Question

A requirement of question- To evaluates reasonableness of a solution by correctly applying a valid sense-making evaluation technique (such as limiting/special case analysis, unit analysis, physical reasonableness argument)

Consider a horizontal linear spring with a total mass M and equilibrium length L; the mass is spread evenly along the length of the spring. The spring is unstretched with its left end fixed while the right end moves with speed vų. Determine the effective mass of the spring, meff by determining the total kinetic energy of the spring, K = = meffuſ. [Hint: The speed of an infinitesimal piece of the spring, which has mass dm, is directly proportional to its distance from the fixed end of the spring.]

Answer 1

da L Ans.) mass of the spring = M =0 equilibrium length = soos mass density speed of the elementary piece (dx) of the spring = = V(9) C2 CA proportionality constant. Now V CEL) - V => CL = c = V/L hence V(2) I mass of the elementary piece = dm = oda Kinetic energy of the elementary piece = dk dk = dm v*(x) - v Via = 1/2 oda v²ex)

t integrating over the length of the spring we get the total kinetic energy of the spring f 8 vya di 늦 sinfx dx Ke th V2 2 w vi - L M र। 러 3 = L (M) vi hence Meff = M/3