Momentum Transfer

• Force applied to the two string endpoints by the mass is given by $f_m(t) = 2Rv(t)$
• From Newton's Law, $f=ma=m\dot v$, momentum $mv$ is the time integral of force:
  $$\begin{eqnarray*} \int_0^t f_m(\tau)\,d\tau &=& 2R\int_0^t v(\tau)\,d\tau \;=\; m\,v_0\left(1-e^{-{\frac{2R}{m}t}}\right) \end{eqnarray*}$$
  
  • Thus, momentum delivered to the string by the mass starts at zero and grows as a relaxing exponential to $mv_0$ as $t\to\infty$

  • Inelastic mass-string collision is not an instantaneous momentum transfer

  • Mass momentum transfers exponentially over time

  • This is why contact width can be zero

• In a real piano, the hammer, which strikes in an upward direction, falls away from the string a short time after collision, but it may remain in contact with the string for a substantial fraction of a period