00:01
Hello students in this question we need to find the lagrangian expression and also the lagrangian equation and also we need to find the frequency of small oscillation of the system.
00:19
This system consists of a block m which is displaced by the xi when its kinetic energy is given by k e is equal to half m x1.
00:38
Dot square and potential energy is equal to p is equal to half k x1 dot square and this mass m is connected to a pendulum of length r and theta the horizontal and vertical displacement of the mass m will be x2 is equal to r sine theta plus x0 and x2 dot is equals to r cost theta theta dot and y2 is equal to r cost theta and y2 dot is equal to minus r sine theta into theta dot here kinetic energy t two can be taken as half m into x1 x2 dot square plus y 2 dot square by substituting the x2 2 dot and y2 dot values we get r square theta dot square and the potential energy v2 is equals to minus m g r cost theta now the lagrangian for the system of masses can be l is equal to t 1 plus t 2 minus v1 plus v2 by substituting the values l can be half m x1 dot square plus half m r square theta dot square minus half k x1 square plus m g r cost theta.
02:59
So this is the lagrangian equation.
03:02
Now to find the lagrangian equation of motion, we have the formula that is d by dt of do l by do x dot minus do l by do x1 which is equals to 0.
03:26
To substitute in this equation, we need to find first do l by do x1 dot.
03:31
And also here the generalized coordinates are x1 and theta.
03:36
Therefore, do l by do x1 dot is equal to m into x1 dot.
03:43
And do l by do x1 is equal to minus k into x1...