A trolley of mass MMM is held in equilibrium on a horizontal low-friction track by two identical springs, each of force constant KKK, attached to fixed supports at either end.
The trolley is displaced a distance x0x_0x0 from its equilibrium position and released so that it undergoes simple harmonic motion. The period of oscillation TTT is related to the mass MMM and the force constant KKK by the equation:
T2=2π2MK T^2 = \frac{2\pi^2 M}{K} T2=K2π2MShow that this equation is homogeneous by reducing both sides to SI base units.
Explain why the magnitude of the resultant force FFF on the trolley when displaced by a distance xxx from equilibrium is given by F=2KxF = 2KxF=2Kx.
A student investigates this relationship by varying the mass MMM on the trolley and measuring the time for 10 complete oscillations to determine the period TTT. The student plots a graph of T2T^2T2 (on the vertical axis) against MMM (on the horizontal axis) and finds that the line of best fit has a gradient of 1.55 s2 kg−11.55\text{ s}^2\text{ kg}^{-1}1.55 s2 kg−1. Determine the force constant KKK of each spring. Include the unit.
