ME3112 Extra Notes

Splitting the terms I The terms for the velocity may be “split” or “combined” in various ways. ~v = r ω (~k × e~r ) = (r ω~k) × e~r = ~k × (r ω e~r ) = (r ~k) × (ω e~r ) = (ω~k) × (r e~r ) But the last expression contains physical quantities: ω ~ = ω~k and ~r = r e~r . Hence, ~v = r ω (~k × e~r ) = (ω~k) × (r e~r ) =ω ~ × ~r 1/2

Splitting the terms II Similarly, r ω2

  ~k × (~k × e~r ) = (r ω 2 ~k) × (~k × e~r ) = (r ω ~k) × (ω ~k × e~r ) = (r ω ~k) × (~k × ω e~r ) = ... = (ω ~k) × (ω ~k × r e~r )

The last expression contains physical quantities, so it makes sense to write   2 ~ ~ ~ k × (k × er ) = (ω ~k) × (ω ~k × r e~r ) rω =ω ~ × (~ ω × ~r )

2/2