Hanging Spring

• A spring of rest length $L$, mass $M$ and spring constant $k$ hangs from two points separated by a horizontal distance $D$.
  1. Write down the differential equation describing the shape of the hanging spring.
  2. In the limit $k=\infty$ show that the curve is of the form $y = y_0 + a\cosh\frac{x-x_0}{a}$. Write down an equation that yields $a$.
  3. For $k=\infty$, find the length $L$ that minimizes the tension in the middle of the spring.
  4. For $L=0$, show that the curve is a parabola and find the tension at the joints and at the hanging bottom of the spring.