A massless platform is kept on a light elastic spring as shown in the figure. When a particle of mass 0.1 kg is dropped on the pan from a height of 0.24m, the particle strikes the pan and the spring is compressed by 0.01m. From what height should the particle be dropped to cause a compression of0.04m?

Let the particle be dropped from a height h and the spring be compressed by y. According to the conservation of mechanical energy, loss in PE of the particle = gain in elastic potential energy of the spring          mg (h+y) = 12  ky2, here m is mass of particle, g is acceleration due to gravity, h is height, y is extent of compression, k is force constantNow, as the particle and spring are same in the  second case also,   mg(h1+y1)mg(h2+y2)=12ky1212ky22 or   (0.24+0.01h2+0.04)=(0.010.04)2Solving we get h2 = 3.96 m