Question

A spring is attached to the ceiling and pulled 8 cm down from equilibrium and released. The damping factor for the spring is determined to be 0.4, and the spring oscillates 12 times each second. Find an equation for the displacement, D(t), of the spring from equilibrium in terms of seconds, t.

a) D(t)=ae^ct sin(wt)

b) D(t)=ae^ct cos(wt)

c) D(t)=ae^ct sin(wt) or D(t)=ae^ct cos(wt)

d) D(t)=ae^−ct sin(wt)

Answer 1

The equation for the displacement, D(t), of the spring from equilibrium in terms of time, t, can be given as D(t) = ae^(-ct) sin(wt). This equation represents a damped simple harmonic motion.

Step-by-step explanation:

The equation for the displacement, D(t), of the spring from equilibrium in terms of time, t, can be given as D(t) = ae-ct sin(wt). In this equation, a is the amplitude of the oscillation, e is the base of the natural logarithm, c is the damping factor, and w is the angular frequency.

The given equation, D(t) = ae-ct sin(wt), represents a damped simple harmonic motion. The exponential term, e-ct, represents the damping effect, causing the amplitude of the oscillation to decrease over time.