Question 9.Q.2: When do the sliders collide? Find the time that elapses befo...

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 \theta}{d t}=\pm\left(\frac{2 g}{a}\right)^{1 / 2}(\sin \theta)^{1 / 2},

and, since θ is an increasing function of t, we take the positive sign. This equation is a first order separable ODE.
Since the sliders collide when θ = π/2, the time τ that elapses is given by

\tau=\left(\frac{a}{2 g}\right)^{1 / 2} \int_{0}^{\pi / 2} \frac{d \theta}{(\sin \theta)^{1 / 2}} \approx 1.85\left(\frac{a}{g}\right)^{1 / 2}.