Maximizing Energy Conservation in Particle Motion on a Helix Wire

[teme92]

Homework Statement

A particle P is free to slide on a smooth wire which has the form of a helix,
with a position vector given by:

r((t)) = a cosθ(t)i + a sinθ(t)j + bθ(t)k​

The particle is released from rest at the point (a, 0, 2∏b). Using energy conservation for conservative forces, or otherwise, show that the speed of P when it reaches the ground at (a, 0, 0) is:​

v = 2sqrt(∏bg)​

Homework Equations

All the equations of motion

The Attempt at a Solution

I know that when you differentiate the position, you get velocity. So I did and got:

v(θ(t))=(a(-sinθ(t)) + (cosθ(t))(1))i + (acosθ(t)) + (sinθ(t))(1))j + (b+θ(t))k

from here I'm stuck.

I let the components for i,j and k equal to one another but I don't know what to do with the results. Please any help would be greatly appreciated.

 

[mfb]

If you are just interested in the final speed, energy conservation is way easier than getting equations of motion.
Did you try this? What is the initial energy, what is the final energy?

 

[Simon Bridge]

I'll second that - otherwise you should probably convert to cylindrical coordinates.

 

[teme92]

Hi mfb and Simon, the help was much appreciated.

I was over complicating the question as you said. I just used the Conservation of Energy and the solution came easily.

PE=KE where;

PE=mgh and KE=0.5(m)(v^2)

h=2(pi)b as the coordinates go from (a,0,2(pi)b) to (a,0,0)

So then I got 2mg(pi)b=0.5(m)(v^2)

simplifying to get my desired answer of v=2sqrt((pi)bg)

Thanks again!

 

[Simon Bridge]

Well done - getting you to realize the advantages of using energy instead of forces is probably the point of the exercise.