Units and Measurements

Units and Measurements form the foundation of physics. This chapter covers SI units, dimensional analysis, and error handling.

Overview

graph TD
    A[Units & Measurements] --> B[SI Units]
    A --> C[Dimensional Analysis]
    A --> D[Errors & Measurements]
    B --> B1[Base Units]
    B --> B2[Derived Units]
    C --> C1[Dimensions of Quantities]
    C --> C2[Applications]
    D --> D1[Types of Errors]
    D --> D2[Significant Figures]

SI Units

Base Units

Quantity	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

Derived Units

Common derived units in physics:

Quantity	Formula	SI Unit	Symbol
Force	$ma$	newton	N = kg·m/s²
Energy	$Fs$	joule	J = N·m
Power	$E/t$	watt	W = J/s
Pressure	$F/A$	pascal	Pa = N/m²
Frequency	$1/T$	hertz	Hz = s⁻¹

Dimensional Analysis

Dimensions of Physical Quantities

Every physical quantity can be expressed in terms of fundamental dimensions:

• [M] - Mass
• [L] - Length
• [T] - Time
• [A] - Electric Current
• [K] - Temperature
• [mol] - Amount of Substance
• [cd] - Luminous Intensity

Important Dimensional Formulas

Quantity	Dimension
Velocity	[LT⁻¹]
Acceleration	[LT⁻²]
Force	[MLT⁻²]
Energy/Work	[ML²T⁻²]
Power	[ML²T⁻³]
Pressure	[ML⁻¹T⁻²]
Momentum	[MLT⁻¹]
Angular Momentum	[ML²T⁻¹]

Applications of Dimensional Analysis

1. Checking equations: Both sides must have same dimensions
2. Deriving relations: Find how quantities depend on each other
3. Converting units: Between different systems

Interactive Demo: Visualize Kinematic Quantities

Explore velocity, acceleration, and displacement graphs - key quantities with different dimensions.

Errors in Measurements

Types of Errors

1. Systematic Errors: Consistent errors in one direction

   • Instrumental errors
   • Personal errors
   • Environmental errors

2. Random Errors: Irregular, unpredictable variations

3. Gross Errors: Mistakes in reading or recording

Error Propagation

For quantities with errors $\Delta a$ and $\Delta b$:

Addition/Subtraction:

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

Multiplication/Division:

$$\frac{\Delta(ab)}{ab} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$

Power:

$$\frac{\Delta(a^n)}{a^n} = n\frac{\Delta a}{a}$$

Significant Figures

Rules

1. All non-zero digits are significant
2. Zeros between non-zero digits are significant
3. Leading zeros are NOT significant
4. Trailing zeros after decimal point ARE significant

Calculations

• Addition/Subtraction: Result has same decimal places as quantity with fewest
• Multiplication/Division: Result has same significant figures as quantity with fewest

Vernier Caliper and Screw Gauge

Vernier Caliper

Least Count = 1 MSD - 1 VSD = Main Scale Division ÷ Number of Vernier divisions

Typically: LC = 1 mm ÷ 10 = 0.1 mm = 0.01 cm

Screw Gauge

Least Count = Pitch ÷ Number of divisions on circular scale

Typically: LC = 1 mm ÷ 100 = 0.01 mm

Practice Problems

1. Check the dimensional correctness of $v = u + at$.

2. The period of a simple pendulum depends on length $l$ and acceleration $g$. Find the formula using dimensional analysis.

3. A physical quantity $P$ is related to four quantities $a$, $b$, $c$, and $d$ as: $P = \frac{a^3b^2}{\sqrt{cd}}$. The percentage errors are 1%, 2%, 3%, and 4% respectively. Find the percentage error in $P$.

Further Reading

• Kinematics - Motion analysis using these units
• Newton’s Laws - Force and dynamics