This must equal the work acquired—that is to say, the kinetic energy of rotation of the body, or
wa (1 − cos ϕ) = ww2k2 /2 g.
Where w is the angular velocity of the body at starting, k the radius of gyration round a, and g the velocity acquired by a falling body in one second. Whence
w2 k2 = 2 ga (1 − cos ϕ),
but w, the angular velocity, is equal to the statical couple applied, divided by the moment of inertia, or,
w = va cos ϕ /k2,
squaring and substituting
v2 = 2g × k2/a × 1 − cos ϕ/cos2 ϕ,
and since the length of the corresponding pendulum is l = k2/a,
v2 = 2gl × 1 − cos ϕ /cos2 ϕ,