 Sphere rolling in a symmetric groove

A sphere of radius $a$ touching both faces of a symmetric groove of opening angle $\varphi$ rotates instantaneously about the line joining its contacts. The centre is a distance $h=a\sin (\varphi /2)$ from this line, so rolling without slipping means $v=h\omega$. For a homogeneous sphere, $K=m{v}^2/2+I{\omega }^2/2=m{v}^2[1+2/(5{\sin }^2(\varphi /2))]/2$. Differentiating this kinetic energy with respect to distance down an incline gives the displayed acceleration. Both contact forces do no work for fixed faces and no slip.