An ideal spring–mass oscillator of period T0 is constructed with a block of mass m attached to a single spring of constant k. A second apparatus is then assembled in which a block of mass 3m is connected to two identical springs, each of constant 2k, arranged in series. Assume oscillations are along a horizontal line and that all surfaces are frictionless.
(1) Derive an algebraic expression for the effective spring constant keff of the two springs when they are connected in series. Express your answer in terms of k.
(2) Using the dependence of period on mass and effective spring constant, obtain an expression for the ratio T0Tnew of the period of the new apparatus to that of the original system. Leave your result in simplest radical form.
(3) Calculate the numerical value of the ratio you derived in part (b).
(4) Indicate whether the new apparatus oscillates faster or slower than the original and provide a physical explanation based on how changes in mass and effective stiffness compete to determine the period.