what is dimensional analysis and how does it work?
It's a method whereby we convert one unit into another. For example, suppose we want to convert 15 inches to feet. We know there are 12 inches/foot so we set it up like this.
15 inches x (1 foot/12 inches) = ??feet. You can see that inches in the numerator cancel with inches in the denominator. So the units we DON'T want to keep cancel. The unit we want to convert things to stay. So 15 inches = 1.25 feet.
You can use this to convert one unit to another (any system), to convert chemical equations from moles of one reagent to another, or to convert rates of reaction on one material to rates of reaction of another. This is a general discussion. Try it yourself; if you get confused post a real problem and tell us what you don't understand about it.
(Note: The above is a working definition of how it is used; another meaning is to set up a problem in chemistry, physics, math, what have you, and use just dimensions with no numbers. Then you see if the units for the problem come out in the units you want. If so it is good evidence that you have set up the problem correctly. For example, if we have a problem to calculate the energy in joules of light of wavelength 500 nanometers, we can set up just the units.
E = hc/wavelength.
h is Planck's constant in Joule*sec = J*s, c is speed of light in meters/sec= m/s and wavelength is in meters =m.
So E = (J*s*m/s)/m.
s cancels, m cancels, to leave J and that is what E is measured in so we can be relatively certain this is the way to attack the problem.) Hope this helps.
thanks
Dimensional analysis is a method used in physics and engineering to check the consistency of mathematical equations and to derive relationships between physical quantities. It involves using the dimensions of various physical quantities (such as length, mass, time, etc.) to analyze and manipulate equations.
The basic idea behind dimensional analysis is that physical variables have units associated with them, and these units can be treated algebraically. The units of measurement for different physical quantities must be compatible in order for the equation to be valid.
To use dimensional analysis, follow these steps:
1. Identify the physical quantities involved in the problem and assign them symbols (e.g., mass as "m" or time as "t").
2. Determine the dimensions of each quantity. For example, mass has the dimension of [M], time has the dimension of [T], length has the dimension of [L], etc.
3. Write down the given equation or relationship between the quantities.
4. Replace each symbol with its corresponding dimension. For example, if the equation involves two variables, "m" and "t", and their dimensions are [M] and [T] respectively, replace them with [M] and [T].
5. Simplify the equation by performing algebraic manipulations on the dimensions, ensuring that the dimensions on both sides of the equation are consistent.
6. Check the derived equation for validity. If the dimensions on both sides of the equation are the same, then it is likely to be correct. If not, there may be an error in the equation.
Dimensional analysis is particularly useful in verifying equations and identifying the relationships between physical quantities. It helps in understanding the underlying physics and in ensuring the correctness of mathematical calculations.