Page 87 - Dynamics of Mechanical Systems

0593_C03_fm Page 68 Monday, May 6, 2002 2:03 PM

68 Dynamics of Mechanical Systems

n
θ n
r
P
r
O θ
R
ω
α
FIGURE 3.7.2
A particle P on the rim of a rotor.

Then, from Eq. (3.7.6), the velocity of P may be expressed as:

V = rω n = rθ n = v n θ (3.7.9)
˙˙
P
θ
θ
Similarly, from Eq. (3.7.7), the acceleration of P may be expressed as:
a = rα n − rω 2 n = ( dv dt) n −( v r) n r
2
P
θ
θ
r
(3.7.10)
˙˙
= rθ n − rθ 2 ˙˙ n r
θ

3.8 Motion in a Plane

Let a point P move in a plane. Let the location of P be defined by the polar coordinates r
and θ as in Figure 3.8.1. As before, let n and n be unit vectors parallel to the radial line
θ
r
and parallel to the transverse direction. Let n be normal to the plane as shown such that:
z
n = n × n (3.8.1)
z r θ

The position vector locating P relative to the fixed point O is then:

p = r n (3.8.2)
r

Y
n θ
n
r
n z
θ
P (r, )
r
θ
FIGURE 3.8.1
Motion of a point in a plane. X