Five particles undergo underdamped harmonic motion with the spring constant k, damping constant b and mass m given below. For which set is the rate of loss of mechanical energy at the initial instant the smallest? (Assume each is started with the same initial conditions.) (A) k = 100 N/m,

Question

Five particles undergo underdamped harmonic motion with the spring constant k, damping constant b and mass m given below. For which set is the rate of loss of mechanical energy at the initial instant the smallest? (Assume each is started with the same initial conditions.) (A) k = 100 N/m, m = 50 g, b = 8 g/s (B) k = 150 N/m, m = 50 g, b = 5 g/s (C) k = 150 N/m, m = 10 g, b = 8 g/s (D) k = 200 N/m, m = 8 g, b = 6 g/s

Accepted Answer

k = 150 N/m, m = 50 g, b = 5 g/s. For a lightly damped oscillator the mechanical energy decays as E(t) = E0 exp(-(b/m) t), so the fractional rate of energy loss is set by the damping ratio b/m (smaller b/m -> slower loss). Computing b/m (in s⁻¹ using consistent units, m in g): (A) 8/50 = 0.16; (B) 5/50 = 0.10; (C) 8/10 = 0.80; (D) 6/8 = 0.75. The smallest b/m is option (B), so it has the smallest rate of energy loss at the initial moment.

Solution

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