We may apply the equations (9) to the case of a solid of revolution rolling with its axis horizontal on a plane of inclination α. If the axis of x be taken parallel to the slope of the plane, with x increasing downwards, we have
Mẍ = Mg sin α − F, 0 = Mg cos α − R, Mκ2θ̈ = Fa,
(10)
where κ is the radius of gyration about the axis of symmetry, a is the constant distance of G from the plane, and R, F are the normal and tangential components of the reaction of the plane, as shown in fig. 74. We have also the kinematical relation ẋ = aθ̇. Hence
| ẍ = | a2 | g sin α, R = Mg cos α, F = | κ2 | Mg sin α. |
| κ2 + a2 | κ2 + a2 |
(11)
The acceleration of G is therefore less than in the case of frictionless sliding in the ratio a2/(κ2 + a2). For a homogeneous sphere this ratio is 5⁄7, for a uniform circular cylinder or disk 2⁄3, for a circular hoop or a thin cylindrical shell 1⁄2.
The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions. It may also be deduced from the principles of linear and angular momentum as embodied in the equations (9). We have
M (ẋẍ + ẏÿ) + lθ̇θ̈ + Xẋ + Yẏ + Nθ̇,
(12)