Dimensions, Units, and Conversions Introduction to Engineering College of Science and Engineering The University of Texas – Pan American

Objectives  Explain the difference between dimensions and units.  Check for dimensional homogeneity.  Explain SI unit prefixes.  Convert between SI and U.S. Customary units.  Explain the difference between mass and weight. Assignment: Handout or visit website.

Difference Between Dimensions and Units  Why are dimensions and units important?  Dimensions are used to describe objects and actions. The three most basic dimensions are length, time, and mass  Units are used to establish the size or magnitude of a dimension. Must be based on some convention with standards

Difference Between Dimensions and Units  Dimensions are divided into fundamental and derived. Fundamental are the most basic or elementary dimensions necessary to describe the physical state of an object. Derived dimensions are defined based upon scientific and engineering equations, and are a combination of fundamental dimensions.

Fundamental and Derived Dimensions

Difference Between Dimensions and Units  Dimensions are fundamental, unchanging characteristics or properties of an object.  Units on the other hand are arbitrary; they can be changed by the vote of a governing body.  History of Units – cubit, meridian mile, foot, etc…

Dimensional Analysis  Equations in Science and Engineering must be dimensionally homogeneous, in other words, the dimensions on each side of the equation should be the same when dimensions (not units) are substituted for the variables and constants.  For example, if you are calculating velocity from the distance traveled in an elapsed time, the dimensions on either side should be equal, i.e., Velocity = Distance traveled / Elapsed time Distance traveled = Length (L) Elapsed time = Time (T) Velocity = Length / Time = L / T

Unit Systems  Systems of units differ in the treatment of mass and force.  In the SI system, mass was chosen as the third fundamental dimension and force is a derived unit.  In the English system, force was chosen as the third fundamental dimension and mass is a derived unit.

The International System of Units  SI units are derived into three classes: base units (seven), derived units, and supplementary units (two).

Base Units

Derived Units

Supplementary Units  Radian is equal to the angle between two radii of a circle that cut off a piece of the circumference whose length is equal to the length of the radius.  Steradian is equal to the solid angle which cuts off, on the surface of a sphere, an area equal to the area of a square whose sides are the same length as the radius of the sphere.

The International System of Units  To avoid very small or very large numbers in the SI system of units, unit prefixes have been developed based on power of ten.

Unit Systems  Fundamental and some important derived dimensions for the three common systems of units.

Unit Systems and Conversions Exact Conversions

The internet provides valuable resources that can be used to obtain a variety of different conversion factors or completely carry out the conversions for you. Please refer to the following website:

Unit Systems and Conversions Example  The employment of the information given in the preceding tables allows for ease of conversion between different units.  For example, if you are traveling at a speed of 65 miles per hour (mi/hr or mph) and wish to know your speed in feet per second (ft/s) and in meters per second (m/s) you would have to carry out the following conversions:

Mass & Weight  The mass of an object is constant.  Weight is the force required to lift or support an object in a gravitational field or an acceleration field.  Acceleration of gravity changes with location. mass weight lower gravitational acceleration  For example, on the Moon, your mass would be the same as here on Earth, yet your weight would be less due to the lower gravitational acceleration present on the Moon.

Open Forum

Quiz  Carry out the following conversions: a) 125 days to seconds b) 16 lb m /ft 3 to kg/m 3 c) 75 slug/min to kg/s d) 15 ft 3 to gallons

Quiz Solutions a) b) Notice that the (ft/m) part is cubed because we cannot cancel out ft 3 with just ft, remember, the dimensions must be the same.

Quiz Solutions c) d)