A frictionless horizontal spring--mass oscillator consists of a block of mass m0 attached to a single ideal spring of constant k0. When the block is displaced from equilibrium and released, it moves with period T0. Consider several modifications made one at a time to the original system.
(1) The block is replaced by another block of mass 4m0 while the spring constant remains k0. Determine the factor by which the period changes, writing your result as T1=c1T0 and stating the value of c1. Show your reasoning.
(2) Starting again from the original system, the spring is replaced by a stiffer spring whose constant is 2k0 while the mass remains m0. Determine the factor by which the period changes, expressing the result in the form T2=c2T0 and justifying each step.
(3) Compare the numerical values of c1 and c2. Which modification—quadrupling the mass or doubling the spring constant—produces the larger absolute change in period? Explain qualitatively why this is the case in terms of how period depends on each variable.