Question

Two blocks of mass m1 and m2 connected by a massless spring of spring constant k are placed on a smooth horizontal table. Determine the equations of motion using lagrangian mechanics.

 write the lagrangian for the system.

Answer 1

The kinetic energy of the system comes from the motion of the blocks and potential energy from the coupling spring. $T=1/2m \dot{x}^{2}+1/2M\dot{y}^{2}$

$V=1/2k(x-y)^{2}$

$L= 1/2m\dot{x}^{2}+1/2M\dot{y}^{2}-1/2k(x-y)^{2}$

$\partial L/ \partial \dot{x}=m\dot{x}, \partial L /\partial x=-k(x-y)$

$\partial L / \partial \dot{y}=M \dot{y} , \partial L/ \partial y=k(x-y)$

Lagrange's equations are written as

$d/dt(\partial L/ \partial \dot{x})-\partial L/ \partial x=0, d/dt(\partial L/ \partial \dot{y})-\partial L/ \partial y=0$

The equations of motion can then be written as

$m\ddot{x}+k(x-y)=0$

$m\ddot{y}+k(y-x)=0$