PX154 - A1 - dimensional analysis

units

SI units
for energy, we will sometimes use the electron volt::

1 e V = 1.6 \times 10^{- 19} J

- visible light photon ~ 1.8-3.1 eV
- room temperature ~ 25 meV

distances
- $1 a n g s t o r m \equiv 10^{- 10} m$
- parsec = 3.26 ; light years ; (3.1 \times 10^{16} m)

units and dimensions

thought experiment :
- playing on a swing
  - me : 80 kg
  - friend : 60 kg
- who swings faster?
  - let us consider a simple pendulum
    - attached to a pivot by a massless string at an angle theta
    - to find: the period of the pendulum (T)
    - could be a function of length (l), mass of the person (m), gravity (g)
    - assumptions + approximations are made
    $T = f (l, m, g)$
    - could invoke mechanics... or...
      - T (LHS) has units of seconds
      - RHS must have a unit of seconds
      - units must match on both sides of the equation
        
        it looks like kg on RHS
        
        for length, can see that we try
        
        $\frac{l}{g} = \frac{m}{m s^{- 2}} = s^{2}$
        
        take: $\sqrt{\frac{l}{g}} = s$
        
        end up with:
        
        $T = 2 π \sqrt{\frac{l}{g}}$ - $2 π$ was not determined by dimensional analysis
        - it is a constant with no units
        
        can write this as:
        
        $2 π = T \sqrt{\frac{g}{l}}$ - $2 π$ is a constant with no units
        $∴ T \sqrt{\frac{g}{l}}$ must also be a constant with no units
        - this is not true for all consistent sets of units
        $⟹$ motivates us to introduce dimensions