Dimension Analysis Homework: Buckingham Theorem

In summary, the question is about the difference between equations a and b in terms of dimensionless variables, and whether the sine function has any influence on the dimension. The answer is that the equations are equivalent, with the only difference being the use of sin(phi) in one and phi in the other. However, the small-angle approximation must be considered for larger values of phi.
  • #1
dirk_mec1
761
13

Homework Statement



http://img21.imageshack.us/img21/613/70858934fn5.png

Homework Equations


Buckingham theorem

The Attempt at a Solution


My question is what is the difference between question a and b? The sine doesn't influence the dimension. Or is it a question to trick me?
 
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  • #2
What I meant is that sin(phi) has the same dimension as phi so the answers of a and b are the same, right?
 
  • #3
sin(phi) is only equivelent to phi (approximately) for very small phi, i.e. very small oscillations of the pendulum, when simple harmonic motion occurs.
Above very small angles, formula (a) is correct, (b) incorrect, as the variation between phi and sin(phi) becomes significant.
Look at Small-angle Approximation on the following link:
http://en.wikipedia.org/wiki/Pendulum_(mathematics )
 
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  • #4
It says in the problem statement that phi is small for part b. The question is about expressing the equations in dimensionless variables.

dirk_mec1 said:
My question is what is the difference between question a and b? The sine doesn't influence the dimension. Or is it a question to trick me?

I think the question really is as easy as it seems: no essential difference between the two equations, other than replacing φ with sin(φ). Once you've converted one equation to a dimensionless form, you basically have the other.
 
  • #5
Thanks for the confirmation redbelly.
 

Related to Dimension Analysis Homework: Buckingham Theorem

What is Dimensional Analysis?

Dimensional analysis is a mathematical technique used to analyze and solve problems that involve multiple physical quantities and their units.

What is the Buckingham Pi Theorem?

The Buckingham Pi Theorem, also known as the Pi Theorem or the Buckingham-Pi method, is a mathematical theorem that states that in any physical system involving a certain number of variables, there exists a certain number of dimensionless quantities that can be used to describe the system.

Why is Dimensional Analysis important?

Dimensional analysis is important because it allows scientists and engineers to understand the relationships between different physical quantities and their units in a given system. This can aid in problem-solving, as well as in the development and testing of theories and models.

How is Dimensional Analysis used in science and engineering?

Dimensional analysis is used in science and engineering to help simplify and solve complex problems involving multiple physical quantities and units. It is also used to check the consistency of mathematical equations and to determine the relationships between different variables in a given system.

What are some common applications of the Buckingham Pi Theorem?

The Buckingham Pi Theorem is commonly used in various fields of science and engineering, such as fluid mechanics, thermodynamics, and electromagnetics. It can be used to determine the conditions for similarity and scale modeling, as well as to simplify and analyze complex systems with multiple variables and units.

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