(Solution) - The plunger of a solenoid is connected to a spring -(2025 Original AI-Free Solution)

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The plunger of a solenoid is connected to a spring. The spring force is given by f = K0 (0.9a ? x), where x is the air-gap length. The inductance of the solenoid is of the form L = L0 (1 ? x/a), and its winding resistance is R. The plunger is initially stationary at position x = 0.9a when a dc voltage of magnitude V0 is applied to the solenoid.
a. Find an expression for the force as a function of time required to hold the plunger at position a/2.
b. If the plunger is then released and allowed to come to equilibrium, find the equilibrium position X0. You may assume that this position falls in the range 0 < X0 < a.