| dv | = | dv | ds | = v | dv | , v | dψ | = v | dψ | ds | = | v2 | , | ||
| dt | ds | dt | ds | dt | ds | dt | ρ |
(12)
where ρ is the radius of curvature of the path at P, the tangential and normal accelerations are also expressed by v dv/ds and v2/ρ, respectively. Take, for example, the case of a particle moving on a smooth curve in a vertical plane, under the action of gravity and the pressure R of the curve. If the axes of x and y be drawn horizontal and vertical (upwards), and if ψ be the inclination of the tangent to the horizontal, we have
| mv | dv | = − mg sin ψ = − mg | dy | , | mv2 | = − mg cos ψ + R. |
| ds | ds | ρ |
(13)
The former equation gives
v2 = C − 2gy,
(14)
and the latter then determines R.
In the case of the pendulum the tension of the string takes the place of the pressure of the curve. If l be the length of the string, ψ its inclination to the downward vertical, we have δs = lδψ, so that v = ldψ/dt. The tangential resolution then gives