Classical Dynamics of Particles and Systems · Problem 10-13

Dynamics of Rigid Bodies — Marion & Thornton

Physics — Mechanics · southerly-deflection-falling-particle

Problem

Refer to Example 10.3 concerning the deflection from the plumb line of a particle falling in Earth's gravitational field. Take

g

to be defined at ground level and use the zeroth order result for the time-of-fall,

T = \sqrt{2h/g}

. Perform a calculation in second approximation (i.e., retain terms in

\omega^2

) and calculate the southerly deflection. There are three components to consider: (a) Coriolis force to second order (

C_1

), (b) variation of centrifugal force with height (

C_2

), and (c) variation of gravitational force with height (

C_3

). Show that each of these components gives a result equal to

C_i \frac{h^2}{g}\omega^2\sin\lambda\cos\lambda

with

C_1 = 2/3

C_2 = 5/6

, and

C_3 = 5/2

. The total southerly deflection is therefore

(4h^2\omega^2\sin\lambda\cos\lambda)/g