**Introduction**

A dimension is a measure of a physical quantity (without numerical values), while a unit is a way to assign a number to that dimension.

### Example:

Length is a dimension that is measured in units such as feet(ft), centimeter(cm), meter, kilometer(km).

**kg , mol , cd , K , A , m , s**

**Primary dimensions and SI units**

Dimensions | Symbol | SI unit |
---|---|---|

Mass | M | kg (kilogram) |

Length | L | m (meter) |

Time | T | s (second) |

Temperature | θ | K (Kelvin) |

Electric Current | I | A (Ampere) |

Amount of Light | C | Cd (candela) |

Amount of Matter | N | M (mole) |

**All non-primary dimensions can be formed by some combination of the seven primary dimensions.**

**Dimensions of the non-primary quantities **

Quantity | Units | Dimensions |
---|---|---|

Area | m^{2} | L^{2} |

Volume | m^{3} | L^{3} |

Velocity | m/s | LT^{ -1} |

Acceleration | m/s^{2} | LT^{ -2} |

Force | neweton (N) | MLT^{ -2} |

Work (or energy) | joule (J) | ML^{2}T^{ -2} |

Power | watt (W) | ML^{2}T^{ -3} |

Pressure | N/m^{2} | ML^{-1}T^{ -2} |

Density | kg/m^{3} | ML^{ -3} |

Frequency | hertz (Hz) | T^{ -1} |

**Dimensional Analysis**

**Dimensional Analysis** is the algebraic conversion of one measurement unit to another using conversion factors.

**Why do we want to do this?**

When making scientific calculations, it is necessary for the numbers used to be dimensionally consistent.

*Consider Newton’s second law*

*Consider Newton’s second law*

- What is the unit label for acceleration?

- If Force is measured in kg m/s
^{2}, what units must**mass**be measured in?

❖* If a mass is given in pounds, the measurement must be converted to kilograms in order for the calculation to be dimensionally consistent.*

**Dimensionless Quantities**

- Some quantities are said to be dimensionless.
- The ratio of one mass m1 to another mass m2 is dimensionless:

Dimension of the fraction,

The dimensions have cancelled and a result is a number.

- Constants are dimensionless.
- The following quantities are important cases of dimensionless quantities.

*Ex: Trigonometric functions, Logarithms, exponential numbers)*

** Summary**…

When it comes to equations the dimensions of the equation must be the same on both sides.

Like this most equations are being equal to both sides.

Velocity ⟶ v = m/s^{-1} ⟶ LT^{ -1} ⎬ Dimensions

Acceleration ⟶ a = m/s^{-2} ⟶ LT^{ -2} ⎬ Dimensions

### Example 01

**Q)** Obtain the fundamental dimensions of,

- Velocity (units m/s)

- Acceleration ( units m/s
^{2})

- Force ( mass x acceleration )

**Worked Examples**

**Worked Examples**

- Bernoulli’s equation is given by

Where P = pressure, ρ=density, v=velocity, z=height, g= acceleration due to gravity. Find the dimensions of the constant.

2. The period T of a pendulum of length l is given by,

**Derive equations using dimensional analysis**

*Example:*

*Example:*

- Consider the oscillation of a simple pendulum, the period T may depend on mass, length, and acceleration due to gravity. If T α L
^{x}× M^{y}× g^{z}, derive an equation for the period T. Consider the value of the constant is 1.

**To be dimentionally consistent, the dimensions of the both sides should be equal. Therfore,**

**Limitations of Dimensional Analysis**

- Dimensional analysis only checks the units.
- Numeric factors have no units and can’t be tested.
- Dimension analysis can not be used to derive the relation involving trigonometrical and exponential functions.
- It doesn’t indicate whether a physical quantity is a scalar or vector.

# Something to know…

## Units for Radiation:

The following units are used in special technologies or disciplines.

Since most people are not familiar with them, they are explained in more detail here.

**Becquerel **

The SI unit for radioactivity symbol (B), which is 1 disintegration per second (dps). 1 Ci = 3.7e10 B.

**Curie (Ci)**

A unit of radioactivity originally based on the disintegration rate of 1 g of radium.

Now a Curie is the quantity of radioactive material that has a disintegration rate of 3.700e10 per second (B). 1 mCi = 1e-3 Ci; 1 microCi = 1e-6 Ci; 1 MCi = 1e6 Ci.

**Gray and Rad**

Radiation dose units. The gray (Gy) is an SI unit for the absorption of 1 J radiation energy by one kg of material.

The rad was a popular unit, which is the absorption of 100 erg of radiation energy by one gram, (1 Gy = 100 rad).

**Roentgen (R)**

A unit for the measure of X-ray and gamma-ray exposure. 1 R = 93 erg per g (1 R = 0.93 rad for X-rays or gamma rays whose energy is above 50 keV).

The unit erg is for energy, 1 J = 10,000,000 erg.

# Review Questions

**1. What is the SI unit and symbol for force? **

Newton (N), he defined force

One N is the gravitational pull of 98 g mass

**2. What is the SI unit and symbol for pressure? **

Pascal (Pa), who studied effect of pressure on fluid

1 atm = 101325 Pa = 101.3 kPa

**3. What physical quantity uses the unit Joule? **

Joule (J) is an energy unit

1 J = 1 N m = 10e7 ergs

**4. Which is the SI unit for temperature? **

Kelvin (K)

1C is the same as 273.15 K

**5. What is the SI unit for measuring the amount of substance? **

mole (mol), derived from Latin, meaning mass

one mole has 6.023e23 atoms or molecules

**6. What are the symbols for the seven basic SI units? **

m, kg, s, A, K, cd, mol

m, k, s, current, temperature, luminous, mole

**7. What is the unit M used for? **

M stands for mol/L, a concentration unit

**8. What is the unit A used for? **

1 C/s, for an electric current

**9. What is the power consumption if the current is 1 A from a source of 10 V? **

10 C/s V (J/s = watt)

watt is the unit for power

**10. What is the SI unit for measuring radioactivity? **

Becquerel (B), he discovered radioactivity

1 Ci = 3.7e10 B

Stay Tuned – Next lesson ➟ Arithmetic Progression, Geometric Progression and Trigonometric Functions

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