Nonuniform Circular Motion Circular motion does not have to be at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of the motion. In uniform circular motion, the particle executing circular motion has a constant speed and the circle is at a fixed radius. If the speed of the particle is changing as well, then we introduce an additional acceleration in the direction tangential to the circle: ar = dt The direction of tangential acceleration is tangent to the circle whereas the direction of centripetal acceleration is radially inward toward the center of the circle. Thus, a particle in circular motion with a tangential acceleration has a total acceleration that is the vector sum of the centripetal and tangential accelerations: a- ac + är. The acceleration vectors are shown below. Note that the two acceleration vectors ac and är are perpendicular to each other, with ac in the radial direction and är in the tangential direction. The total acceleration ä points at an angle between äc and är. a ä The centripetal acceleration points toward the center of the circle. The tangential acceleration is tangential to the circle at the particle's position. The total acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular.
Total Acceleration during Circular Motion A particle moves in a circle of radius r = 2.0 m. During the time interval from t = 1.5 s to t = 4.0 s its speed varies with time according to v (t) = c1 - C2 C1=4.0m/s, c2=6.0m.s. What is the total acceleration of the particle at t = 2.0 s? Strategy We are given the speed of the particle and the radius of the circle, so we can calculate centripetal acceleration easily. The direction of the centripetal acceleration is toward the center of the circle. We find the magnitude of the tangential acceleration by taking the derivative with respect to time of |v (t) | and evaluating it at t = 2.0 s. We use this and the magnitude of the centripetal acceleration to find the total acceleration. Solution Centripetal acceleration is v(2.0s) = (4.0- 6.0 (2.0)2 m/s = 2.5 m/s ac =2 2.0 m =3.1 m/s2 (2.5 m/s)2 directed toward the center of the circle. Tangential acceleration is dv dt 2c2 87 12.0 (2.0)3m/s2 = 1.5 m/s2. ar = = Total acceleration is ??| = v3.12 + 1.52m/s2 = 3.44 m/s2 and 0 = tan 1 21 = 64" from the tangent to the circle. a, (j?] = 1.5 m/s2) 64° a a (a = 3.1 m/s2) The tangential and centripetal acceleration vectors. The net acceleration a is the vector sum of the two accelerations. Significance The directions of centripetal and tangential accelerations can be described more conveniently in terms of a polar coordinate system, with unit vectors in the radial and tangential directions. This coordinate system, which is used for motion along curved paths, is discussed in detail