work power energy question 3

clearly v_com = 0

now assuming that velocity of a point at a distance x frm com is directly proportional to x ,

v = c x

c = some constant
let at the given instant spring length = L

so v at L/2 = v₀,

v_o = c L /2 , hence c = 2v_o /L

so v = 2v_ox /L

let mass of a differential element of length dx at a distance x be dm

also dm = MdxL

KE of dm
dK= dmv²2 = (Mdx2L) ( 2v_oxL )² = 2Mv_o²L³x²dx

so K = 2Mv_o²L³ (x³3)

and limits are x= -L/2 to L /2

so K = 2Mv_o²L³ (2L³24)

= Mv_o²6