Dimensional Analysis Explained - MIT 8-01

[Aspchizo]

In http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/lecture-1/ at about 23 minutes in, he starts talking about dimensional analysis. Can someone help expand on this a bit? I don't understand the Alpha, Beta and Gamma terms he uses.

Thanks.

 

[grzz]

The professor is making the assumption that the time t of fall of an object depends on the height h from which it falls, the mass m of the object and the acceleration g due to gravity.

He is using dimensional analysis to derive the required formula.

The powers of h, m and g are still unknown. Hence he is using the symbols $\alpha$,$\beta$ and $\gamma$ for these powers.

 

[Aspchizo]

So they are just algabraic placeholders?

 

[grzz]

The $\alpha$.$\beta$ and $\gamma$ are, as yet, unknown values to be determined later by equating the dimensions on each side of the resulting equation.

 

[pmacias]

Hi there,

Dimensional analysis is a powerful tool used in science and engineering to understand the relationships between physical quantities. It involves analyzing the dimensions (such as length, time, mass, etc.) of different physical quantities and using them to derive equations and make predictions.

In the lecture you mentioned, the professor introduces the concept of dimensional analysis by discussing the Buckingham Pi theorem, which states that the number of independent variables in a physical problem can be reduced by the number of fundamental dimensions involved. This means that instead of using multiple variables to describe a physical phenomenon, we can use a single dimensionless quantity (known as a pi term) to represent it.

The Alpha, Beta, and Gamma terms mentioned in the lecture refer to the exponents used in the dimensional analysis process. These exponents are determined by equating the dimensions of different physical quantities in an equation, and they help us understand the relationships between these quantities.

For example, let's say we have an equation that relates the force (F), mass (m), and acceleration (a) of an object: F = ma. By analyzing the dimensions of these quantities (force = mass * acceleration), we can see that the dimensions of force are equal to the dimensions of mass times the dimensions of acceleration. This can be represented as [F] = [m][a], where the brackets indicate dimensions. To make this equation dimensionless, we can divide both sides by a reference quantity (such as the mass of the object) to get [F]/m = [a]. This is known as the pi term for acceleration, and the exponents of the dimensions in this term (alpha = 0, beta = 1) represent the relationship between force and acceleration.

In summary, dimensional analysis helps us understand the relationships between physical quantities and reduces the complexity of physical problems by using dimensionless quantities. The Alpha, Beta, and Gamma terms are used to represent the exponents of the dimensions in these relationships. I hope this helps clarify the concept of dimensional analysis for you.