When you rapidly remove the slack, you start with a high velocity and it moves to zero in a short distance. You don't have enough information to figure out the stress on the rope unless you know the distance of deflection (the immediate moment when slack was removed to rope at full stretch under that acceleration) or the time it takes to stretch that deflection. It's not a simple problem. If you can find the modulus of elasticity for the rope at temperature and for that crossectional area, you have a start. Unfortunately, this is also a dynamic problem so some integration is needed... fortunately, that integration has been done before. From that modulus of elasticity, you can figure out the spring-rate of the rope (and somewhat ignore the damping rate) which will be really high. k (spring rate) = A (cross sectional area) * E (modulus of elacity) / L (effective length of rope - distance between the two secure ends).
You'll know initial velocity (when the slack is taken up but no stress on the rope) and final felocity (zero). That'll convert to potential at full extention, so KE (kinetic energy) = 1/2 m(mass)*v(velocity)^2. PE (potential energy) = (1/2)k(spring rate)*x(displacement)^2.
Assuming 100% conservation of kinetic energy to potential energy, KE=PE, you can figure out displacement as long as you know the spring rate: 1/2(m)*v^2=1/2(k)*x^2=1/2(A*E/L)*x^2
You know: v, k(possibly, from AE/L) and m. Use algebra to solve for x (I don't feel like doint it all) you can multiply x*k to get the force on the rope. If you want stress, divide by cross sectional area.
. figure out the acceleration applied to the mass. from there you can get force. for stress on the rope, divide the force by the cross-sectional area.
