Q) An insect crawls up a hemispherical surface very slowly.The coeffiecient of friction is $\mu$ between surface and insect.If line joining the centre of hemispherical surface to the insect makes an angle $\alpha $ with the vertical, find the maximum possible value of $\alpha$.
With the force method, the solution can be found as at the highest point the frictional force would be equal to gravitational force.Therefore, $$\mu mg\cos\alpha=mg \sin\alpha$$ $$\implies \cot \alpha=1/\mu$$
However, when I tried to do this by energy conservation,equating the total frictional force with potential energy the answer was different.
Let $\theta$ be angle covered by it and $d\theta$ be a small angle covered by it.
$$mgr(1-\cos\alpha)=\int_0^\alpha \mu (mg\cos\theta )*rd\theta$$
$$mgr(1-\cos\alpha)=\mu mgr \sin\alpha$$
$$2\sin^2\frac{\alpha}{2}=\mu 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}$$
$$\cot\frac{\alpha}{2}=1/\mu$$
Why is the answer different if I used force or if i use energy conservation?