Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.
When studying physics, it's very common to use multiple equations embedded in each other in order to solve a problem. This lesson will detail how to check if the final equation in a derivation is correct in terms of units.
When solving physics problems, it's sometimes easier to do all of the equation derivation first before plugging in any numerical values; however, this can lead to errors in terms of units or dimension. The dimensional equation process is helpful in determining if an equation has been derived properly. In any dimensional equation, M is used to represent mass, L is used to represent length, and T is used to represent time. Let's go through some examples using dimensional equations.
A good starting point for gaining understanding of this process is to determine the dimensional equation for the volume of a cube. Let's now take a moment to look at some examples:
The equation for the volume of a cube is
and we plug M, L, and T into this equation to get
Since anything to the zero-power is 1, the only part left in our dimensional equation is length cubed, which is volume.
Now we can look at a more advanced derived equation.
The equation to determine the horizontal distance a projectile travels is given as:
This is where R is the range or horizontal distance a projectile travels, v0 is the initial velocity of the projectile, θ is the angle of launch, and g is the acceleration due to gravity.
So, in plugging in M, L, and T we get:
As you may suspect, this equation requires simplification. We realize that the sine of an angle has no unit, so sin(2θ) can be left out of the equation. M0 also drops out of the equation, because as we saw earlier, it equals 1.
Notice that our resulting dimension is length which is exactly what the horizontal range of a projectile is.
There are limitations to using a dimensional equation to verify that an equation gives the proper units. There are units in physics which are derived using mass, length, time, and other variables. Examples of these types of derived units are force, which is measured in newtons, and energy, which is measured in joules. The newton is a kilogram meter-per-second squared (kgm/s2). The joule is a newton-meter which can be represented by kgm2/s2. When these units are represented in a dimensional equation, the resulting simplification's units may lose the context of the equation. This makes it hard to determine if the original equation is correct, defeating the purpose of a dimensional equation.
There is another process which is very similar to a dimensional equation, called dimensional analysis, in which the actual units are plugged into the variables of an equation. Simplifying the units can determine whether the equation is correct or not.
Let's wrap this lesson up by look at two more examples:
Let's go back to the range equation for a projectile and plug in units instead of M, L, and T:
We can see how similar this is to our original dimensional equation, but the resulting unit leftover after simplification is the meter, the international metric system unit for displacement, instead of L which just represents length.
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Dimensional analysis is very helpful when an equation has a physical constant in it such as the universal gravitational constant G which has the unit Nm2/kg2. The orbital period of a satellite equation includes G. Let's use dimensional analysis to show that the correct unit of time comes out of the equation. We'll plug in units instead of dimensions.
In the second line, the very last expression is as simple as possible in terms of units, despite there being quite a few. All of them are pure and cannot be broken down into anything else. The 2π is left out of the dimensional analysis because it's a constant. Now we can get to work canceling out units, resulting as you can see, with the square root of s squared, which is obviously s.
The leftover unit is seconds, which is definitely a measure of time.
The velocity of a satellite at apogee (furthest distance from the earth in an elliptical orbit) is given by
Plugging in the units for the variables and constants and then simplifying, we get these equations:
Line two shows all the pure units, and lines three and four show the progressive canceling of units until nothing is left that will cancel: that's why it ends in the square root of meters squared per seconds squared, which is simply meters per second. Velocity has the units meters per second, or m/s, so our dimensional analysis proves that this equation is correct.
Let's take a couple of moments to review what we've learned. When solving physics problems, it's very common to have to plug one equation into another equation. Algebra's used to simplify this new equation. To determine if the resulting equation is correct, one of two processes can be used: a dimensional equation or dimensional analysis.
A dimensional equation uses the dimensions of mass (M), length (L), and time (T), and plugs them in as the equation's variables. If there's no variable for a dimension, then it is entered to the zero-power, which is effectively canceling out that dimension all together. More simplification is done and the resulting dimension(s) should be what the equation states it to be.
Dimensional analysis is similar to a dimensional equation, but is a process whereby the actual units are plugged into an equation. The units can then be simplified, and what's left over must show what the equation represents.
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