Using Dimensional Analysis to Check an Equation's Correctness

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  • 0:04 Dimensional Equations
  • 2:07 Dimensional Analysis
  • 3:02 Two More Examples
  • 5:07 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser
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.

Dimensional Equations

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:

Example 1

The equation for the volume of a cube is


volume

and we plug M, L, and T into this equation to get


vol_dim_eq

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.

Example 2

The equation to determine the horizontal distance a projectile travels is given as:


Range

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:


range

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.


range_simplified

Notice that our resulting dimension is length which is exactly what the horizontal range of a projectile is.

Dimensional Analysis

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:

Example 3

Let's go back to the range equation for a projectile and plug in units instead of M, L, and T:


range_with_units

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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