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fluidistic
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Homework Statement
I'm doing past a past exam (2003) and I'm stuck on the first exercise. Here it is:
Consider a helix centered in the z-axis, of radius R and fixed step [itex]a[/itex], given in cylindrical coordinates by [itex]z=\frac{a\theta }{2 \pi }[/itex], [itex]r=R[/itex].
A particle of mass m slides without rolling over the helix under the action of gravity [itex]-g \hat z[/itex], being at rest in [itex]t=0[/itex] and at a height [itex]h[/itex] over the x-y plane.
1)How many degrees of freedom does the system have? Does it has constraints? If so, of which kind? Write down the Lagrangian modified in order to use the method of Lagrange's multipliers.
2)Write down the conserved quantities.
3)Calculate the position of the particle for all [itex]t \geq 0[/itex].
4)Calculate the force of constraint over the particle for all [itex]t \geq 0[/itex].
Homework Equations
Coming in part 3)
The Attempt at a Solution
I was trying to solve the exercise until I realized I didn't use the constraints and I realized I don't understand something important.
Here is what I've done:
1)3 degrees of freedom, the motion is in 3d and requires 3 coordinates to be described (now I'm unsure of this since z is in function of theta, it would mean only 2 degrees of freedom, although since r is constant, maybe only 1 after all. Sigh)
Yes it has constraints, the motion of the particle must follow the helix. I don't know if the constraint is holonomic. I'd say it isn't, because the force applied to the mass will vary with its velocity, so I'd say the constraint is non-holonomic.
[itex]L=T-V[/itex]. [itex]T=\frac{ m \dot {\vec r}^2}{2}[/itex]. Writing [itex]\vec r =x \hat i +y \hat j + z\hat k[/itex], converting x, y and z into cylindrical coordinates and doing [itex]\vec r \cdot \vec r[/itex], I reached that [itex]T= \frac{m(R^2 \dot \theta ^2 + \dot z ^2)}{2}[/itex]. While [itex]V=mgz[/itex]. So that [itex] L=m \left ( \frac{R^2 \dot \theta ^2 +\dot z ^2 }{2} -gz \right )[/itex].
I calculated the generalized forces, [itex]F _r = \frac{d}{dt} \left ( \frac{\partial T }{\partial \dot r } \right )- \frac{\partial T }{\partial r}=-R \dot \theta ^2 m[/itex] which indeed has units of Newton in the SI.
I reached [itex]F_ \theta=R^2 \ddot \theta m[/itex] with units of Newton times meter (torque units) and [itex]F_z =m(\ddot z +g )[/itex] with units of Newton.
I had read that the values of Lagrange multipliers are worth the forces, so I'm guessing I can use the values I just calculated as values for [itex]\lambda[/itex]. Is what they ask for of the form [itex]L _{\text {modified} }=L- \lambda \Phi[/itex] where [itex]\Phi[/itex] is a constraint equation, like [itex]r=R[/itex], [itex]z=\frac{a \theta }{2 \pi}[/itex]?
After I had done this, I realized I never used the fact that [itex]z= \frac{a\theta }{2 \pi }[/itex]. Should I use it now? [itex]F_z[/itex] would disappear...
Also, I don't understand how can [itex]z=\frac{a\theta }{2 \pi }[/itex] describe a helix. First of all, in cylindrical coordinates theta has a range of [itex]2 \pi[/itex]. This seriously limits z. Does that mean that the helix is "cut", starting from [itex]z=0[/itex] to [itex]z=a[/itex]? So that [itex]0<h<a[/itex]. If not... then I do not understand how this equation (along with [itex]r=R[/itex]) describe a helix since theta is limited in cylindrical coordinates.
Any help is appreciated!