- #1
DannyJ108
- 25
- 2
- Homework Statement
- Find the solutions to the equations of motion of a simple pendulum with a mass which varies in time
- Relevant Equations
- ##L= T -V =\frac 1 2 m(t) l^2 {\dot \theta}^2 -m(t)gl\cos\theta##
Hello,
I've got to rationally analice the form of the solutions for the equations of motion of a simple pendulum with a varying mass hanging from its thread of length ##l## (being this length constant).
I approached this with lagrangian mechanics, asumming the positive ##y## direction is pointing down, I get that:
##L= T -V =\frac 1 2 m(t) l^2 {\dot \theta}^2 -m(t)gl\cos\theta##
I work out my Euler-Lagrange equations with respect to ##\theta## and get:
##\dot m(t) l \dot \theta + m(t)l \ddot \theta - m(t)g\sin \theta = 0##
If i assume the mass varies slowly with time and use small angle approximation (##\sin\theta \approx \theta##) I thought to proceed like this:
##\frac d {dt} (m(t) \dot \theta) = m(t) \frac g l \theta##
I don't know how to proceed from here. I think I don't have to give a explicit solution to this equation, but rather just interpret how it'll be. Either way if you could help me out finding a solution it would be great.
Thank you!
I've got to rationally analice the form of the solutions for the equations of motion of a simple pendulum with a varying mass hanging from its thread of length ##l## (being this length constant).
I approached this with lagrangian mechanics, asumming the positive ##y## direction is pointing down, I get that:
##L= T -V =\frac 1 2 m(t) l^2 {\dot \theta}^2 -m(t)gl\cos\theta##
I work out my Euler-Lagrange equations with respect to ##\theta## and get:
##\dot m(t) l \dot \theta + m(t)l \ddot \theta - m(t)g\sin \theta = 0##
If i assume the mass varies slowly with time and use small angle approximation (##\sin\theta \approx \theta##) I thought to proceed like this:
##\frac d {dt} (m(t) \dot \theta) = m(t) \frac g l \theta##
I don't know how to proceed from here. I think I don't have to give a explicit solution to this equation, but rather just interpret how it'll be. Either way if you could help me out finding a solution it would be great.
Thank you!