Answer
$$W=0.0010 \mathrm{\ J}$$
Work Step by Step
$$
d U=(m / L) g|y| d y=-(m / L) g y d y
$$
since $y$ is negative valued (we have $+y$ upward and the source is at the table-top). The total
potential energy change is
$$
\Delta U=-\frac{m g}{L} L_{L / 4}^{0} y d y=\frac{1}{2} \frac{m g}{L}(L / 4)^{2}=m g L / 32
$$
The work required to pull the chain onto the table is therefore
$$W=\Delta U=m g L / 32=(0.012 \mathrm{kg})\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)(0.28 \mathrm{m}) / 32$$$$=0.0010 \mathrm{\ J}$$
