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0593_C08_fm Page 247 Monday, May 6, 2002 2:45 PM

Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 247

Ω

n 3
n
n 2 θ
O r
θ
n 1
P(m)
FIGURE 8.5.1 n r
A rotating tube with an interior particle P. R
tangential unit vectors, respectively, and let n and n be horizontal and vertical unit vectors,
3
2
respectively, fixed in T.
Using the principles of kinematics of Chapter 4 we see that the acceleration of P in an
inertial frame R, in which T is spinning, is (see Eq. (4.10.11)):

a = a + a +2 ωω × νν (8.5.1)
R P T P R P * R T T P

*
where P is that point of T that coincides with P. P moves on a horizontal circle with
*
radius r sinθ. The acceleration of P in R is, then,
*
˙
R P * =−Ω rsinθ n − Ω 2 rsinθ
a n (8.5.2)
1 2
where n is a unit vector normal to the plane of T.
1
The velocity and acceleration of P in T are:
νν= rθn θ and a = r − θ n r + rθn θ (8.5.3)
T P ˙ T P ˙ 2 ˙˙

By substituting from Eqs. (8.5.2) and (8.5.3) into Eq. (8.5.1) the acceleration of P in R
becomes:

˙˙
r n + θ
˙
r sinθ
P
R a =− θ ˙ 2 r n − Ω n
r θ 1
(8.5.4)
− rΩ 2 sinθ n + 2Ω n × θ ˙ n
4
2 3 θ
R
T
where ωω ωω is identified as being Ωn . The unit vectors n and n may be expressed in terms
2
3
3
of n and n as:
θ
r
n = sinθ n + cosθ n
2 r θ
(8.5.5)
n =−cosθ n + sinθ n
3 r θ
By substituting into Eq. (8.5.4) and by carrying out the indicated addition and multipli-
R P
cation, a becomes:
r (
r (
˙
R a =− Ω sinθ − 2 rΩθ ˙ cosθ n ) +− θ ˙ 2 − rΩ 2 sin 2 n r ) θ
P
1 (8.5.6)
˙˙
θ
r −
+( θ rΩ 2 sin cosθ n ) θ