Final answer:
The equation of motion for a mass attached to a spring and a dash-pot undergoing damped harmonic motion is given by m(d2x/dt2) + c(dx/dt) + kx = 0, with initial conditions x(0) = x0 and v(0) = v0.
Step-by-step explanation:
The equation of motion for a mass attached to both a spring with spring constant k and a dash-pot with damping constant c is derived from Newton's second law of motion, which states that the sum of the forces acting on an object equals the mass of the object multiplied by its acceleration (F = ma). When the mass is displaced from its equilibrium position and released, it undergoes damped harmonic motion. The forces acting on the mass are the restoring force of the spring (Fspring = -kx) and the damping force of the dash-pot (Fdamping = -cv), where k is the spring constant, c is the damping constant, x is the displacement, and v is the velocity of the mass. The resulting equation of motion is m(d2x/dt2) + c(dx/dt) + kx = 0, where m is the mass of the object. The initial conditions are given as x(0) = x0 and v(0) = v0.