- #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

##\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!