BTEC | Mechanical Engineering Academy # Engineering Maths – Dimensional Analysis

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.

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**

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.**

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

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

- 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.*

- 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

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

- Velocity (units m/s)

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

- Force ( mass x acceleration )

- 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,

- 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,**

- 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.

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.

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

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.

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).

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.

Newton (N), he defined force

One N is the gravitational pull of 98 g mass

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

1 atm = 101325 Pa = 101.3 kPa

Joule (J) is an energy unit

1 J = 1 N m = 10e7 ergs

Kelvin (K)

1C is the same as 273.15 K

mole (mol), derived from Latin, meaning mass

one mole has 6.023e23 atoms or molecules

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

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

M stands for mol/L, a concentration unit

1 C/s, for an electric current

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

watt is the unit for power

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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