how will you test the correctess of an equation in physics F=ma by using dimensional analysis?
Answers
When solving physics problems, it is 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.
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
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.
Plugging in M, L, and T we get
range
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.
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.
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 will plug in units instead of dimensions.
This shows the cancelation of the mass.
Answer:
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