It exhibits centripetal force,F= mv^2/r.we know, F= ma.ma= mv^2/r.a= v^2/r.

The expression for acceleration of a particle performing uniform circular motion can be derived by considering the motion along the circular path. Here, we will break down the derivation into the following steps:

Step 1: Understanding Uniform Circular Motion
Uniform circular motion refers to the motion of an object moving in a circular path with a constant speed. In this type of motion, the object always experiences an inward force called centripetal force, which keeps it moving in a circular path.

Step 2: Defining Acceleration
Acceleration is defined as the rate of change of velocity. In the case of uniform circular motion, the velocity of the particle is constantly changing due to the change in direction. Therefore, there must be an acceleration acting on the particle.

Step 3: Analyzing Velocity Vectors
In uniform circular motion, the velocity vector of the particle is always tangent to the circular path at any given point. To determine the acceleration, we need to analyze the change in the direction of the velocity vector.

Step 4: Analyzing Position Vectors
In uniform circular motion, the position vector of the particle always points towards the center of the circle. This position vector is perpendicular to the velocity vector at any given point.

Step 5: Determining Change in Velocity
To determine the change in velocity, we consider two vectors: the initial velocity vector (v1) and the final velocity vector (v2). The change in velocity (∆v) is the difference between these two vectors.

Step 6: Using Vector Subtraction
To find the change in velocity (∆v), we can subtract the initial velocity vector (v1) from the final velocity vector (v2). This can be represented as follows:
∆v = v2 - v1

Step 7: Determining Change in Direction
As mentioned earlier, the change in velocity (∆v) is due to the change in direction. Since ∆v is perpendicular to the initial velocity vector (v1), it must be in the direction of the position vector, pointing towards the center of the circle.

Step 8: Finding Acceleration
The acceleration (a) can be defined as the change in velocity (∆v) divided by the time taken (∆t) to undergo that change. In the case of uniform circular motion, ∆t refers to the time taken to move through a small arc length (∆s) on the circle.

Step 9: Calculating Arc Length
The arc length (∆s) is related to the radius (r) and the angle (∆θ) subtended by the arc. The arc length (∆s) can be given as:
∆s = r∆θ

Step 10: Expressing Acceleration
Combining all the above steps, we can express the acceleration (a) as:
a = (∆v / ∆t) = (∆v / ∆s) * (∆s / ∆t) = (∆v / ∆s) * (1 / ∆t) = (∆v / ∆s) * (1 / (∆s / v)) = (∆v / ∆s) * (v / ∆s) = v² / r