Final answer:
To find the acceleration of the 4m mass in the double Atwood machine, set up the Lagrangian using two suitable generalized coordinates. Solve the equations of motion to find the values of the tensions and accelerations. The top pulley rotates because the tensions in the two strings connected to it are different.
Step-by-step explanation:
To set up the Lagrangian for the double Atwood machine, we need to define two suitable generalized coordinates. Let's use the vertical positions of the two masses as our generalized coordinates. Let y_1 be the position of the 3m mass and y_2 be the position of the 4m mass.
The Lagrangian is given by L = T - V, where T is the kinetic energy and V is the potential energy. The kinetic energy is T = (1/2)m(v_1^2 + v_2^2), where v_1 and v_2 are the velocities of the masses. The potential energy is V = mg(y_1 + y_2).
Using the Lagrange equations, we can find the equations of motion for the generalized coordinates. The equation for y_1 is m(ay_1'' - g) = 3mg - T_1, where T_1 is the tension in the string connected to the 3m mass. The equation for y_2 is 4m(ay_2'' - g) = T_1 - T_2, where T_2 is the tension in the string connected to the 4m mass.
To find the acceleration of the 4m mass when the system is released, we need to solve the equations of motion. Solving these equations will give us the values of T_1, T_2, and the accelerations y_1'' and y_2''. Finally, we can find the acceleration of the 4m mass by substituting the values into the equation for y_2''.
The top pulley rotates even though it carries equal weights on each side because the tensions in the two strings connected to the pulley are different. This creates a net torque on the pulley, causing it to rotate.