a. Assuming the motion is such that gravity is not involved (horizontal motion) the basic equation isA point particle of mass m is attached to a spring of spring constant k which is pulled with constant acceleration A. The spring lies along the direction of the acceleration A and the system is immersed in a fluid, so that when the mass moves with velocity v, it is subject to a viscosity force fv = βv, which opposes its motion.

a. Write down, without solving it, the equation of motion for the mass in a reference frame moving with the system.

∑F = m*d²x/dt²

The forces are the spring force in the positive direction = k*x and the viscosity force in the negative direction = -β*v.

k*x - β*v = m*d²x/dt²

Since v = dx/dt the equation becomes

m*d²x/dt² + β*dx/dt - k*x = 0

or

d²x/dt² + β/m*dx/dt - k/m*x = 0

EDIT: The reference is the anchor point of the spring. The mass is not stationary with respect to that.