An object of mass 2 grams hanging at the bottom of a spring with a spring constant 3 grams per second square is moving in a liquid with damping constant 4 grams per second. Denote by y vertical coordinate, positive downwards, and y = 0 is the spring-mass resting position. (a) Write the differential equation y" = f(y, y') satisfied by this system. f(y, y')= -2yp-3/2y Note: Write t for t, write y for y(t), and yp for y' (t). (b) Find the mechanical energy E of this system. E(t)= yp^2+(3y^2)/2 Note: Write t for t, write y for y(t), and yp for y' (t). (c) Find the differential equation E' = F(y') satisfied by the mechanical energy. F(y')= Σ Note: Write t for t, write y for y(t), write yp for y' (t), write E for E(t). M Σ

An object of mass 2 grams hanging at the bottom of a spring with a spring constant 3 grams per second square is moving in a liquid with damping constant 4 grams per second. Denote by y vertical coordinate, positive downwards, and y = 0 is the spring-mass resting position. (a) Write the differential equation y" = f(y, y') satisfied by this system. f(y, y')= -2yp-3/2y Note: Write t for t, write y for y(t), and yp for y' (t). (b) Find the mechanical energy E of this system. E(t)= yp^2+(3y^2)/2 Note: Write t for t, write y for y(t), and yp for y' (t). (c) Find the differential equation E' = F(y') satisfied by the mechanical energy. F(y')= Σ Note: Write t for t, write y for y(t), write yp for y' (t), write E for E(t). M Σ