Page 85 - Dynamics of Mechanical Systems

0593_C03_fm Page 66 Monday, May 6, 2002 2:03 PM

66 Dynamics of Mechanical Systems

3.6 Geometric Interpretation of Acceleration
Consider again a point P moving along curve C as in Figure 3.6.1. As before, let C be fixed
in reference frame R. Then, from Eqs. (3.5.1) and (3.5.2) the velocity and acceleration of P
in R are:

V = v n and a = dv n v d n (3.6.1)
+
P
P
dt dt
where n is a unit vector tangent to C at P and where v is the magnitude of V . From Eq.
P
(3.5.6), we see that dn/dt is perpendicular to n. Therefore, the two terms of a in Eq. (3.6.1)
P
are perpendicular to each other. The first term arises from a change in the speed of P. The
second arises from a change in the direction of P.
Recall that a vector may be characterized as having magnitude and direction; hence,
changes in a vector can arise from changes in either the magnitude of the vector or the
direction of the vector, or both. Therefore, because a is the time rate of change of v , the
P
P
terms of a arise due to changes in the magnitude of V (that is, dv/dt n) and due to
P
P
P
changes in the direction of V (that is, v dn/dt).

3.7 Motion on a Circle

To illustrate these concepts, consider the special case of a point moving on a circle.
Specifically, let C be a circle with radius r and center O. Let P be a point moving on C as
in Figure 3.7.1. Let θ measure the inclination of the radial line OP. Let n and n be unit
θ
r
vectors parallel to OP and parallel to the tangent of C at P, as shown. Let n be a unit
z
vector normal to the plane of C such that:
n = n × n (3.7.1)
z r θ
The position vector p locating P relative to O is:

p = r n (3.7.2)
r

Y n θ
n n
C r
P n
z P
r
R O θ X

C

FIGURE 3.6.1 FIGURE 3.7.1
A point moving on a curve. A point P moving on a circle C.