Consider a particle of mass m that is in motion on a helicoid. In terms of cylindrical polar coordinates {r, θ, z}, the equation of the right helicoid is z = βθ, where β is a constant. A gravitational...

Consider a particle of mass m that is in motion on a helicoid. In terms of cylindrical polar coordinates {r, θ, z}, the equation of the right helicoid is

z = βθ,

where β is a constant. A gravitational force −mgE₃
acts on the particle.

(a) Consider the following curvilinear coordinate system for E³:

Show that

and that

(b) Consider a particle moving on the smooth helicoid. The generalized coordinates for the particle are θ and r.

(i) What is the constraint on the motion of the particle, and what is a prescription for the constraint force Fc enforcing this constraint?

(ii) What is the kinematical line element ds for the helicoid?

(iii) Show that the equations governing the unconstrained motion of the particle are

(iv) Prove that the angular momentum H_O· E₃
is not conserved.

(c) Suppose the constraint

is imposed on the particle. Establish a second-order differential equation for r(t), a differential equation for θ (t), and an equation for the constraint force enforcing the constraint. Indicate how you would solve these equations to determine the motion of the particle and the constraint forces acting on it.