- #1
carllacan
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Homework Statement
A masspoint finds itself under the influence of gravity and constrained to move on a (inverted) circular cone. Using D'Alembert's Principle find the equations of motion on cylindric coordinates.
Homework Equations
D'Alembert's Principle: ([itex]\vec{F_a}[/itex] -m·[itex]\vec{a}[/itex])·[itex]\delta[/itex][itex]\vec{r}[/itex]=0
The Attempt at a Solution
Chose as generalized coordinates l and m, which measure, respectively, "how high" is the particle on the cone and the angle coordinate.
Write F = -mgz, where z is the vector for the vertical cylindric coordinate. Write [itex]\delta[/itex][itex]\vec{r}[/itex] as the total differential dr minus the dt term, i.e. [itex]\delta[/itex][itex]\vec{r}[/itex] = [itex]\stackrel{d\textbf{r}}{dl}[/itex] [itex]\delta[/itex]l + [itex]\stackrel{d\textbf{r}}{dm}[/itex] [itex]\delta[/itex]m
Then, as the generalized coordinates are independent we can equate the coeficients of [itex]\delta[/itex]l and [itex]\delta[/itex]m to zero. Which should give us the equations of motion. Th problem is that I obtain one equation according to which the angular acceleration is 0 (as expected) and another one that reads: a_r ·tg([itex]\alpha[/itex])+a_z = -g, where a_r and a_z are the radial and vertical coordinates of the acceleration.