2.B Dimensional Analysis
Performing a Dimensional Analysis
Given independent variables \(x,y,z\) and dependent variable \(w\):
- Write down the hypothesized “generalized product” \(w = k x^a y^b z^c\)
- Write & simplify the dimensional version \([w] = [x]^a [y]^b [z]^c\)
- List the dimensions of all quantities
- Invoke dimensional compatibility to solve.
- Rewrite your equation, marvel at your awesomeness!
Activities
Vocal Chord Frequency
The frequency \(f\) of the sound of an organism’s vocal chords depend on their length \(\ell\), tension \(s\) (a force) and mass density \(\mu\) (mass per unit length). Find a formula for the frequency \(f = f(\ell, s, \mu).\)
Solutions
We have \[ \begin{array}{rcl} f &=& k \ell^a s^b \mu^c \\ [f] &=& [\ell]^a [s]^b [\mu]^c \\ T^{-1} &=& \left( L\right)^a \left(MLT^{-2}\right)^b \left( M L^{-1} \right)^c \\ M^0L^0T^{-1} &=& M^{b+c} L^{a+b-c} T^{-2b} \end{array} \] and so we have \[ 0 = b+c \qquad 0 = a+b-c \qquad -1=-2b. \] Therefore \(b=1/2\) and \(c=-1/2\) and \(a=-1\). Our formula is \[ f = k \ell^{-1} s^{1/2} \mu^{-1/2} = k \frac{1}{\ell} \sqrt{\frac{s}{\mu}}. \]
We have \[ \begin{array}{rcl} A &=& k P^a D^b H^c \\ [A] &=& [P]^a [D]^b [H]^c \\ T^{-1} &=& \left( ML^2T^{-3}\right)^a L^b \left( M L^{-1} T^{-1} \right)^c \\ M^0L^0T^{-1} &=& M^{a+c} L^{2a+b-c} T^{-3a-c} \end{array} \] and so we have \[ 0 = a+c \qquad 0 = 2a+b-c \qquad -1 = -3a-c \] which means that \(a=1/2\) and \(c=-1/2\) and \(b=-3/2\). So our final equation is \[ A = k a^{1/2} b^{-3/2} c^{-1/2} = k \sqrt{\frac{P}{D^3H}} \]
We have \[ \begin{array}{rcl} V &=& k P^a g^b \mu^c \\ [V] &=& [P]^a [g]^b [\mu]^c \\ LT^{-1} &=& (T)^a (LT^{-2})^b (ML^{-3})^c \\ L^1 T^{-1} M^0 &=& L^{b-3c} T^{a-2b} M^{c} \end{array} \] so we have the equations \[ 1 = b - 3c \qquad -1 = a-2b \qquad 0 = c \] which means that \(c=0\) and \(b=1\) and \(a=1\). So our final equation is \[ V = k Pg \]