- #1
Tomtam
Homework Statement
A very thin disk of mass m and radius R rolls without slipping along a horizontal plane. The disk is constrained to remain vertical. Let ψ be the angle between the plane of the disk and the x-axis of a fixed frame and θ be the angle measuring spinning of the disk about its center. By neglecting the gravitational force, show that the center of mass of the disk moves in a circle, if dψ/dt(t=0) ≠ 0 ; and along a straight line, if dψ/dt(t=0) = 0.
Homework Equations
Iθ and Iψ are moment of inertia.
The Attempt at a Solution
I've tried to solve this problem, but I couldn't find any clues from what I've got.
what I've got are
constrain equations:
f1 = dx/dt - R(dθ/dt)cosψ = 0
f2 = dy/dt - R(dθ/dt)sinψ = 0
Lagrangian :
L = ½m(dx/dt)2+½m(dy/dt)2+½Iθ(dθ/dt)2+½Iψ(dψ/dt)2
Lagrange's equations:
m(d2x/dt2) = -(dλ1/dt)
m(d2y/dt2) = -(dλ2/dt)
Iψ(d2ψ/dt2) = λ1R(dθ/dt)sinψ-λ2R(dθ/dt)cosψ
Iθ(d2θ/dt2) = -λ1R(dψ/dt)sinψ+(dλ1/dt)Rcosψ+λ2R(dψ/dt)cosψ+(dλ2/dt)Rsinψ
I don't know what I should do next.