Consider a particle of mass m that is free to move on the smooth inner surface of a hemisphere of radius R 0 The particle is under the influence of a gravitational force −mgE 3 . (a) Using a spherical...

Consider a particle of mass m that is free to move on the smooth inner surface of a hemisphere of radius R₀
The particle is under the influence of a gravitational force −mgE₃.

(a) Using a spherical polar coordinate system, what is the constraint on the motion of the particle? Give a prescription for the constraint force acting on the particle.

(b) Using Lagrange’s equations, establish the equations of motion for the particle and an expression for the constraint force.

(c) Prove that the total energy E and the angular momentum H_O· E₃
of the particle are conserved.

(d) Show that the normal force acting on the particle can be expressed as a function of the position of the particle and its initial energy E₀:

(e) Numerically integrate the equations of motion of the particle and show that there are instances for which it will always remain on the surface of the hemisphere.