= Sphere rolling in a symmetric groove
{title2=$a_{\rm cm}=g\sin\theta/[1+2/(5\sin^2(\phi/2))]$}
A <sphere> of <radius> $a$ touching both faces of a symmetric groove of opening angle $\phi$ rotates instantaneously about the line joining its contacts. The centre is a distance $h=a\sin(\phi/2)$ from this line, so <rolling without slipping> means $v=h\omega$. For a homogeneous <sphere>, $K=mv^2/2+I\omega^2/2=mv^2[1+2/(5\sin^2(\phi/2))]/2$. 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.
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