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0593_C08_fm Page 249 Monday, May 6, 2002 2:45 PM

Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 249

Therefore, the scalar governing equations are:

˙
N =− mr sinθ − 2 mrΩθ ˙ cosθ (8.5.13)
Ω
1
N =− mgcosθ − mrθ ˙ 2 − mrΩ 2 sin θ (8.5.14)
2
r
θ
˙˙ θ − Ω 2 sin cosθ +( ) sinθ = 0 (8.5.15)
gr
Equations (8.5.13), (8.5.14), and (8.5.15) are three equations for the unknowns N , N ,
r
1
and θ. Observe that Eq. (8.5.15) involves only θ. Hence, by solving Eq. (8.5.15) for θ we
can then substitute the result into Eqs. (8.5.13) and (8.5.14) to obtain N and N . Observe
1
r
further that if Ω is zero, Eq. (8.5.15) takes the same form as Eq. (8.4.4), the pendulum
equation. We will consider the solution of Eq. (8.5.15) in Chapters 12 and 13.

8.6 Inertia Forces on a Rigid Body

For a more general example of an inertia force system, consider a rigid body B moving
in an inertial frame R as depicted in Figure 8.6.1. Let G be the mass center of B and let P
i
be a typical point of B. Then, from Eq. (4.9.6), the acceleration of P in R may be expressed as:
i
a = a + αα × r + ωω ×(ωω × r (8.6.1)
R P i RG i)
i
where G is the mass center of B, r is the position vector of P relative to G, αα αα is the angular
i i
acceleration of B in R, and ωω ωω is the angular velocity of B in R.
Let B be considered to be composed of particles such as the crystals of a sandstone. Let
P be a point of a typical particle having mass m . Then, from Eq. (8.2.5), the inertia force
i i
on the particle is:
F =− m a P i (nosumon i) (8.6.2)
*
R
i i
The inertia forces on B consist of the system of forces made up of the inertia forces on
the particles of B. This system of forces (usually a very large number of forces) may be
represented by an equivalent force system (see Section 6.5) consisting of a single force F *

FIGURE 8.6.1
A rigid body moving in an inertial
reference frame.