NCERT 11th Class : Motion in a Plane - IV : Uniform Circular Motion

Uniform Circular Motion : When an object follows a circular path at a constant speed, the motion of the object is called uniform circular motion. The word “uniform” refers to the speed, which is uniform (constant) throughout the motion. Since the velocity of the object is changing continuously in direction, the object undergoes acceleration. The acceleration of an object in uniform circular motion is always directed towards the centre of the circle.

Let r and r′ be the position vectors and v and v′ the velocities of the object when it is at point P and P′ as shown in Fig.(a). By definition, velocity at a point is along the tangent at that point in the direction of motion. The velocity vectors v and v′ are as shown in Fig.(a1). Δv is obtained in Fig.(a2) using the triangle law of vector addition.
Since the path is circular, v is perpendicular to r and so is v′ to r′ . Therefore, Δv is perpendicular to Δr. Since average acceleration is along Δv, a= Δv/Δt, the average acceleration a is perpendicular to Δr. If Δv is placed on the line that bisects the angle between r and r′ , it can be seen that it is directed towards the centre of the circle. Figure (b) shows the same quantities for smaller time interval. Δv and hence a is again directed towards the centre.
In Fig.(c), Δt tends to 0 and the average acceleration becomes the instantaneous acceleration. It is directed towards the centre. Thus, it is found that the acceleration of an object in uniform circular motion is always directed towards the centre of the circle. The magnitude of a is, by definition, given by

Let the angle between position vectors r and r′ be Δθ. Since the velocity vectors v and v′ are always perpendicular to the position vectors, the angle between them is also Δθ . Therefore, the triangle CPP′ formed by the position vectors and the triangle GHI formed by the velocity vectors v, v′ and Δv are similar. Therefore, the ratio of the base-length to side-length for one of the triangles is equal to that of the other triangle.