(2)
if T0 be the tension corresponding to ψ = 0. This illustrates the resistance to dragging of a rope coiled round a post; e.g. if we put μ = .3, ψ = 2π, we find for the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared, and so on.
Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane. Let ψ denote the inclination to the horizontal, and wδs the weight of an element δs. The tangential and normal components of wδs are −s sinψ and −wδs cosψ. Hence
δT = wδs sin ψ, Tδψ = wδs cos ψ + Rδs.
(3)
If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have δy = sin ψ δs, whence δT = wδy. If the string be uniform, w is constant, and
T = wy + const. = w (y − y0),
(4)
say; hence the tension varies as the height above some fixed level (y0). The pressure is then given by the formula
| R = T | dψ | −w cos ψ. |
| ds |