- #1
stunner5000pt
- 1,461
- 2
consider a bead of mass m constrained to move on a fricitonless wire helix whose equations in cylindrical polar coords is
z = a phi where a is some constant
the bead is acted upon by a force which deends on the distance from the cneter only.
Formulate the problem using s the distance along the helix as your generalized coordinate.
for distance s along the helix [itex] s = a \phi_{0} [/itex]
but r would not be constnat
[tex] T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\phi_{0}}^2 + \dot{z}^2) [/tex]
z dot is zero and phi0 is a constnat soso
[tex] T = \frac{1}{2} m \dot{r}^2 [/tex]
the force is dependant on the distance from the center only
thus [tex] \vec{F} = -k\vec{r} = -\nabla V [/tex](say)
then [tex] V = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + a^2 \phi_{0}^2) [/tex]
[tex] L = T - V = \frac{1}{2} m\dot{r}^2- \frac{1}{2} k (r^2 + a^2 \phi_{0}^2) [/tex]
is this formulation correct?
Or am i totally off?
z = a phi where a is some constant
the bead is acted upon by a force which deends on the distance from the cneter only.
Formulate the problem using s the distance along the helix as your generalized coordinate.
for distance s along the helix [itex] s = a \phi_{0} [/itex]
but r would not be constnat
[tex] T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\phi_{0}}^2 + \dot{z}^2) [/tex]
z dot is zero and phi0 is a constnat soso
[tex] T = \frac{1}{2} m \dot{r}^2 [/tex]
the force is dependant on the distance from the center only
thus [tex] \vec{F} = -k\vec{r} = -\nabla V [/tex](say)
then [tex] V = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + a^2 \phi_{0}^2) [/tex]
[tex] L = T - V = \frac{1}{2} m\dot{r}^2- \frac{1}{2} k (r^2 + a^2 \phi_{0}^2) [/tex]
is this formulation correct?
Or am i totally off?