A particle of mass m is placed on a fixed, smooth sphere of radius...

Let the normal reaction exerted by the sphere on the particle be $N$.

The forces acting on the particle are its weight $mg$ acting vertically downwards and the normal reaction $N$ acting radially outwards.

Resolving the weight along the radial direction, the component is $mg \cos 30^{{}^{{}^{\circ}}}$.

The equation of motion in the radial direction is given by $mg \cos 30^{{}^{{}^{\circ}}} - N = \dfrac{mv^2}{R}$.

Just after the release, the velocity of the particle is zero, so $v = 0$.

Substituting $v = 0$ into the equation, we get $mg \cos 30^{{}^{{}^{\circ}}} - N = 0$.

Therefore, $N = mg \cos 30^{{}^{{}^{\circ}}} = \dfrac{\sqrt{3}mg}{2}$.