PX154 - A3 - applying dimensional analysis

• two more concepts required:
  • dimensionless groups
  • fundamental independent quantities

dimensionless groups

• for our pendulum, we had

- $T\sqrt{\frac{g}{l}}$ is a dimensionless group

• more generally, if we have several variables, we may form more than one dimensionless group
  • instead of $\mathrm{\Pi }$, we have ${\mathrm{\Pi }}_1,{\mathrm{\Pi }}_2,{\mathrm{\Pi }}_3,...$
    • for our variables: $Q_1=f(Q_2,Q_3,Q_4,...)$
    • we would write: $\mathrm{\Pi }=\varphi ({\mathrm{\Pi }}_1,{\mathrm{\Pi }}_2,{\mathrm{\Pi }}_3,...)$
  • eg: gravity waves on water might need inclusion of depth (L) and wavelength(L)
• the number of dimensionless groups is called $\tilde{N}$
  • in general, we would need to know $\tilde{N}$ in order to proceed: it can be determined by "[buckingham's pi method](PX154 - A2 - dimensions#^5abdae)."
  $$\tilde{N}=N-m$$$m=$ number of "fundamental independent quantities"

fundamental independent quantities

• to find $m$:
  • for our pendulum:

• • firstly, $T$ can depend on other variables $⟹$ NOT independent
  • so, discard $T$ and consider remaining variables
    $⟹$ in this example, there is no way to express any of these as a function of the others
  • hence, the number of independent variables is $3$ and so, $m=3$
  • hence, $\tilde{N}=N-m=4-3$
    $\therefore \tilde{N}=1$
    • we would stick with $\tilde{N}$ in an exam
    • check moodle handout for more examples

the procedure

• • step 1: make a list of variables that that we think may be needed
  • step 2: use [buckingham's pi method](PX154 - A2 - dimensions#^5abdae) to determine $\tilde{N}$
  • step 3: consider the exponents of the function that relates them
• eg:
  • speed of waves on water, for deep water:
    • we assume that the depth ($h$) is much greater than the wavelength ($\lambda$)
      • ie: $h>>>\lambda$
    • step 1: variables:
      • speed $[V]=L{T}^{-1}$
      • wavelength $[\lambda ]=L$
      • gravity $[g]=L{T}^{-2}$
      • density $[\rho ]=M{L}^{-3}$
      • we anticipate $v=f(\lambda ,g,\rho )$
    • step 2: find $\tilde{N}$
      $N=4$
      $m=3$
      $\therefore \tilde{N}=1$
    • step 3: considering the exponents
      • dimensionless group
        $\mathrm{\Pi }=v^{\alpha }{g}^{\beta }{\lambda }^{\gamma }{\rho }^{\delta }$
        $[\mathrm{\Pi }]={[L{T}^{-1}]}^{\alpha }[L{T}^{-2}{]}^{\beta }[L{]}^{\gamma }[M{L}^{-3}{]}^{\delta }$
      • exponents:
        $L:0=\alpha +\beta +\gamma -3\delta$
        $T:0=-\alpha -2\beta$
        $M:0=-3\delta$
      • taking $\alpha =1$:
        $\beta =-\frac{1}{2}$
        $\delta =0$
        $\gamma =-1/2$
      • finally,
        $[\mathrm{\Pi }]={[L{T}^{-1}]}^1[L{T}^{-2}{]}^{-\frac{1}{2}}[L{]}^{-\frac{1}{2}}[M{L}^{-3}{]}^0$
        $⟹\mathrm{\Pi }=v.g^{-\frac{1}{2}}.{\lambda }^{-\frac{1}{2}}$
        $or,v=\mathrm{\Pi }\sqrt{g\lambda }$
        $\therefore v\propto \sqrt{g\lambda }$
• PDFs on moodle: shallow and intermediate water