Question

Three spheres of equal mass m are constrained to move in one dimension along the line connecting their centers. The three spheres are connected by three springs, as shown in the figure. The three springs have equal spring constants k In equilibrium, all three of the springs are at their respective natural lengths a. Choose a reasonable set of coordinates and find the equations of motion. b. Find the normal-mode frequencies.

Answer 1

Using Newton's and Hook's law for first mass, m d^2 x_1/dt^2 = -k (x_1 - x_2) (1) T^nd mass, m d^2 d_3/dt^2 = -k (x_2 - x_3) - k (x_2 - x_1((2) 3^rd mass, n d^2 x_3/dt^2 = -k (x_3 - x_2) (3) are the equation of motion. from (1), d^2 x_1/dt^2 = -k/m (x_1 - x_2) (2), d^2 x_2/dt^2 = -2k/m x_2 + k/m (x_1 + x_3) (3), d^2 x_3/dt^2 = -k/m (x_3 - x_2) Let constant a matrix V' = (x_1 x_2 x_3)v (x_1 x_2 x_3) \r\n where V = k/m (1 -1 0 -1 2 -1 0 -1 1) diagonaling V, Vertical-bar v - lambda I Vertical-bar = 0 and assume lambda is required is its f k/m Vertical-bar 1 - lambda -1 0 -1 2 - lambda -1 0 -1 1 - lambda Vertical-bar = 0 (1 - lambda) ((2 - lambda) (1 - lambda) - 1) + 1 (- (1 - lambda)) = 0 (1 - lambda) ((2 - lambda) (1 - lambda) -1 -1) = 0 (1 - lambda) (lambda^2 - 3 lambda) = 0