Problems with Dimensional Analysis.

[Col Kilgore]

Homework Statement

1) A string is stretched between two walls. When its plucked, a transverse wave travels from one end to the other. The length of the string is l, the mass of the string is m. There is a tensional force of F Newtons present in the string. The velocity of the resultant wave is v m/s. It is reasonable to assume that there is some relationship between the wave velocity and the other parameters such as:
v=k.F^x.l^y.m^z (where k is some dimensionless constant)

Using dimensional analysis, determine the likely relationship.

2) A sphere of radius a is dropped into a viscous liquid with a coefficient of viscosity n and its velocity at an instant is v. The frictional force can be partly found by dimensional analysis.

F=k.a^x.n^y.v^z (where k is a dimensionless constant)

The dimensions of the variables are:
[F] = MLT^-2
[a] = Lk
[n] = ML^-1T^-1
[v] = LT^-1

Use dimensional analysis to find a likely relationship between the variables.

The Attempt at a Solution

This isn't coursework or homework, I have just been looking at past papers and a question similar to the ones I've posted always comes up. Most questions, I have some sort of an idea about, but dimensional analysis questions always stump me. Any help with methods on doing these types of questions would be appreciated.

 

[cepheid]

At this point it's just algebra. For instance, in your second problem, you know that the left-hand side must have dimensions of force, and so you just tweak the right-hand side until it matches:

(I'm going to assume that [a] = L, since "Lk" doesn't make sense to me).

MLT^-2 = kL^x(ML^-1T^-1)^y(LT^-1)^z

It helps if you first combine powers that have the same base:

MLT^-2 = kL^x-y+zM^yT^-y-z

Now you just have to match exponents up by inspection. In order for the right-hand side to have the same dimensions as the left, it must be true that:

x-y+z = 1

y = 1

-y-z = -2

substituting the second equation into the third, you get:

-1 - z = -2

z = 1

substituting this into the first equation, you get:

x - 1 +1 = 1

x = 1

Therefore, it seems that the answer is:

F = kanv

which might be plausible. I imagine that the solution method to most other problems of this type would be very similar.