A uniform wire of length L is bent in the form of a circle.  The shift in its centre of mass is

A circle is formed by curving the uniform wire. The circle's circumference is equal to the wire's length. The value of a circle's boundary is equal to its circumference. In plain English, if we convert the circle into a straight wire, the wire's length will match the circle's circumference.
The following formula yields the circle's circumference.

$C=2\mathrm{\pi R}$

We can determine from the information provided that the wire's length is "L".
Therefore, the circle's circumference is equal to the wire's length.

$$\begin{gathered} C=L \\ \Rightarrow L=2\mathrm{\pi R} \\ \Rightarrow \mathrm{R}=\frac{\mathrm{L}}{2\mathrm{\pi }} \end{gathered}$$

Hence the correct answer is $\frac{\mathrm{L}}{2\mathrm{\pi }}.$