CALC A Spring with Mass. ?The preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely massless. To find the effect of the spring’s mass, consider a spring with mass ?M?, equilibrium length Lo, and spring constant ?k?. When stretched or compressed to a length ?L?, the potential energy is , what is ?M?¿ in terms of ?M??
Solution 103CP Step 1: Consider a spring having mass M and a maximum stretchable length L. Consider L and l 0 are the length of the spring at some particular position while stretching . Step 2: a) dm is the small portion of the spring M is the mass and L is the total mass then M dm = ( L ) dl Velocity at portion l , v(l ) = (L) v) Kinetic energy over the small portion is L dKE = 1 dm v 2 0 2 Step 3: Putting the values of dm and v on the above equation L dKE = 1 ( ( M ) dl .( ( ) v) ) 0 2 L L L = 1 m v l dl 2 L3 0 = 1 m v [3 ]L 2 L3 3 0 = M6l3 L3 The kinetic energy of the spring in terms of M and v = Mv2 6