Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

Let's have a disc which moves along a straight curve on a plane in a uniform gravitational field. There need to discover the stability of it's trajectory.

I represented the possible deviation of the center of masses position by the vertical oscillations around the axis, which is parallel to direction of movement and which contains a point of contact between the cylinder and the plane (look to the picture).

enter image description here

So, with $X$-axis parallel to the direction of movement, $Y$-axis antiparallel to the gravitational field (look to the picture)

enter image description here

I got for the center of masses coordinates $$ y_{c} = R(cos(\varphi ) - 1), \quad z_{c} = Rsin(\varphi ), $$ and $$ L = \frac{m\dot {x}^{2}}{2} + \frac{I_{3}\omega_{z}^{2}}{2} + \frac{(I_{1} + mR^{2})\omega_{x}^{2}}{2} + mgRcos(\varphi ). $$ I need to associate $ \omega_{x, z}$ with $x_{c}, \varphi $. But I don't know how to use Poinsot formula, $$ \mathbf v_{surface} = \mathbf v_{c} + [\mathbf \omega \times \mathbf R_{cs}], $$

where $\mathbf R_{cs}$ is a vector from center of masses to contact point, for the case of rotation near the axes do not pass through one point. Can you help me?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.