Fast Particle in Bowl
المؤلف:
Sidney B. Cahn And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 14
28-7-2016
1332
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)
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