Dimensional anaylsis problem involving trigonometry

[mileena]

Homework Statement

Dimensional Analysis:

A displacement is related to time as:

x = A sin (2∏ft), where A and f are constants.

Find the dimensions of A. (Hint: a trigonometric function appearing in an equation must be dimensionless.)

Homework Equations

t = seconds
The domain of a sine must be an angle.

The Attempt at a Solution

x = A sin (2∏ft), where A and f are constants.

m = A sin (2∏fs)

A = m/[sin (2∏fs)]

Therefore, f, a constant, must be in terms of s^-1, in order to cancel out the s next to it, as the input of sine in an equation must be dimensionless.

Therefore, A = m

Thanks for any help!

 

[rock.freak667]

Yes that would be correct.

 

[Emilyjoint]

Correct

 

[mileena]

I am absolutely stunned that I got the question correct (other than failing to use base dimensions instead of units in the SI system, as Emilyjoint posted in my other thread here!).

Thank you!

 

[Nickie]

I would like to clarify that dimensional analysis is a method used to check the correctness of a mathematical equation by ensuring that the dimensions on either side of the equation are consistent. It is not a method for solving equations or finding constants. In this case, the given equation is already dimensionally consistent, so there is no need for further analysis.

However, I can provide some additional information regarding the dimensions of A. Since x is a displacement (in meters), A must also have the dimensions of length. This means that the constant f must have the dimensions of inverse time (s-1), as you correctly stated. Therefore, A cannot be equal to m, as m is a unit for mass and does not have the correct dimensions for A.

In summary, the dimensions of A are length, and the constant f must have the dimensions of inverse time for the given equation to be dimensionally consistent. I hope this helps clarify the concept of dimensional analysis in this context.