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. Suppose an object is moving with uniform speed v in a circle of radius R as shown in Fig. Since the velocity of the object is changing continuously in direction, the object undergoes acceleration.
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, the average acceleration a is perpendicular to Δr. If we place Δv on the line that bisects the angle between r and r′ , we see 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 zero and the average acceleration becomes the instantaneous acceleration. It is directed towards the centre.
Thus, we find that the acceleration of an object in uniform circular motion is always directed towards the centre of the circle.
a = change in velocity /time.
Let the angle between position vectors r and r′ be Δ θ. Since the velocity vectors v and v′ are always perpendicular to the position vectors, the angle between them is also Δ θ . Therefore, thetriangle CPP′ formed by the position vectors and the triangle GHI formed by the velocity vectors v, v′ and Δv are similar (Fig. a). Therefore, the ratio of the base-length to side-length for
one of the triangles is equal to that of the other triangle. That is :
Therefore, the centripetal acceleration ac=v2/RThus, the acceleration of an object moving with speed v in a circle of radius R has a magnitude v2/R and is always directed towards the centre.
This is why this acceleration is called centripetal acceleration . “Centripetal” comes from a Greek term which means ‘centre-seeking’. Since v and R are constant, the magnitude of the centripetal acceleration is also constant. However, the direction changes, pointing always towards the centre. Therefore, a centripetal acceleration is not a constant vector.
Related Posts :
Linear motion
Acceleration Velocity
One dimensional motion
The previous post of the blog deals with Projectile motion.
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