This chapter covers the fundamental rules for measuring physical quantities — the SI system of units, significant figures, dimensions, and dimensional analysis. You will find formulas for writing dimensional equations, checking the consistency of physical equations, and deducing relations between quantities.
Each formula below is grouped by topic, with the meaning of every symbol, when to use it, and original worked examples that show you how to apply it. For a full explanation of the concepts and derivations, see the Units and Measurement notes.
Formulas at a Glance
| Purpose | Formula |
|---|---|
| Dimensional formula of a physical quantity | \( [Q] = [M^a L^b T^c \ldots] \) |
| Checking dimensional consistency of an equation | Each term in the equation must have the same dimensions |
| Time period of a simple pendulum (from dimensional analysis) | \( T = 2\pi \sqrt{\frac{l}{g}} \) |
| Multiplication / division significant figures | Result has same number of significant figures as the least precise input |
| Addition / subtraction significant figures | Result has same number of decimal places as the least precise input |
| Relative error in product | \( \frac{\Delta A}{A} = \frac{\Delta a}{a} + \frac{\Delta b}{b} \) (for \(A = a\times b\)) |
All Formulas, Grouped by Topic
Dimensions and Dimensional Formulae
Every physical quantity can be expressed in terms of the base dimensions: mass [M], length [L], time [T], electric current [A], thermodynamic temperature [K], amount of substance [mol], and luminous intensity [cd]. In mechanics, only [M], [L] and [T] are needed.
\[ [V] = [M^0 L^3 T^0] \quad \text{(volume)} \]
\[ [v] = [M^0 L T^{-1}] \quad \text{(speed / velocity)} \]
\[ [a] = [M^0 L T^{-2}] \quad \text{(acceleration)} \]
\[ [F] = [M L T^{-2}] \quad \text{(force)} \]
\[ [\rho] = [M L^{-3} T^0] \quad \text{(mass density)} \]
\[ [E] = [M L^2 T^{-2}] \quad \text{(energy)} \]
The general form of a dimensional formula is:
\[ [\text{Physical quantity}] = [M^a L^b T^c A^d K^e \text{mol}^f \text{cd}^g] \]
where the exponents \(a, b, c, \ldots\) are integers (often zero, positive or negative). (NCERT, p. 7)
Principle of Homogeneity of Dimensions
In a correct physical equation, the dimensions of every term on both sides must be identical. This is used to check whether an equation is dimensionally consistent.
\[ [\text{Term}_1] = [\text{Term}_2] = \cdots = [\text{RHS}] \]
If an equation fails this test, it is certainly wrong. Passing it does not guarantee the equation is correct — only that it is dimensionally possible. (NCERT, p. 8)
Deducing Relations Using Dimensional Analysis
If a physical quantity \(Q\) depends on variables \(x_1, x_2, x_3\), we can write:
\[ Q = k \, x_1^a \, x_2^b \, x_3^c \]
where k is a dimensionless constant. Taking dimensions on both sides gives a system of equations that determines the exponents \(a, b, c\). The constant k cannot be found by dimensional analysis alone.
Example: For a simple pendulum:
\[ T = k \, l^x \, g^y \, m^z \]
\[ [L^0 M^0 T^1] = [L^1]^x [L^1 T^{-2}]^y [M^1]^z \]
From this we get \(x = \frac12,\; y = -\frac12,\; z = 0\), so:
\[ T = k \sqrt{\frac{l}{g}} \]
Experiment shows \(k = 2\pi\), giving:
\[ T = 2\pi \sqrt{\frac{l}{g}} \]
(NCERT, p. 9-10)
Significant Figures and Arithmetic Operations
- Multiplication / Division: The result should have as many significant figures as the number with the least number of significant figures among the inputs.
- Addition / Subtraction: The result should have as many decimal places as the number with the least number of decimal places.
- Rounding off: Drop the insignificant digit. If the digit to be dropped is >5, raise the preceding digit by 1. If it is 5, round to the nearest even preceding digit.
(NCERT, p. 5-6)
Combination of Errors (Uncertainty in Results)
For a product \(A = a \times b\) or a quotient \(A = a/b\):
\[ \frac{\Delta A}{A} = \frac{\Delta a}{a} + \frac{\Delta b}{b} \]
For a sum or difference \(A = a + b\) or \(A = a – b\):
\[ \Delta A = \Delta a + \Delta b \]
where \(\Delta a\) is the absolute error in \(a\). (NCERT, p. 6-7)
What Each Symbol Means
| Symbol | Meaning | Unit / Dimension |
|---|---|---|
| \( [M] \) | Dimension of mass | — (base dimension) |
| \( [L] \) | Dimension of length | — (base dimension) |
| \( [T] \) | Dimension of time | — (base dimension) |
| \( a, b, c, \ldots \) | Exponents (powers) in a dimensional formula | dimensionless integers |
| \( T \) | Time period of a pendulum | s |
| \( l \) | Length of the pendulum string | m |
| \( g \) | Acceleration due to gravity | \( \text{m s}^{-2} \) |
| \( m \) | Mass of the bob | kg |
| \( \Delta A \) | Absolute error in quantity \(A\) | same unit as \(A\) |
| \( \frac{\Delta A}{A} \) | Relative error in quantity \(A\) | dimensionless |
When to Use Each Formula
- Dimensional formula: Use to write the dimensions of a quantity quickly, or to check the homogeneity of an equation. Always write the formula in the form \([M^a L^b T^c]\).
- Principle of homogeneity: Apply to any equation you derive or are given in a problem — if the dimensions do not match, the equation is wrong.
- Dimensional analysis for relation: Use when you need to derive an approximate relation between up to three variables, and you know the dependence is a product of powers. The method gives the form but not the constant.
- Significant figures rules: Use when reporting the result of a calculation involving measured values. Decide the number of significant figures based on the least precise measurement.
- Error combination: Use when you have measured values with uncertainties and need to find the uncertainty in a derived quantity.
Worked Examples
Example 1: Checking the Dimensional Consistency of an Equation
Check whether the equation \(v^2 = u^2 + 2as\) is dimensionally correct, where \(v\) and \(u\) are velocities, \(a\) is acceleration, and \(s\) is displacement.
Step 1: Write the dimensions of each term.
\([v^2] = [L T^{-1}]^2 = [L^2 T^{-2}]\)
\([u^2] = [L T^{-1}]^2 = [L^2 T^{-2}]\)
\([2as] = [L T^{-2}] [L] = [L^2 T^{-2}]\) Step 2: Compare dimensions.
All three terms have the same dimensions \([L^2 T^{-2}]\).
Conclusion: The equation is dimensionally correct.
Example 2: Significant Figures in Multiplication
A rectangle has length 5.62 cm and breadth 3.2 cm. Find its area, reporting the answer to the correct number of significant figures.
- Step 1: Identify significant figures: length 5.62 cm (3 sig fig), breadth 3.2 cm (2 sig fig).
- Step 2: Multiply: \(5.62 \times 3.2 = 17.984\).
- Step 3: The least precise input has 2 sig fig, so round the result to 2 sig fig: 18 cm².
Final answer: \(18 \text{ cm}^2\) (2 significant figures).
Example 3: Deducing a Relation Using Dimensional Analysis
The period \(T\) of a simple pendulum depends on the length \(l\) and the acceleration due to gravity \(g\). Find the form of the relation.
- Step 1: Write \(T = k \, l^x \, g^y\), where \(k\) is a dimensionless constant.
- Step 2: Write dimensions: \([T] = [T^1]\), \([l] = [L^1]\), \([g] = [L^1 T^{-2}]\).
\([T^1] = [L^1]^x [L^1 T^{-2}]^y = [L^{x+y} T^{-2y}]\) Step 3: Equate powers of \(L\) and \(T\):
\(x + y = 0\) (for L), \(-2y = 1\) (for T).
\(\Rightarrow y = -\frac12,\; x = \frac12\).
Step 4: Hence \(T = k \, l^{1/2} \, g^{-1/2} = k \sqrt{\frac{l}{g}}\).
Final form: \(T = k \sqrt{\dfrac{l}{g}}\). Experiment shows \(k = 2\pi\), so \(T = 2\pi \sqrt{\dfrac{l}{g}}\).
Common Mistakes to Avoid
| Mistake | Correct Rule | How to Check Your Answer |
|---|---|---|
| Adding or subtracting quantities with different dimensions | Only quantities with the same dimensions can be added or subtracted. | Write the dimensions of each term; if they differ, the operation is invalid. |
| Assuming dimensional consistency guarantees correctness | Dimensional consistency is necessary but not sufficient — dimensionless constants and functions (like trig, log) are not tested. | After dimensional check, verify with a known numerical example. |
| Confusing significant figures with decimal places | For multiplication/division, use the number of significant figures; for addition/subtraction, use the number of decimal places. | Identify the operation; apply the rule accordingly. |
| Forgetting that trailing zeros in a number without a decimal point are not significant | Trailing zeros in a number without a decimal are not significant. Use scientific notation to avoid ambiguity. | Rewrite the number as \(a \times 10^b\); the base number \(a\) shows the significant figures. |
| Ignoring the condition \(d \neq 0\) when deriving relations (not applicable here, but common in other chapters) | Dimensional analysis works for any exponents; always check that the relation is physically meaningful. | Ensure that the derived relation matches known experimental results. |
Frequently Asked Questions
What is the dimensional formula of a physical quantity?
It is the expression showing how the base dimensions (mass, length, time, etc.) are combined to represent that quantity. For example, force has dimensions \([M L T^{-2}]\).
How do I convert a value from one system of units to another using dimensions?
Write the dimensional formula of the quantity. Then, for each base unit, replace the original unit with the equivalent in the new system, using the conversion factors. For example, to convert \(1\, \text{kg m}^2\text{s}^{-2}\) to \(\text{g cm}^2\text{s}^{-2}\), substitute \(1\text{ kg} = 10^3\text{ g}\), \(1\text{ m} = 10^2\text{ cm}\), giving \(1\text{ J} = 10^7\text{ erg}\).
Does dimensional analysis give the exact numerical constant in a formula?
No. Dimensional analysis can only determine the form of the relation (the exponents of the variables). The dimensionless constant (like \(2\pi\), \(1/2\), etc.) must be found by experiment or by a detailed theoretical derivation.
When should I use scientific notation for reporting measurements?
Scientific notation is recommended whenever the number of significant figures could be ambiguous — especially with trailing zeros. For example, write 4700 as \(4.700 \times 10^3\) to show four significant figures clearly.
Reference: NCERT Class 11 Physics textbook, chapter Units and Measurement.
For more detailed explanations and derivations, visit the Units and Measurement notes page. To practice the textbook questions, see the NCERT Solutions for Units and Measurement.
Explore Class 11 Physics Formulas
More for this chapter:
- Units and Measurement Notes
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Official source: download the NCERT textbook free from ncert.nic.in.