A light inextensible smooth string is in contact with a smooth – pulley (fixed at C) through an angle $\varphi$ as shown in figure. If T is the tension in the string, then force with which the string presses the pulley vertically is $\mathrm{xTsin}(\varphi /2)$ then x is______

Both the tensions acting at the end points will be making an angle of $\frac{\varphi }{2}$ with horizontal.

And the components acting along the horizontal will cancel out.

The components acting towards 'C" will add up to $2\mathrm{Tsin}\frac{\mathrm{\varphi }}{2}$.

So, the force with which the string presses the pulley vertically is $\text{F=\hspace{0.17em}2Tsin\hspace{0.17em}}\frac{\varphi }{2}$