Answer
$\theta=51.2^{\circ}$ stale equilibrium, $\theta=4.71^{\circ}$ unstable equilibrium
Work Step by Step
The required angle $\theta$ can be determined as follows:
According to the potential energy equation
$V=V_g+V_e=Wy+\frac{1}{2}ks^2$
$\implies V=2(10)(9.81)sin\theta+\frac{1}{2}(1.5)(1000cos\theta-600)^2$
At equilibrium, $\frac{dV}{d\theta}=0$
$\implies \frac{d[2(10)(9.81)sin\theta+\frac{1}{2}(1.5)(1000cos\theta-600)^2]}{d\theta}=0$
$\implies \frac{dV}{d\theta}=49050cos\theta-1500sin\theta(1000cos\theta-600)=0$
This simplifies to:
$\theta_1=51.2^{\circ}$ and $\theta_2=4.71^{\circ}$
Now, $\frac{d^2V}{d\theta}=-49050sin\theta-1500[cos51.2 \cdot (100(51.2)-600-1000 sin51.2^{\circ})^2]=848778.726\gt 0$
which implies stable equilibrium.
For $\theta_2=4.71^{\circ}$
$\implies \frac{d^2V}{d\theta^2}=-49050sin(4.71)-1500[cos(4.71)(1000cos (4.71)-600)-1000 sin 4.71)^2]=-586820.2\lt 0$
which implies unstable equilibrium.
