Introduction to Measurement
Physics is an experimental science that relies on measurements. A measurement consists of a numerical value and a unit.
SI Units
The International System of Units (SI) has seven base units:
| Base Quantity | SI Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric Current | ampere | A |
| Temperature | kelvin | K |
| Amount of Substance | mole | mol |
| Luminous Intensity | candela | cd |
Fundamental and Derived Units
Fundamental units: The seven base units defined above are independent of each other.
Derived units: Units derived from fundamental units using algebraic relationships. Examples: velocity (m/s), acceleration (m/s^2), force (kg m/s^2 = N), pressure (N/m^2 = Pa).
SI Prefixes
| Prefix | Symbol | Factor |
|---|---|---|
| giga | G | 10^9 |
| mega | M | 10^6 |
| kilo | k | 10^3 |
| centi | c | 10^(-2) |
| milli | m | 10^(-3) |
| micro | mu | 10^(-6) |
| nano | n | 10^(-9) |
Dimensional Analysis
Dimensions represent the nature of a physical quantity in terms of base quantities.
Dimensions of common quantities:
- Velocity:
[LT^(-1)] - Acceleration:
[LT^(-2)] - Force:
[MLT^(-2)] - Energy:
[ML^2 T^(-2)] - Pressure:
[ML^(-1) T^(-2)]
Principle of Homogeneity of Dimensions
In a correct physical equation, each term has the same dimensions.
Uses of Dimensional Analysis
- Checking correctness of equations: Verify dimensions on both sides are equal.
- Deriving relations: Example: time period of a pendulum
T prop sqrt(l/g). - Converting units: Convert 1 J to erg using dimensions.
Example: Check s = ut + (1/2)at^2.
LHS: [L]. RHS: [LT^(-1)][T] + [LT^(-2)][T^2] = [L] + [L] = [L]. Homogeneous.
Limitations of Dimensional Analysis
- Does not determine dimensionless constants.
- Cannot distinguish between quantities with same dimensions (e.g., torque and work).
- Applicable only to quantities where dimensions are independent.
- Cannot derive trigonometric, exponential, or logarithmic functions.
Significant Figures
Significant figures indicate the precision of a measurement.
Rules for Counting Significant Figures
- All non-zero digits are significant.
- Zeros between non-zero digits are significant (e.g., 1005 has 4 SF).
- Leading zeros are NOT significant (e.g., 0.0025 has 2 SF).
- Trailing zeros after decimal are significant (e.g., 2.500 has 4 SF).
- Trailing zeros without decimal are ambiguous (e.g., 2500 may have 2, 3, or 4 SF).
Rounding Off
- If digit after last significant digit is less than 5, discard it.
- If greater than 5, increase last significant digit by 1.
- If exactly 5, round to nearest even digit.
Arithmetic with Significant Figures
- Addition/Subtraction: Result should have same decimal places as least precise quantity.
- Multiplication/Division: Result should have same significant figures as least precise quantity.
Errors in Measurement
Types of Errors
Systematic errors: Consistent and repeatable. Due to instrument calibration, method, or personal bias. Can be minimised but not eliminated.
Random errors: Fluctuate unpredictably. Can be minimised by taking multiple measurements and averaging.
Absolute Error
Delta a = a_i - a_(mean)
Mean Absolute Error
Delta a_(mean) = (sum |Delta a_i|)/n
Relative Error
delta a = Delta a_(mean)/a_(mean)
Percentage Error
% error = delta a * 100%
Propagation of Errors
- Addition/Subtraction: Absolute errors add.
- Multiplication/Division: Relative errors add.
- Power: Relative error is multiplied by the power.
Worked Examples
Example 1: Check the correctness of F = mv^2/r.
Solution: RHS [M][LT^(-1)]^2/[L] = [MLT^(-2)] = [F]. Correct.
Example 2: Express 5 g/cm^3 in SI units.
Solution: 5 g/cm^3 = 5 * 10^(-3) kg/(10^(-2) m)^3 = 5 * 10^(-3)/10^(-6) kg/m^3 = 5000 kg/m^3.
Common Mistakes
- Dimensions of angle and solid angle are zero: They are dimensionless.
- Constants with dimensions: Gravitational constant G has dimensions
[M^(-1)L^3T^(-2)]. - Significant figures of constants: Use exact values (pi, e) with required SF.
ISC Exam Focus
- Theory (70%): SI base units, dimensional formulas, uses and limitations of dimensional analysis.
- Practical/Application (30%): Significant figures, error analysis, unit conversion problems.
- ISC frequently asks: "Check dimensional correctness of ...", "Find dimensions of ...".
- Error analysis problems typically carry 2-3 marks.
Self-Test Questions
Q1: What are the SI base units of pressure?
Answer: Pa = N/m^2 = kg m^(-1) s^(-2).
Q2: Find the dimensions of Planck's constant h.
Answer: From E = hf, h = E/f. Dimensions [ML^2T^(-2)]/[T^(-1)] = [ML^2T^(-1)].
Q3: How many significant figures in 0.05030? Answer: 4 significant figures (5, 0, 3, and trailing zero).
Q4: A student measures length as 5.24 cm, 5.26 cm, and 5.22 cm. Find mean absolute error.
Answer: Mean = (5.24+5.26+5.22)/3 = 5.24. Absolute errors: 0, 0.02, 0.02. Mean absolute error = (0+0.02+0.02)/3 = 0.0133 cm.
Q5: Check if T = 2pi sqrt(l/g) is dimensionally correct.
Answer: LHS [T]. RHS sqrt([L]/[LT^(-2)]) = sqrt([T^2]) = [T]. Correct.
Q6: Convert 1 N to dyne using dimensional analysis.
Answer: [F] = [MLT^(-2)]. 1 N = 1 kg * 1 m/s^2 = 1000 g * 100 cm/s^2 = 10^5 g cm/s^2 = 10^5 dyne.
