Figure 4.17 The motion of a particle along an arbitrary curved path lying in the xy plane. If the velocity vector v (always tangent to the path) changes in direction and magnitude, the component vectors of the acceleration a are a tangential component at and a radial component ar.
Now let us consider a particle moving along a curved path where the velocity changes both in direction and in magnitude, as shown in Figure 4.17. As is always the case, the velocity vector is tangent to the path, but now the direction of the acceleration vector a changes from point to point. This vector can be resolved into two component vectors: a radial component vector ar and a tangential component vector at . Thus, a can be written as the vector sum of these component vectors:
The tangential acceleration causes the change in the speed of the particle. It is parallel to the instantaneous velocity, and its magnitude is
The radial acceleration arises from the change in direction of the velocity vector as described earlier and has an absolute magnitude given by
where r is the radius of curvature of the path at the point in question. Because ar and at are mutually perpendicular component vectors of a, it follows that a = √(ar2 + at2). As in the case of uniform circular motion, ar in nonuniform circular motion always points toward the center of curvature, as shown in Figure 4.17. Also, at a given speed, ar is large when the radius of curvature is small (as at points A and B in Figure 4.17) and small when r is large (such as at point C). The direction of at is either in the same direction as v (if v is increasing) or opposite v (if v is decreasing).
In uniform circular motion, where v is constant, at = 0 and the acceleration is always completely radial, as we described in Section 4.4. (Note: Eq. 4.18 is identical to Eq. 4.15.) In other words, uniform circular motion is a special case of motion along a curved path. Furthermore, if the direction of v does not change, then there is no radial acceleration and the motion is one-dimensional (in this case,
ar = 0, but at may not be zero).
Figure 4.18 (a)Descriptions of the unit vectors r̂ and θ̂. (b) The total acceleration a of a particle moving along a curved path (which at any instant is part of a circle of radius r) is the sum of radial and tangential components. The radial component is directed toward the center of curvature. If the tangential component of acceleration becomes zero, the particle follows uniform circular motion.
It is convenient to write the acceleration of a particle moving in a circular path in terms of unit vectors. We do this by defining the unit vectors r̂ and θ̂ shown in Figure 4.18a, where r̂ is a unit vector lying
along the radius vector and directed radially outward from the center of the circle and θ̂ is a unit vector tangent to the circle. The direction of θ̂ is in the direction of increasing θ, where θ is measured
counterclockwise from the positive x axis. Note that both r̂ and θ̂ “move along with the particle” and so vary in time. Using this notation, we can express the total acceleration as
These vectors are described in Figure 4.18b. The negative sign on the v2/r term in Equation 4.19 indicates that the radial acceleration is always directed radially inward, opposite r̂.