A particle $P$ is initially at rest on the top pf a smooth hemispherical surface which is fixed on a horizontal plane. The particle is given a velocity $u$ horizontally. Radius of spherical surface is $a$.

If the particle leaves the sphere, when it has fallen vertically by a distance of $\frac{a}{4}, u=\frac{\sqrt{ga}}{2}$

If $u=0$ and the particle just slides down the hemispherical surface, it will leave the surface when $\cos \theta =\frac{2}{3}$

If the particle leaves the sphere at angle $\theta$ (fig) where $\cos \theta =\frac{\sqrt{3}}{2}$, then $u=\frac{\sqrt{ag}}{3}$

The minimum value of $u$, for the object to leave the sphere without sliding over the surface is $\sqrt{ag}$