Final answer:
To find the displacement of the mass at any time t in a mass-spring system, we use the displacement function u(t) = A * cos(ωt + φ). The position of the mass at t = 12π/8π can be found by substituting the time into the displacement function and solving. The velocity at t = 163π/8π can be determined by taking the derivative of the displacement function with respect to time and evaluating it at that time. The mass passes through the equilibrium position when its displacement is equal to zero.
Step-by-step explanation:
First, we determine the displacement function u(t) for the mass-spring system. The displacement function for Simple Harmonic Motion (SHM) is given by u(t) = A * cos(ωt + φ), where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle.
1. To find the position of the mass at t = 12π/8π, substitute the given time into the displacement function u(t) = A * cos(ωt + φ). Here, A = 6 inches and ω = 2π/T, where T is the period of the system. Since T = 2π/ω, we can find ω = 2π/[(6 inches * 2) / 20 pounds] = 10π radians/s. φ is the phase angle, which can be determined from the initial condition u(0) = -6 inches.
2. The velocity of the mass is the derivative of the displacement with respect to time, which is given by v(t) = -A * ω * sin(ωt + φ). Plug in t = 163π/8π, ω = 10π, and φ obtained from the initial condition to find the velocity at that time. The direction of the mass's motion can be determined by considering the sign of the velocity.
3. The mass passes through the equilibrium position when its displacement u(t) is equal to zero. Set the displacement function u(t) = 0 and solve for t to find the time when the mass passes through the equilibrium position.