Université des Sciences et Technologies de Lille

Niveau: Supérieur, Master, Bac+5
Université des Sciences et Technologies de Lille 1 2011/2012 Master degree in Mathematical Engineering Refresher Course in Physics Semester 3 Classical Mechanics Exercise 1 A body (assimilated to a point M) of mass m is submitted to the action of a central force of the form ~F = f(r)~ur, We use (r, ?) for the polar coordinates of a point M in (xOy) and (~ur, ~u?) for the associated orthonormal basis (see the course for precise definition). (1) Show that ~? = ~OM ? (m~v) and r2?˙ do not depend on t. (2) Denote C = r2?˙. Give an expression of f(r) in terms of m, C, u = 1/r and d 2u d?2 . (3) Suppose M desribes a spiral of the form r? = K. Give an expression for f(r). (4) Suppose M desribes a spiral of the form r = e?a?, where a > 0. Give an expression for f(r). Exercise 2 A body (assimilated to a point M) of mass m has initially velocity v0 and is falling on the vertical direction (on the surface of the earth). We make the assumption that the friction due to the air acts on M as ~F = ?bm~v, where b is a positive constant.

• friction


• d? √


• ?02 sin


• limit velocity


• sin2n ?02


• sin ?2


• sin ?


• dm dt


• positive constant