Answer:
Hello your question is incomplete attached below is the complete question
Answer : x ( acceleration of mass 4m ) = [tex]\frac{g}{7}[/tex]
The top pulley rotates because it has to keep the center of mass of the system at equilibrium
Explanation:
Given data:
mass suspended = 4 meters
mass suspended at other end = 3 meters
first we have to express the kinetic and potential energy equations
The general kinetic energy of the system can be written as
T = [tex]\frac{4m}{2} x^2 + \frac{3m}{2} (-x+y)^2 + \frac{m}{2} (-x-y)^2[/tex]
T = [tex]4mx^2 + 2my^2 -2mxy[/tex]
also the general potential energy can be expressed as
U = [tex]-4mgx-3mg(-x+y)-mg(-x-y)+constant=-2mgy +constant[/tex]
The Lagrangian of the problem can now be setup as
[tex]L =4mx^2 +2my^2 -2mxy +2mgy + constant[/tex]
next we will take the Euler-Lagrange equation for the generalized equations :
Euler-Lagrange equation = [tex]4x-y =0\\-2y+x +g = 0[/tex]
solving the equations simultaneously
x ( acceleration of mass 4m ) = [tex]\frac{g}{7}[/tex]
The top pulley rotates because it has to keep the center of mass of the system at equilibrium
