Question

Problem #1 It is known that the displacement x(t) of a cart from the position of equilibrium is described by the mas spring model. The value of x(t) is the solution of the initial value problem as follows Calculate the mass m of the cart for which the maximum of the VELOCITY of the cart is equal to s. Round the value of the mass m you just found to three figures and provide your result below (16 points) (your numerical result must be written here) abo ve: Also, you must provide some intermediate results obtained by you while solving the problem

Answer 1

Given differential equation is m x + 900 x = 0 its equivalent auxiliary equations is m k^2 + 900 = 0 implies k^2 = -900/m k = plusminus i 30/squareroot m Roots of the Auxiliary equation are imaginary hence the general solution of x (t) is x (t) = A cos (30/squareroot m t) + D sin (30/squareroot m t) (for some constant A, B) given x (0) = 1 implies 1 = A cos (0) + B sin (0) 1 = A x^1 (t) = -30/squareroot m sin (30/squareroot m t) + B(30/squareroot m) cos(30/squareroot m t) x^1 (0) = 4 implies 4 = -30/squareroot m (0) + B 30/squareroot m B= 2 squareroot m/15 Hence displacement x (t) = cos (30/squareroot m t) + 2 squareroot m/15 sin (30/squareroot m t) velocity x^1 (t) = -30/squareroot m sin (30/ squareroot m t) + 4 cos (30/ squareroot m t)