A system consists of three ideal springs arranged as follows: Springs E and F are attached in parallel between a rigid vertical wall and a junction point J. Spring G is attached between point J and a 0.50 kg cart on a horizontal frictionless track. The force constants are kE=120N/m, kF=80N/m, and kG=200N/m.
(a) Determine the single spring constant that could replace the parallel combination of Springs E and F.
(b) Determine the overall single spring constant that could replace the entire three-spring network between the wall and the cart.
(c) Using your result from part (b), derive an algebraic expression for the angular frequency ω of small oscillations of the cart in terms of kE, kF, kG, and the cart mass m. Show all necessary steps.
(d) Calculate the period of small oscillations of the cart using the given numerical values.