Dimensions, Units, and Conversions Introduction to Mechanical Engineering The University of...
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Transcript of Dimensions, Units, and Conversions Introduction to Mechanical Engineering The University of...
Dimensions, Units, and Dimensions, Units, and ConversionsConversions
Introduction to Mechanical EngineeringIntroduction to Mechanical Engineering
The University of Texas-Pan AmericanThe University of Texas-Pan American
College of Science and EngineeringCollege of Science and Engineering
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.
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
Dimensional Analysis – Example 1
essDimensionlM
TL
L
M
T
LL
LT
ML
M
T
LL
3
3
vD
Re
The Reynolds number is given by
where D = pipe diameter, v = fluid velocity, ρ = fluid density, and µ = viscosity (M/LT). Show that the Reynolds number is dimensionless.
Dimensional Analysis – Example 2
The pressure in a column of fluid is given by
where P = pressure, ρ = fluid density, g =acceleration of gravity, and h = height of fluid column. Is this equation dimensionally homogeneous?
hgP
YESTL
ML
T
L
L
M
TL
M2232
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).
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 ConversionsExact 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:
http://www.onlineconversion.com/
Unit Systems and ConversionsExample
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. For example, on the Moon, your massmass would be
the same as here on Earth, yet your weightweight would be less due to the lower gravitational lower gravitational accelerationacceleration present on the Moon.
Quiz
Carry out the following conversions:
a) 125 days to seconds
b) 16 lbm/ft3 to kg/m3
c) 75 slug/min to kg/s
d) 15 ft3 to gallons
Quiz Solutions
a)
b)
Notice that the (ft/m) part is cubed because we cannot cancel out ft3 with just ft, remember, the dimensions must be the same.