- #1
Davidllerenav
- 424
- 14
- Homework Statement
- A particle of mass ##m_1## hangs from a rod of negligible mass and length ##l##, whose support point consists of another particle of mass ##m_2## that moves horizontally subject to two springs of constant ##k## each one. Find the equations of motion for this system.
- Relevant Equations
- ##L=T-V##
What I first did was setting the reference system on the left corner. Then, I said that the position of the mass ##m_2## is ##x_2##. I also supposed that the pendulum makes an angle ##\theta## with respect to the vertical axis ##y##. So the generalized coordinates of the system would be ##x_2## and ##\theta##. Thus, the coordinates of ##m_1## are:
- ##x_1=x_2+l\sin\theta##
- ##y_1=-l\cos\theta##
- ##\dot x_1=\dot x_2+l\dot\theta\cos\theta##
- ##\dot y_1=l\dot\theta\sin\theta##
- ##T=\frac{1}{2}m_2\dot x_2^2+\frac{1}{2}m_1(\dot x_2^2+l^2\dot\theta^2+2\dot x_2\dot\theta l)##
- ##V_1=-m_1gl\cos\theta##
- ##V_2=\frac{1}{2}k x_2^2+\frac{1}{2}k(-x_2)^2=kx_1^2##