Finding the Lagrangian of a bead sliding along a wire

[Rumor]

Homework Statement

"A bead with mass m slides without friction on a wire which lies in a vertical plane near the earth. The wire lies in the x-z plane and is bent into a shape conforming to the parabola az = x², where a is a positive known constant. (X is horizontal and z is vertical) The particle moves in the x-z plane but its motion is constrained by the requirement that it remains on the wire.

Find the kinetic energy in terms of m, a, x, x-dot. (Eliminate z and z-dot using the constraint)"

Homework Equations

z = x²/a and z-dot = (x-dot)²/a
(I think these are the restraint equations, unless I did these wrong.)

The Attempt at a Solution

I know that K = (1/2)mv², so in this case wouldn't that translate to K = (1/2)*m*((x-dot)² + (z-dot)²)?

If so, then continuing on, K = (1/2)*m*(x-dot)²+((x-dot)²/a)²) or K = (1/2)*m*(x-dot)²+[(x-dot)⁴/a²]

Am I doing this right?

 

[rude man]

1.

Homework Equations

z = x²/a and z-dot = (x-dot)²/a
(I think these are the restraint equations, unless I did these wrong.)


z-dot = what now?

 

[Rumor]

It's not specified within the problem given. I just rearranged the given equation ( az = x^2 ) and applied it to z-dot as well, since part of the problem later requires eliminating z and z-dot.

 

[rude man]

Well, I would have written z-dot = 2x(x-dot)/a.

Would you like to see how I got that? Might be a chance for you to make a fool out of me. It happens often enough! :-)

 

[paranormal]

Yes, you are on the right track. However, the constraint equation should be z = x^2/a, not z = x^2. So, the correct expression for z-dot would be z-dot = (2x*x-dot)/a.

Using this, the kinetic energy can be written as K = (1/2)*m*((x-dot)^2 + (2x*x-dot)^2/a^2). Simplifying this further, we get K = (1/2)*m*((1 + 4x^2/a^2)*(x-dot)^2).

So, the final expression for the kinetic energy in terms of m, a, x, and x-dot would be K = (1/2)*m*(1 + 4x^2/a^2)*(x-dot)^2.