## Uniform Circular Motion Centripetal Acceleration

We know that the axis of rotation of a rotating particle passes through the centre of the circular path and also lies normally on the plane of rotation. In the case of uniform circular motion, the particle travels equal distances in equal intervals of time, i.e., the speed of the particle remains constant.

**Uniform Circular Motions Centripetal Acceleration Definition:** if the angular velocity of a particle rotating in a circular path remains constant, then its motion is called a uniform circular motion.

- The direction of motion of a particle under uniform circular motion changes at every moment, and hence, the direction of the linear velocity v of the particle changes continuously. However due to the constant speed of the particle, the magnitude of linear velocity always remains constant. Since linear velocity is a vector quantity, it has both magnitude and direction.
- So, in this case, it cannot be said that the linear velocity of the particle is constant. Hence, uniform circular motion is an example of uniform speed but not of uniform velocity.
- The direction of the linear velocity of the particle at every point on the circular path is along the tangent drawn at that point. For this reason, during the revolution of a stone tied with a thread, when the thread snaps, the stone flies off along the tangent.
- When a bicycle moves along a muddy road speedily, the mud particles seem to fly off from the wheel of the bicycle tangentially. Sparks of fire seem to fly off tangentially when knives, scissors, etc., are sharpened on a rotating sharpening machine.
- We have learnt that the rate of change of velocity of a body is called its acceleration. In a uniform circular motion, the linear velocity of a body varies in direction and hence, the body possesses an acceleration.

**Read and Learn More: Class 11 Physics Notes**

With the help of the following calculation, we can determine the magnitude and direction of the acceleration of the body.

**Calculation Of Centripetal Acceleration:** Suppose a particle of mass m is rotating along a circular path of radius r with uniform speed v. In a very small time interval t, the particle moves from A to B and as a result, the angular displacement θ(=∠AOB) is very small. So, the angular velocity of the particle is, \(\omega=\frac{\theta}{t}\).

- At point A, the direction of linear velocity v is along the tangent AP. The component of this velocity along the radius AO is zero because AO and AP are mutually perpendicular.
- At point B, the velocity v of the particle is along the tangent BQ. This velocity is resolved into two mutually perpendicular components. The component of this velocity parallel to AP and in the direction BR is vcosθ and the component parallel to AO and in the direction BS is vsinθ.
- If θ is very small, then sinθ ≈ θ and cosθ ≈ 1. Again, if ∠AOB is very small, then the straight lines BR and BS will just coincide with the straight lines AP and AO, respectively.

So, the initial velocity of the particle in the direction AP = v and its final velocity = vcosθ = v.

Hence, change in velocity = v- v = 0

or, acceleration = \(\frac{\text { change in velocity }}{\text { time }}=0\)

So, in the direction AP, i.e., along the tangent of the circle, the particle has no acceleration.

Again, the initial velocity of the particle along AO = 0

Its final velocity = vsinθ = vθ

∴ Change in velocity = vθ – 0 = vθ

or, acceleration = \(\frac{\text { change in velocity }}{\text { time }}=\frac{v \theta}{t}=v \omega\)

= \(\omega r \cdot \omega=\omega^2 r=\left(\frac{v}{r}\right)^2 r=\frac{v^2}{r}\)

So, along the direction AO (radially towards the centre of the circle), the particle has an acceleration whose value is (ω²r or\(\frac{v^2}{r}\). This acceleration is called the radial or, normal or centripetal acceleration.

**Calculation Of Centripetal Acceleration Definition:** When a particle moves along a circular path with constant speed, it possesses an acceleration towards the centre of the circle and this acceleration is called its radial normal or centripetal acceleration.

It is to be mentioned here that the velocity of the particle is always along the tangent of the circle but its acceleration is always towards the centre. So, in the case of uniform circular motion, velocity and acceleration are always perpendicular to each other.