# unit and dimensions

Saurabh Kumar
9 years ago
Units and Dimensions:
An experiment is performed in which exactly 2.3 gallons of fluid is heated and the temperature at which the fluid boils as noted as 253° F. Restate this finding using standard SI units (cubic meters and K).When measuring a quantity, the units we use are just as important as the numerical value we obtain. Stating that the mass of an object is one (1) says very little about the actual mass of the object. It is perfectly reasonable for an instructor to respond to such an answer: "One what???"

To be complete,
every measurement should be
expressed with the appropriate units.

Base quantity examples of units
length centimeter (cm), inch (in), foot (ft), meter (m), kilometer (km)
mass gram (g), kilogram (kg)
time second (s), minute (min), hour (hr)
temperature Fahrenheit (°F), Kelvin (K), Celsius (°C)
electric current ampere (A)
amount of matter mole (mol)
luminous intensity candela (cd)

The units in bold font are the base units used in the SI System of Units the recommended scientific system of units. In addition to length, mass, time and temperature, the list of base quantities includes three others you may not have seen before: electric current (ampere), luminous intensity (candela) and amount of a chemical substance (mole). The last unit (mole) is constantly used in chemistry and explained on the stoichiometry page.
More complicated quantities can be measured (e.g., area, volume, density, velocity, acceleration) -- so you need to understand how to handle combinations of units.

Different people use different units (e.g., inches versus centimeters)-- so you need to be able to convert from one unit to another.

Combinations of Units

When studying chemistry you will measure many quantities with units that are actually combinations of the base units. Be sure to include the units in your notes when you encounter a new term. Here is a list of quantities that you should already be familiar with from previous classes. Follow the link to a short description for those you don't recall.

Quantity Unit(s) Quantity Unit(s)
Area m ∗ m = m2 Volume m ∗ m ∗ m = m3
Density kg / m3 = kg m-3 Velocity m / s = m s-1
Acceleration ( m / s ) / s = m / s2 = m s-2 Force kg ∗ ( m / s2 ) = kg m s-2
Energy kg ∗ ( m / s2 ) ∗ m = kg m2 s-2

Dimensional analysis:

1. In engineering the application of fluid mechanics in designs make much of the use of empirical results from a lot of experiments. This data is often difficult to present in a readable form. Even from graphs it may be difficult to interpret. Dimensional analysis provides a strategy for choosing relevant data and how it should be presented.

This is a useful technique in all experimentally based areas of engineering. If it is possible to identify the factors involved in a physical situation, dimensional analysis can form a relationship between them.

The resulting expressions may not at first sight appear rigorous but these qualitative results converted to quantitative forms can be used to obtain any unknown factors from experimental analysis.

2. Dimensions and units

Any physical situation can be described by certain familiar properties e.g. length, velocity, area, volume, acceleration etc. These are all known as dimensions.

Of course dimensions are of no use without a magnitude being attached. We must know more than that something has a length. It must also have a standardised unit - such as a meter, a foot, a yard etc.

Dimensions are properties which can be measured. Units are the standard elements we use to quantify these dimensions.

In dimensional analysis we are only concerned with the nature of the dimension i.e. its quality not its quantity. The following common abbreviation are used:

length = L

mass = M

time = T

force = F

temperature = Q

In this module we are only concerned with L, M, T and F (not Q). We can represent all the physical properties we are interested in with L, T and one of M or F (F can be represented by a combination of LTM). These notes will always use the LTM combination.

The following table lists dimensions of some common physical quantities:

Quantity SI Unit Dimension
velocity m/s or ms-1 LT-1
acceleration m/s2 or ms-2 LT-2
force Nkg m/s2 kg ms-2 M LT-2
viscosity N s/m2kg/m s N sm-2kg m-1s-1 M L-1T-1
surface tension N/mkg /s2 Nm-1kg s-2 MT-2

3. Dimensional Homogeneity

Any equation describing a physical situation will only be true if both sides have the same dimensions. That is it must be dimensionally homogenous.

For example the equation which gives for over a rectangular weir (derived earlier in this module) is,

The SI units of the left hand side are m3s-1. The units of the right hand side must be the same. Writing the equation with only the SI units gives

i.e. the units are consistent.

To be more strict, it is the dimensions which must be consistent (any set of units can be used and simply converted using a constant). Writing the equation again in terms of dimensions,

Good Luck:..............................................................................