Question

Two blocks and three springs are configured as in Fig. All motion is horizontal. When the blocks are at rest, all springs are unstretched. k2 k3 m1 m2 (a) Choose as generalized coordinates the displacement of each block from its equilibrium position, and write the Lagrangian. (b) Find the T and V matrices (c) Suppose m1 2m m2-mk4k Find the frequencies of small oscillations. (d) Find the eigenvectors and be sure to normalize them. See the hint below from the lecture. Eigenvector satisfying Va = λTa a We fix the normalization by declaring

Answer 1

(a) K. e of system T = 1/2 m_1 x_1^2 + 1/2 m_2 x_2^2 Position energy of system V = 1/2 R_1 x_1^2 + 1/2 k_2 (x_1 - x_2)^2 + 1/2 R_3 x_2^2 (b) the T & V matrices L = T - V = 1/2 m_1 x_1^2 + 1/2 m_2 x_2^2 - 1/2 R_1 x_1^2 + 1/2 R_2 (x_1 - x_2)^2 + 1/2 R_3 x_2^2 L = 1/2(m_1 x_1^2 + m_2 x_2^2) - 1/2(R_1 x_1^2 + R_2(x_1 - x_2)^2 + R_3 middot x_2^2) \r\n (c) m_1 = 2m m_2 = m k_1 = 4k k_2 = k k_3 = 2k from eqn (b) L = m_1 middot x_1^2 + m_2 middot x_2^2 - 1/2 (4k_1 x_1^2 + k(x_1 - x_2)^2 + 2kx_2^2) L = 2m middot x_1^2 + m_2 middot x_2^2 - 1/2(4k middot x_1^2 + k(x_1 - x_2)^2 + 2kx_2^2) equation of motion will be d/dt (dL/dx_1) - dL/dx_1 = 0 d/dt(2m x_1) + 4kx_1 + k(x_1 - x_2) = 0 2mx_1 + 5k x_1 - k(x_1 - x_2) = 0