- #1

koopernikos

- 1

- 0

## Homework Statement

A semi-infinite string (constant tension T and density ρ) lies along the positive

x-axis. The motion of the string is described by the linear wave equation

\begin{equation}

h_{xx}-c^2h_{tt}=0

\end{equation}

with

c² ≡ T/ρ, where h(x,t) is the transverse displacement. Attached to the end of the string at

x = 0 is a mass m. Attached to the mass in the vertical direction is a spring with stiffness

constant k that has its other end attached to a rigid surface. The vertical displacement of

the mass m is designated by ξ (t) ≡ h(0,t). When the system is in equilibrium (i.e.,

when the spring is unextended and everything is at rest) ξ (t) ≡ 0. The radian frequency

of the free oscillator (when the string is not attached) is σ = k /m .

(a) By applying Newton's law to the motion of the mass at x = 0, find the boundary

condition on the string at x = 0. That is, find the equation of motion of the mass in

terms of ξ (t) . Ignore horizontal forces on the mass, and ignore gravity.

[HINT: the boundary condition should not have any terms showing h(0,t).]

(b) Prove that the total energy of the mechanical system (oscillator and string) is constant.

(c) Suppose that at t = 0 the initial conditions are

## The Attempt at a Solution

a)

There are two forces (neglecting $\vec{g}$) acting on the mass: one due to the displacement (spring) and one stemming from the tension in the string.

\begin{align}

F_{net}

&=

\sum_i F_i\\

f_{net}

&=

\sum_i f_i\\

\ddot{\xi}

&=

-\sigma^2 \xi

+ c^2 h(0,t)_{xx}

\end{align}

b)

?

c)

?

...aaand that is about as much as i can come up with..

i played around with seperating variables a bit.. i don't even know what type of solution to expect.. something like

\begin{equation}

h(x,t)=e^{-At} (B \sin(kx-\omega t)+ C \cos(kx-\omega t))

\end{equation}

?

obviously

\begin{equation}

\sigma \neq \omega

\end{equation}

please help! I've been staring at the problem for 4 hours