[Work check] Parametric frictionless wire

[Zinggy]

Homework Statement

A friction less wire is wound into a shape described by the parametric equations:
x = acosαλ, y = asinαλ, z = bλ where zis the vertical axis and a, b, and α are positive
constants.

Find the Lagrangian of the bead in terms of λ
using Lagrangian equations of motion, find and expression for λ d^2/dt^2

Relevant Equations

x = acosαλ, y = asinαλ, z = bλ

for my formatting, (dot) implies a single time derivative with respect to the variable

Kinetic Energy = T = (1/2) m (x(dot)² +y(dot)² + z(dot)²

Plug in respective values for x y and z -> T= (1/2) m (a² α²sin²(αλ) λ(dot) +a² α²cos²(αλ) λ(dot) + b²λ(dot)

After canceling out Sin and cos -> (1/2)m (a² α² + b²)λ(dot)² = Kinetic Energy = T

Potential Energy = U = mg(z - z₀)

Plug in value for z U = mgb(λ-λ₀)

So the full Lagrangian is L= T-U = (1/2)m (a² α² + b²)λ(dot)² - mgb(λ-λ₀)Part 2) Using the equation d/dt(dL/dλ(dot))-dL/dλ = 0

Take necessary derivatives and plug in -> d/dt[m(a² α²+b²)λ(dot)]+mgb
->m(a² α²+b²)λ(double dot)+mgb = 0
Do the Algebra to solve for λ(double dot). -> λ(double dot) = -gb/(a² α²+b²)

Terribly sorry for the formatting on the dots, couldn't figure out how to do that.

Thanks for your time, please point out anything I've done wrong!

 

[Zinggy]

If anyone could take the time to check my work I would appreciate it!

 

[Zinggy]

Assignment is due tomorrow, would really appreciate another set of eyes!