## Physics (10th Edition)

Because there is no work due to nonconservative forces, the total mechanical energy $E_0$ at the top of the plane is the same as the total mechanical energy $E_f$ at the bottom of the plane. Therefore, $$KE_0+PE_0=KE_f+PE_f$$ At the top of the plane, $\omega_0=0$ and $v_0=0$ so $KE_0=0$. At the bottom of the plane, $h_f=0$, so $PE_f=0$. Therefore, $$KE_f=PE_0$$ As we can see, the total kinetic energy in the end depends only the initial potential energy. Because both hoops start off at the same height, they have the same $PE_0$, so their $KE_f$ will be the same, too.