Relations in the tautological ring of $M_g$

Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of $M_g$ (and in fact that of $M_{g,2}$). One immediate consequence of these relations is that the classes ${\kappa }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\kappa }_{[g\mathrm{/}3]}$ generate the tautological ring of $M_g$, which has been conjectured by Faber, and recently proven at the level of {\em cohomology} by Morita.