Problems deal with a nuiss−spring−dashpot system having position function x(t) satisfying...

Problems deal with a nuiss−spring−dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x′(0) and recall that p = c/(2m), ω02 = k/m, and ω12 = ω02 − p2. The system is criticallydamped, overdamped, or underdamped, as specified in each problem.

(Underdamped) Let x1 and x2 be two consecutive local maximum values of x(t). Deduce from the result of Problem 32 that

The constant is called the logarithmic decrement of the oscillation. Note also that because

Note: The result of Problem provides an accurate method for measuring the viscosity of a fluid, which is an important parameter in fluid dynamics but is not easy to measure directly. According to Stokes’s drag law, a spherical body of radius a moving at a (relatively slow) speed through a fluid of viscosity μ experiences a resistive force FR = 6π μav. Thus if a spherical mass on a spring is immersed in the fluid and set in motion, this drag resistance damps its oscillations with damping constant c = 6π aμ. The frequency ω1 and logarithmic decrement Δ of the oscillations can be measured by direct observation. The final formula in Problem then gives c and hence the viscosity of the fluid.