I’ve always been amazed at the power of dimensional analysis. To me the best example is the problem of calculating the period of an oscillating mass on a spring. The relevant values are the spring constant K (kg/s^2) and the mass M (kg), and the period T is in (s). The only way to combine K and M to obtain a value with dimensions of (s) is sqrt(M/K), and that’s the correct form of the actual answer—no calculus required!
Actually, there’s another parameter, the displacement. It turns out that the spring period does not depend on the displacement, but that’s a miracle that is special to springs. Instead, look at the pendulum. The same dimensional analysis gives the square root of the length divided by gravitational acceleration. That’s off by a dimensionless constant, 2π. Moreover, even that is only approximately correct. The real answer depends on the displacement in a complicated way.
This is a good point. At best you can figure out that period is proportional to (not equal to) sqrt(M/K) multiplied by some function of other parameters, say, one involving displacement and another characterizing the non-linearity (if K is just the initial slope, as I’ve seen done before). It’s a fortunate coincidence if the other parameters are unimportant. You can not determine based solely on dimensional analysis whether certain parameters are unimportant.
I’ve always been amazed at the power of dimensional analysis. To me the best example is the problem of calculating the period of an oscillating mass on a spring. The relevant values are the spring constant K (kg/s^2) and the mass M (kg), and the period T is in (s). The only way to combine K and M to obtain a value with dimensions of (s) is sqrt(M/K), and that’s the correct form of the actual answer—no calculus required!
Actually, there’s another parameter, the displacement. It turns out that the spring period does not depend on the displacement, but that’s a miracle that is special to springs. Instead, look at the pendulum. The same dimensional analysis gives the square root of the length divided by gravitational acceleration. That’s off by a dimensionless constant, 2π. Moreover, even that is only approximately correct. The real answer depends on the displacement in a complicated way.
This is a good point. At best you can figure out that period is proportional to (not equal to) sqrt(M/K) multiplied by some function of other parameters, say, one involving displacement and another characterizing the non-linearity (if K is just the initial slope, as I’ve seen done before). It’s a fortunate coincidence if the other parameters are unimportant. You can not determine based solely on dimensional analysis whether certain parameters are unimportant.