dimentional analysis

Overview of Units in Engineering Mechanics

In this unit, the focus will be primarily on the foundational quantities used in engineering mechanics:

• Time: Measured in seconds (s), represented by the symbol t.

• Length: Measured in meters (m), represented by the symbol l.

• Mass: Measured in kilograms (kg), represented by the symbol m.

These quantities serve as the base units that form the foundation for many topics in engineering mechanics.

Derived Units

Derived units are combinations of base units.

• Examples include:

  • Speed: Derived unit is meters per second (m/s) with dimensions of length divided by time: $ext{Speed} = rac{l}{t}$

Common Derived Units

Various derived units and their relationships to base units include:

• Kilogram (kg):

  • Equals 1,000 grams.

• Kilonewton (kN):

  • Equals 1,000 newtons.

• Megapascal (MPa):

  • Equals 10^6 pascals.

• Millimeter (mm):

  • Equals one-thousandth of a meter (1 mm = 0.001 m).

Unit Conversion

In certain contexts, unit conversion is necessary to apply more typical measurement units.

• Example of Conversion:

  • A vehicle traveling at 2,000,000 millimeters per minute (mm/min) can be converted to kilometers per hour (km/h).

• Conversion Factors:

  • 1 km = 1,000,000 mm

  • 1 hour = 60 minutes

Conversion Process

• Start with the value to be converted: 2,000,000 mm/min.

• To convert:

  1. Eliminate 'mm' by multiplying with the conversion factor $rac{1 ext{ km}}{1,000,000 ext{ mm}}$ (this is placed in the denominator).

  2. Eliminate 'min' by multiplying with the conversion factor $rac{60 ext{ min}}{1 ext{ hour}}$ (this is placed in the numerator).

Distance Equation

The equation that describes distance traveled by an object under constant acceleration is given by:
$s = ut + rac{1}{2} at^2$
Where:

• s = distance (length)

• u = initial velocity (length/time)

• a = acceleration (length/time^2)

• t = time (time)

Dimensional Analysis

• Each term of the equation must maintain dimensional homogeneity, meaning:

  • Left side has units of length (s)

  • Right side terms maintain the length dimension:

  • $u imes t$ has dimensions of length ($rac{l}{t} imes t = l$)

  • $rac{1}{2} a t^2$ also results in length ($rac{l}{t^2} imes t^2 = l$)

Reynolds Number and Dimensional Analysis

The Reynolds number is defined as a dimensionless parameter crucial in fluid mechanics.

• It relates to:

  • Fluid Density (mass/volume)

  • Fluid Velocity (length/time)

  • Characteristic Length (length)

  • Fluid Viscosity (mass/(length time))

Determining Dimensions of the Reynolds Number

• The dimensions of the Reynolds number can be derived by substituting the known quantities:

• After simplification, it is shown that the Reynolds number is dimensionless (no dimensions cancel out).

• Similarly, using units instead of dimensions confirms that the Reynolds number has no units.

Summary

• Focus on SI units and base dimensions in engineering mechanics: time, length, and mass.

• Derived units stem from base units with common equivalents and metric prefixes for succinct representation.

• Conversion between units may be necessary to reflect standard practices.

• Importance of dimensional homogeneity in ensuring valid physical equations with consistent units and dimensions across all terms.

• Dimensional analysis aids in understanding and determining unknown variables in engineering contexts.