Final answer:
The constant k in the differential equation d2q/dt2 = -kq for an LC circuit can be derived using the analogy between mechanical and electrical systems. The constant k is found to be 1/LC, which relates the inductance L and capacitance C of the circuit.
Step-by-step explanation:
The differential equation d2q/dt2 = -kq suggests that the charge q oscillates similarly to a mass-spring system where the force is proportional to displacement.
To find the constant k in terms of L (inductance) and C (capacitance), we look to the analogy between mechanical and electrical oscillating systems.
The mechanical spring constant k is analogous to 1/C, and the mass m in the mechanical system to the inductance L in the electrical system.
From the angular frequency ω of the LC circuit oscillation, which is given by ω = 1/√(LC), we can derive the constant k. We know that for simple harmonic motion, ω = √(k/m).
By relating these two expressions and replacing m with L (from the mechanical-electrical analogy), we find that k = 1/LC.