Fast Particle in Bowl

A particle constrained to move on a smooth spherical surface of radius R is projected horizontally from a point at the level of the center so that it's an

Figure 1.1

gular velocity relative to the axis is ω (see Figure 1.1). If ω²R >> g show that its maximum depth z below the level of the center is approximately.

SOLUTION

Introduce cylindrical coordinates ρ, φ, z where z is positive down (see Figure 1.2). We can write the Lagrangian

(1)

From (1), we can see that, as usual, the angular momentum is conserved:

 Figure 1.2

Using the constraint ρ² = R² – z² which follows from the fact that the particle moves along the spherical surface, we have

(2)

From the same constraint,

(3)

Energy is of course also conserved:

(4)

Substituting  and  from (2) and (3) into (4), we obtain

(5)

The condition leads to Therefore we can approximately write

 (6)

Or

(7)