Answer
$4\ sec.$
Work Step by Step
As the telescope rotates at the same rate as Earth, it will turn $360^\circ$ in $24$ hours. Assume it takes $t$ seconds to turn an angle of $1'$, we have $\frac{t\ sec}{1'}=\frac{24\times60\times60\ sec}{360\times60'}$, thus we have $t=4\ sec.$
