Page 291 - Dynamics of Mechanical Systems

0593_C08_fm Page 272 Monday, May 6, 2002 2:45 PM

272 Dynamics of Mechanical Systems

˙
θ
Show that this equation can be integrated by multiplying by ; that is,
˙˙˙
θθ +( ) l θ ˙ sinθ = 0
g
leads to:

d ( θ 2) + d − ( cosθ) =
˙ 2
dt dt 0

so that

θ 2 − cosθ = C
˙ 2

where C is a constant.
P8.4.6: See Problem P8.4.5. Suppose a simple pendulum of length 3 ft is displaced through
an angle of 60° and released from rest. Find the speed of the bob when θ is zero, the lowest
position.
P8.4.7: Consider the small-amplitude oscillations of a simple pendulum (see Eq. (8.4.5)).
Show that the general solution of the governing equation may be written in the forms:

t
t
t
θ = Acos ω + Bsin ω = Ccos ( ω + φ)
where ω is g l and A, B, C, and φ are constants (where C is the amplitude; φ, the phase
angle; and ω, the circular frequency).
P8.4.8 See Problem P8.4.7. Express the amplitude C and phase in terms of the constants
A and B.

Section 8.5 A Smooth Particle Moving Inside a Vertical Rotating Tube
P8.5.1: A 6-oz smooth particle P moves inside a 2.5-ft-radius tube, which is rotating about
a vertical diameter at a uniform rate Ω of 3 ft/sec as depicted in Figure P8.5.1. Determine
the n , n , and n components of the inertia force F on P for an instant when the angle θ
*
1
2
3
locating P is 60° and P is moving at a uniform speed v of 3 ft/sec clockwise relative to
the tube.
Ω

T
n 3
n 2
n 1
r
θ

FIGURE P8.5.1
A smooth particle P inside a rotating tube T.