Question

Problem 1: Greens Functions In class on Wednesday we found that we could solve the position, r(t), of a mass and spring when an arbitrary force is applied using Green's Functions, r(t)= | G(t,T)f(r)dr and that the particular Green's Function for this case is Gu.T)--sin ) wo where w, h/m and Θ(p) is a step function at p = 0. A: Find the position of the mass as a function of time if a force fo is applied for a short time Δι << T where T is the period of the mass-spring system. 2π Check that your answer has sensible physical units B Imagine the force on a mass spring system is 0, then is constant for the ime of one natural period, and then is 0 again.

Answer 1

The position is given by
$x(t)=\int G(t,\tau)f(\tau)~d\tau$
where, the Green's function is given by
$G(t,\tau)=\frac{1}{\omega_0}\Theta(t-\tau)\sin(\omega_0(t-\tau))$

A)

Now for
$f(\tau)=f_0, ~\text{for}~0<\tau<\Delta t,\text{where},~\Delta t <<T$
we get
  $x(t)=\frac{1}{\omega_0}\int_{-\infty}^{\infty} f(\tau)\Theta(t-\tau)\sin(\omega_0(t-\tau))~d\tau$
$\Rightarrow x(t)=\frac{1}{\omega_0}\int_{-\infty}^{t} f(\tau)\sin(\omega_0(t-\tau))~d\tau$
$\Rightarrow x(t)=\frac{f_0}{\omega_0}\int_{0}^{\Delta t} \sin(\omega_0(t-\tau))~d\tau$
$\Rightarrow x(t)=\frac{f_0\Delta t}{\omega_0} \sin(\omega_0(t-\frac{\Delta t}{2}))$

B)

And for
$f(\tau)=\left\{\begin{matrix} f_0, & \text{for}~0<\tau<T \\ 0, & \text{otherwise} \end{matrix}\right.$
where,
$\omega_0 T = 2\pi$ .
So, we get
  
  $\Rightarrow x(t)=\frac{1}{\omega_0}\int_{-\infty}^{t} f(\tau)\sin(\omega_0(t-\tau))~d\tau$
  $\Rightarrow x(t)=\frac{f_0}{\omega_0}\int_{0}^{T} \sin(\omega_0(t-\tau))~d\tau$
$\Rightarrow x(t)=\frac{f_0}{\omega_0}\int_{0}^{T} \big[\sin\omega_0 t \cos\omega_0 \tau -\cos\omega_0 t \sin\omega_0 \tau\big]~d\tau$
$\Rightarrow x(t)=\frac{f_0}{\omega_0} \big[\sin\omega_0 t \int_{0}^{T}\cos\omega_0 \tau~d\tau -\cos\omega_0 t \int_{0}^{T}\sin\omega_0 \tau~d\tau\big]$
We now use the result
$\int_{0}^{T}\cos\omega_0 \tau~d\tau =\int_{0}^{T}\sin\omega_0 \tau~d\tau=0$
So, we get
  $\Rightarrow x(t)=0$

This is the averaged value.