than three quantities, or where the dependence is not purely multiplicative (e.g., involving addition or subtraction of terms with different dimensions).
• It cannot distinguish between quantities with the same dimensions (e.g., Work, Torque, Energy all have dimensions [ML$^2$T$^{-2}$]).
• It cannot handle equations involving trigonometric, exponential, or logarithmic functions, as these are dimensionless or introduce complexity that dimensional analysis alone cannot resolve.