Page 169 - Standard Handbook Of Petroleum & Natural Gas Engineering

154 General Engineering and Science

unit vectors .Z, tangent to the path of motion, and fi, inwardly perpendicular to the
path of motion.
The velocity is always tangent to the path of motion, and thus the velocity vector
has only one component (Equation 2-20).

v = vz (2-20)
The acceleration vector has a component tangent to the path at = d IV l/dt, which is
the rate at which the magnitude of the velocity vector is changing, and a component
perpendicular to the path an = I v I z/p, which represents the rate at which the direction
of motion is changing (Equation 2-21).

dlvl A \VI' -
a = a,Z+a,n = -z+-n (2-21)
dt P

In Equation 2-21 p is the local radius of curvature of the path. The normal
component of acceleration can also be expressed as an = p I w I or an = 1 v I I w I where
o is the angular velocity of the particle.

Example 2-5

A car is increasing in speed at a rate of 10 ft/s2 when it enters a curve with a radius
of 50 ft at a speed of 30 ft/s. What is the magnitude of its total acceleration?

dlvl
a, = - = 10 ft/s2
dt

la1 = (lo2+ 1S2)05 = 20.6 ft/s2

In addition to the Cartesian and normal/tangential coordinate systems, the
cylindrical (Figure 2-1 1) and spherical (Figure 2-12) coordinate systems are often used.
When dealing with the motions of rigid bodies or systems of rigid bodies, it is
sometimes quite difficult to directly write out the equations of motion of the point in
question as was done in Examples 2-6 and 2-7. It is sometimes more practical to
analyze such a problem by relative motion. That is, first find the motion with respect
to a nonaccelerating reference frame of some point on the body, typically the center
of mass or axis of rotation, and vectorally add to this the motion of the point in
question with respect to the reference point.
Example 2-6

Consider an arm 2 ft in length rotating in the counterclockwise direction about a
fixed axis at point A at a rate of 2 rpm (see Figure 2-13a). Attached to the arm at