Question 9.Q.3: Displacement at time t Find the displacement of the rope at ...
Displacement at time t
Find the displacement of the rope at time t.
Since this system has only one degree of freedom, the motion can be found from energy conservation alone. From the energy conservation equation, it follows that
\frac{d x}{d t}=\pm n x^{1 / 2}(x+2 b)^{1 / 2},
where n² = g/a. Since x is an increasing function of t, we take the positive sign.
This equation is a first order separable ODE.
It follows that
\begin{aligned}n t &=\int \frac{d x}{x^{1 / 2}(x+2 b)^{1 / 2}} \\&=2 \sinh ^{-1}\left(\frac{x}{2 b}\right)^{1 / 2}+C,\end{aligned}
on using the substitution x = 2b sinh² w. The initial condition x = 0 when t = 0 implies that C = 0 and, after some simplification, we obtain
x=b(\cosh n t-1)
as the displacement of the rope after time t. As before, this formula holds while there is still some rope left on the the table top.