Question

A plane pendulum of length L and mass m is suspended from a block of mass M. The block moves without friction and is constrained to move horizontally only (i.e. along the x axis). You may assume all motion is confined to the xy plane. At t = 0, both masses are at rest, the block is at   $x_{M}=x_{0}$ , and the pendulum has angular deflection   $\theta = \theta_{0}$ with respect to the y axis.

a) Using $x_{M}$ and $\theta$ as generalized coordinates, find the Lagrangian of the system.

b) Obtain the Lagrange equations of motion of the system.

c) Assuming $\theta_{0} \ll 1$ , find $x_{M}$ and $\theta$ as functions of time.

Answer 1

let us suppose a pendulum of length L and mass M is suspended from a block or mass M. The Block moves without friction and is constrained to move horizontally ie, it is confined to xy plane. At, t = 0 both masses are at rest, the block is at x_m = x_0, and the pendulum has angular deflection theta = theta_0 with respect to y axis. Lagrangian of the system: (x_m_1 y_m_1) (x_m_2 y_m_2) (x_m_1 y_m) be coordinates of Top bob, bottom bob and cart respectively x_m = x: y_m = 0. \r\n x_m_1 = (x + b sin theta): y_m_1 = (h-b cos theta_1) x_m_2 = (x + b sin theta_1 + b sin theta_2): y_m_2 = (h-b cos theta_1-b cos theta_2) Total kinetic Energy, T = 1/2{M(x'_m^2 + y_m^2) + [x'^2_m_1 + y^2_m_1 + x^2_m_2 + y^2_m_2]m_1} x_, = x: y_m = 0 x_m_1 = x + b cos theta_1 theta_1: y_m_1 = b sin theta_1 theta_1 x_m_2 = x + b cos