Motion of a Particle

Extract

Suppose that a particle of mass m is attached to a fixed point O, by means of a light inextensible string of length a, and when hanging at rest is suddenly-given a velocity u₀ at right angles to the string. The earlier phases of the motion are dealt with in standard textbooks, but I do not recall having seen a complete analysis of the later stages in the case when the particle leaves its circular path. It is well known that if u₀² ≤2ag, the particle executes oscillations like a simple pendulum (this will be referred to later as the case of simple oscillations), while if u₀²≥5ag, it makes complete revolutions in a circle about O. If 2ag<₀²< 5ag, the particle leaves its circular path and then travels in a parabolic path until the string becomes taut again. The textbooks do not follow the motion beyond this point, perhaps because it is realised that no real string fulfils the condition of being inextensible. If, however, we assume that the ideal string exists, the resulting motion has some rather surprising features. Owing to the obvious loss of energy, the particle when it first passes through the lowest point again will have a smaller velocity that it had initially, and so will either execute simple oscillations or will leave its circular path at a lower point than previously and lose further energy when the string tightens a second time. It would at first appear that ultimately the energy must decrease to such a value that the particle will finally be moving with simple oscillations. This is not always true, as the following analysis will show.