Dimensional Analysis: Solving Confusing Steps

[promise899]

Homework Statement

I have a question which has two part described below:

Let the force P to be described by the surface area A between the ice and the buildings. The paramater for ice is comp. str. Y, which has the dimesion of stress. i) Proove the max. force per unit area P/A is independent of A. ii) Ice is a brittle material. That suggest an alternative model, that relevant materia parameter might not be comp stress Y but might instead be fracture toughness K(FL^-3/2). Show that in that case P/A is not independent of A and find how it depends on A?

Relevant Equations

i )pi(1) = P^X1 * A^Y1 * Y^Z1

I tried to use dimensional analysis, there is variable for part i) m= P,A and Y also parameter is used in analysis is n=3(M,L,T). So m-n=0 number of dimensionless analysis group. I am confused at this step however I did this calculation to reach solution:

i) P=MLT-2 (Ice Force) m-n=4-3=1(number of dimensionless group)
γ =ML-2T-2 (Specific Weight) A=L2 (Contact Surface Area)

pi(1) = P^X1 * A^Y1 * Y^Z1

pi(1) =( P/A*Y)

How it shows P/A is not dependent of A? I could skip some points in this question also I stuck at part ii. Thanks for help!

 

[Orodruin]

Homework Statement:: I have a question which has two part described below:

Let the force P to be described by the surface area A between the ice and the buildings. The paramater for ice is comp. str. Y, which has the dimesion of stress. i) Proove the max. force per unit area P/A is independent of A. ii) Ice is a brittle material. That suggest an alternative model, that relevant materia parameter might not be comp stress Y but might instead be fracture toughness K(FL^-3/2). Show that in that case P/A is not independent of A and find how it depends on A?
Relevant Equations:: i )pi(1) = P^X1 * A^Y1 * Y^Z1

is n=3(M,L,T). So m-n=0 number of dimensionless analysis group.

This is not correct as T and M are not independent dimensions here (ie, they always appear in the same combination, which is M/T^2). Hence, in this case, n=2.

 

[promise899]

This is not correct as T and M are not independent dimensions here (ie, they always appear in the same combination, which is M/T^2). Hence, in this case, n=2.

Force equal to P=ML/T^2 so we use M,L,T . In this way I think n=3

 

[Orodruin]

Force equal to P=ML/T^2 so we use M,L,T . In this way I think n=3

Again, no - you think wrong. M only appears along with 1/T^2. If you know M appears to the power of k in a dimension you know that T will appear to the power -2k. That it also appears with an L in force is irrelevant. You only have two independent dimensions. This is also clear from the simple fact that you can construct a dimensionless quantity.

 

[bhobba]

OP made an error in the question and will do a new post. Thread now locked.

Thanks
Bill