Two clastic strings. ab and a'b', have the same natural length / and the same modulus of elasticity. the ends a and a' are attached to two fixed points in the same vertical line, with 4 a distance i above a'. the ends b and b' are both attached to a particle p of mass m. when phangs freely, both strings are extended. show that mig > .. the particle is pulled down a distance d below its equilibrium position and then released from rest. given that in the subsequent motion both strings remain taut, show that 2d2 < img - 2). given also that 2 = mg/3, find the potential energy stored in cach string when p is at its lowest position in the above motion.

Lake happy times contains 100 km3 of pure water. it is fed by a river at a rate of 50 km3/yr. at time zero, there is a factory on one shore of lake happy times that begins introducing a pollutant to the lake at a rate of 2km3/yr. there is another river that is fed by lake happy times at a rate that keeps the volume of lake happy times constant. this means that the rate of flow from lake happy times into the outlet river is 52km3/yr. in turn, the flow from this outlet river goes into another lake, lake sad times, at an equal rate. finally, there is an outlet river from lake sad times flowing at a rate that keeps the volume of lake sad times at a constant 100 km3 . (a) find the amount of pollutant in lake sad times at the end of 3 months. (b) at the end of 3 months, observers close the factory due to environmental concerns and no further pollu- tant enters lake happy times. how long will it take for the pollutant in lake sad times (found in part (a)) to be cut in half?