Question

3. A frictionless and massless pulley with radius Ri and string of li is suspended as shown in Fig. 1. One end connects with a mass mi and the other with another frictionless pulley of mass m with radius R2 and string /2, joining two masses ms and ms at the two ends. (a) Find the equations of holonomic constraints. (b) Find the equations of motion for masses mi, m2, m3, and m4. (c) Find the solutions for the undetermined Lagrange multipliers λ| and i2. (d) Find the components of the generalized forces due to the constraints. (25 points) X2 ng X4 Fig. 1

Answer 1

"Consider the figure below, Let the distance of mass m_1, m_2, m_3 and m_4 from reference level be x_1, x_2, x_3 and x_4 correspondingly Length of strings is l_1 and l_2 respectively a) Holonomic constraints: - As the total length of both the string will be constant So, x_1 + x_2 = l_1 -----(1) forst equations (x_3 - x_2) + (x_4 - x_2) = l_2 ---(2) second equation differentiate eq""(1) and (2), you will get holonomic constraint equation: - x_1^.. + x_2^.. = 0 and x^.._3 + x^.._4 - 2x^.._2 = 0 rightarrow second derivation or dx_1 + dx_2 = 0 amd dx_3 +dx_4 - 2dx_2 = 0 rightarrow derivative multiply both equations with language multiplier lambda_1 and lambda_2 and add then, you will get lambda_1 dx_1 + (lamnda_1 -- 2 lambda_2) dx_2 + lambda_2 dx_3 + lambda_2 dx_4 = 0 \r\n (b) Equations of motion of masses: - Lagrangian L = T - V T = kinetic energy V = potential energy T = 1/2 m_1x^.2_1 + 1/2 m_2 x^.2_2 + 1/2 m_3 x_3^.2 + 1/2 m_4 x^.2_4 V = -g(m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4) L = 1/2 m_1 x^.2_1 + 1/2 m_2 x^.2_2 + 1/2"