An object is said to be moving in uniform circular motion when it maintains a constant speed while traveling in a circle. Remember that since acceleration is a vector quantity comprised of both magnitude and direction, objects can accelerate in any of these three ways: 1. constant direction, changing speed (linear acceleration); 2. constant speed, changing direction (centripetal acceleration); 3. change in both speed and direction (angular acceleration). In this lesson, we will be investigating centripetal acceleration and uniform circular motion - that is, objects moving in circular paths at constant speeds. While moving in a circular path, an object is constantly being pulled "towards the center" of the circle away from its tangential path. Envision a stopper on the end of a string being twirled over your head in a horizontal circle. If the string were to break, the stopper would "fly off at a tangent." The tension in the string is forcing the stopper to constantly be pulled back towards the center to follow a circular, instead of a linear, path. As shown in the diagram above, in a certain amount of time, Δt, an object traveling in a circular path would move from position A at time t_{1} where its velocity is labeled v_{o} to position B at time t_{2} where its velocity is labeled v_{f}. Note that the magnitude of v_{f} equals that of v_{o} since we are only changing the direction of the velocity, not the object's speed. Remember that acceleration equals Δv/Δt. To diagram this acceleration, we must be able to diagram the resultant change in velocity, or Δv. Thus we must recognize the orientation of the vector -v_{o}. Since the vector v_{o} points to the right, the vector -v_{o} would have the exact same magnitude but point in the opposite direction. The direction of the acceleration that an object experiences during an interval of time, Δt, is illustrated in the next diagram by showing the direction of v_{f} - v_{o}. To diagram the vector resultant v_{f} - v_{o}, we will use the head-to-tail method of vector addition where
**Δv = v**_{f} - v_{o} = v_{f} + (- v_{o})
Notice that the resultant velocity, Δv, starts at the beginning of the vector v_{f} and terminates at the end of the vector -v_{o}. This relation can also be seen in the following diagram when we merely rearranged the vector equation **Δv = v**_{f} - v_{o} to read **v**_{o} + Δv = v_{f} . Notice that v_{f} is now the resultant vector since v_{f} starts at the beginning of the vector v_{o} and terminates at the end of the vector Δv. Note that in both cases, Δv points to the center of the circle reflecting that the acceleration is also directed towards the center of the circle. |