Derive an expression for the magnitude of the centripetal acceleration ac in an uniform circular motion (UCM). Do not forget to draw a diagram of UCM and relate your derivation with the diagram.
Answers
Explanation:
We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an .”

Figure 1. The directions of the velocity of an object at two different points are shown, and the change in velocity Δv is seen to point directly toward the center of curvature. (See small inset.) Because ac = Δv/Δt, the acceleration is also toward the center; ac is called centripetal acceleration. (Because Δθ is very small, the arc length Δs is equal to the chord length Δr for small time differences.)
The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii r and Δs are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds v1 = v2 = v. Using the properties of two similar triangles, we obtain