# Chapter 1: Units and Measurement

### Measurement
The process of comparing an unknown quantity with a known standard quantity is called measurement.

### Systems of Measurements
1.  **CGS system:** In this system, Length is measured in centimetre (cm), mass is measured in gram (g), and time is measured in seconds (s).
2.  **MKS system:** In this system, Length is measured in metre (m), mass is measured in kilogram (kg), and time is measured in second (s).
3.  **FPS System:** In this system, Length is measured in foot (ft), mass is measured in pound (lb), and time is measured in second (s).
4.  **SI system:** In this system, Length is measured in metre (m), mass is measured in kilogram (kg), time is measured in second (s), temperature is measured in kelvin (K), electric current is measured in ampere (A), luminous intensity in candela (cd), and quantity of matter in mole (mol).

### Physical Quantity
The quantity that can be measured directly or indirectly is called a physical quantity.
-   Physical Quantity = Numerical Value × Unit

**Types of Physical Quantity:**
1.  Fundamental Quantity
2.  Derived Quantity

> **Note:** The quantity used as a standard of measurement is called a unit.

### Characteristics of a Standard Unit
* It should be well-defined.
* It should remain constant.
* It should be easily available and reproducible.
* It should be universally agreed upon.

### Fundamental and Supplementary Quantities with Units and Symbols

**Fundamental Quantities**
| Fundamental Quantity  | Unit      | Symbol |
| --------------------- | --------- | ------ |
| Mass                  | Kilogram  | kg     |
| Length                | metre     | m      |
| Time                  | second    | s      |
| Luminous Intensity    | candela   | cd     |
| Current               | Ampere    | A      |
| Amount of Substance   | mole      | mol    |
| Temperature           | Kelvin    | K      |

**Supplementary Quantities**
| Supplementary Quantity | Unit      | Symbol |
| ---------------------- | --------- | ------ |
| Plane angle            | Radian    | rad    |
| solid angle            | Steradian | sr     |

### Quantities in Terms of Power '10' (Prefixes)
| Prefix      | Symbol | Multiplying Factor |
| ----------- | ------ | ------------------ |
| zeta        | Z      | $10^{21}$          |
| exa         | E      | $10^{18}$          |
| peta        | P      | $10^{15}$          |
| tera        | T      | $10^{12}$          |
| giga        | G      | $10^{9}$           |
| mega        | M      | $10^{6}$           |
| kilo        | k      | $10^{3}$           |
| deci        | d      | $10^{-1}$          |
| centi       | c      | $10^{-2}$          |
| milli       | m      | $10^{-3}$          |
| micro       | µ      | $10^{-6}$          |
| nano        | n      | $10^{-9}$          |
| pico        | p      | $10^{-12}$         |
| femto/fermi | f      | $10^{-15}$         |
| atto        | a      | $10^{-18}$         |

### Definitions of SI Base Units
* **Metre (m):** A metre is defined as the length of the path travelled by light in a vacuum during a time interval of $1/299,792,458$ of a second.
* **Second (s):** The second is the duration of $9,192,631,770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.
* **Kilogram (kg):** The kilogram is the mass equal to the mass of the standard platinum-iridium (90% platinum and 10% iridium) cylinder kept at the International Bureau of Weights and Measures in Paris.

### Advantages of SI-Units
* It is a rational system of units.
* It is a coherent system of units.
* It is a decimal system.
* It is highly useful in practical and theoretical work.

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## Dimensional Analysis

* **Dimension:** The power to which the fundamental quantities are raised to express any physical quantity is called dimension.
* **Dimensional Formula:** The expression of any physical quantity in terms of its basic units.
* **Dimensional Equation:** An equation containing physical quantities with their dimensional formula is known as a dimensional equation.

**Dimensions of Base Quantities:**
* Dimension of Mass = $[M]$
* Dimension of Length = $[L]$
* Dimension of Time = $[T]$
* Dimension of Temperature = $[\theta]$ or $[K]$
* Dimension of Current = $[A]$ or $[I]$

### Dimensional Formula of Some Physical Quantities
1.  **Area** = $L \times b = [L^2] = [M^0L^2T^0]$
2.  **Volume** = $L \times b \times h = [L^3] = [M^0L^3T^0]$
3.  **Speed** = $\frac{\text{distance}}{\text{time}} = \frac{L}{T} = [M^0LT^{-1}]$
4.  **Velocity** = $\frac{\text{displacement}}{\text{time}} = \frac{L}{T} = [M^0LT^{-1}]$
5.  **Acceleration** = $\frac{\text{change in velocity}}{\text{time}} = \frac{LT^{-1}}{T} = [M^0LT^{-2}]$
6.  **Force** = $m \times a = [M][LT^{-2}] = [MLT^{-2}]$
7.  **Density** = $\frac{m}{V} = \frac{M}{L^3} = [ML^{-3}T^0]$
8.  **Surface Tension** = $\frac{F}{L} = \frac{MLT^{-2}}{L} = [ML^0T^{-2}]$
9.  **Impulse** = $F \times t = [MLT^{-2}][T] = [MLT^{-1}]$
10. **Momentum** = $m \times v = [M][LT^{-1}] = [MLT^{-1}]$
11. **Frequency** = $\frac{1}{T} = [M^0L^0T^{-1}]$
12. **Angle** = $\frac{\text{arc}}{\text{radius}} = \frac{L}{L} = [M^0L^0T^0]$
13. **Torque / Work** = $F \cdot d = [MLT^{-2}][L] = [ML^2T^{-2}]$
14. **Gravitational Constant (G)** = $\frac{Fd^2}{m_1 m_2} = \frac{[MLT^{-2}][L^2]}{[M^2]} = [M^{-1}L^3T^{-2}]$
15. **Specific Heat Capacity** = $[M^0L^2T^{-2}K^{-1}]$
16. **Power** = $\frac{W}{t} = \frac{[ML^2T^{-2}]}{T} = [ML^2T^{-3}]$
17. **Coefficient of Viscosity ($\eta$)** = $\frac{F}{A \cdot (\text{velocity gradient})} = \frac{MLT^{-2}}{[L^2][T^{-1}]} = [ML^{-1}T^{-1}]$
18. **Moment of Inertia** = $m \times d^2 = [ML^2T^0]$
19. **Angular Momentum** = Moment of Inertia $\times$ Angular Velocity = $[ML^2][T^{-1}] = [ML^2T^{-1}]$

### Principle of Homogeneity
It states that the dimensions of each of the terms of a dimensional equation on both sides should be the same. For an equation $z = x + y$, the dimensions of $z$ must be equal to the dimensions of $x$ and $y$.

### Four Categories of Physical Quantities
1.  **Dimensional Variables:** Physical quantities which have dimensions but do not have a fixed value. (e.g., work, energy)
2.  **Dimensionless Variables:** Physical quantities which have neither dimensions nor a fixed value. (e.g., strain, specific gravity)
3.  **Dimensional Constants:** Physical quantities which have dimensions and a fixed value. (e.g., Gravitational constant, velocity of light)
4.  **Dimensionless Constants:** Quantities which do not possess dimensions but have a fixed value. (e.g., $\pi$, counting numbers)

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### Uses of Dimensional Equations
1.  To check the correctness of a physical relation.
2.  To derive the relation between various physical quantities.
3.  To convert the value of a physical quantity from one system of units to another.
4.  To find the dimension of constants in a given relation.

#### 1. To Check the Correctness of a Physical Relation
**Example:** Check the correctness of the formula $v^2 = u^2 + 2as$.
* **LHS Dimension:** $[v^2] = (LT^{-1})^2 = [L^2T^{-2}]$
* **RHS Dimension:**
    * For $u^2$: $[u^2] = (LT^{-1})^2 = [L^2T^{-2}]$
    * For $2as$: $[2as] = [L T^{-2}][L] = [L^2T^{-2}]$ (The constant 2 is dimensionless)
* Since the dimensions of each term on the LHS and RHS are the same, the formula is dimensionally correct.

#### 2. To Convert Units from One System to Another
The measure of a physical quantity is $x = nu$. For two systems, $n_1u_1 = n_2u_2$.
If the dimensions are $[M^aL^bT^c]$, then:
$n_2 = n_1 \left[ \frac{M_1}{M_2} \right]^a \left[ \frac{L_1}{L_2} \right]^b \left[ \frac{T_1}{T_2} \right]^c$

**Example:** Convert 1 Newton (SI) into dyne (CGS).
* The dimensional formula for Force (F) is $[MLT^{-2}]$. So, $a=1, b=1, c=-2$.
* **SI System (1):** $n_1=1$, $M_1=1$ kg, $L_1=1$ m, $T_1=1$ s.
* **CGS System (2):** $n_2=?$, $M_2=1$ g, $L_2=1$ cm, $T_2=1$ s.

$n_2 = 1 \left[ \frac{1 \text{ kg}}{1 \text{ g}} \right]^1 \left[ \frac{1 \text{ m}}{1 \text{ cm}} \right]^1 \left[ \frac{1 \text{ s}}{1 \text{ s}} \right]^{-2}$
$n_2 = \left[ \frac{1000 \text{ g}}{1 \text{ g}} \right] \left[ \frac{100 \text{ cm}}{1 \text{ cm}} \right] [1]^{-2}$
$n_2 = 1000 \times 100 = 10^5$
Therefore, **1 Newton = $10^5$ dyne**.

#### 3. To Derive the Relation Between Physical Quantities
**Example:** Find the formula for the time period (T) of a simple pendulum if it depends upon mass (m), length (l), and acceleration due to gravity (g).
1.  Assume the relation: $T \propto m^a l^b g^c$ or $T = k m^a l^b g^c$, where k is a dimensionless constant.
2.  Write the dimensions for each quantity:
    $[T] = [M]^a [L]^b [LT^{-2}]^c$
    $[M^0L^0T^1] = [M^a L^{b+c} T^{-2c}]$
3.  Equate the powers of M, L, and T by the principle of homogeneity:
    * For M: $a = 0$
    * For T: $1 = -2c \implies c = -1/2$
    * For L: $0 = b+c \implies b = -c \implies b = 1/2$
4.  Substitute the values of a, b, and c back into the equation:
    $T = k m^0 l^{1/2} g^{-1/2}$
    $T = k \sqrt{\frac{l}{g}}$
    (Experimentally, $k=2\pi$)

### Limitations of Dimensional Analysis
* It does not give information about dimensionless constants.
* It does not distinguish between scalar and vector quantities.
* Formulae cannot be derived if the physical quantity depends upon trigonometric, exponential, or logarithmic functions.
* A formula cannot be derived if the quantity depends upon more than three factors having dimensions.
* A formula cannot be derived if the physical relation contains more than one term (e.g., sums or differences).

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### Other Definitions

* **Precision:** The degree of exactness, which gives the limitation of the measuring instrument.
* **Accuracy:** The degree to which the observed value approaches the true value.
* **Significant Figures:** The number of digits in a measurement in which we are reasonably sure.

### Error in Measurement
* Instrumental error
* Systematic error
* Personal error
* External error
* Imperfection error
* Random error
* Least Count Error

### Notes
* The SI unit is called a **coherent system** of units because it is based on a set of basic units from which all derived units are obtained by multiplication or division without introducing numerical factors.
* The SI unit is called a **rational system** of units because it uses only one unit for a given physical quantity. For example, all forms of energy are measured in Joules.