if we give some constant tangential acceleration along with velocity to a particle which is performing uniform circular motion then is its centripetal acceleration will increases and if yes/no then why?

what i think is that this increase angular acceleration

as a_t = (r)×(angular acceleration)

when a particle has angular acceleration then its angular velocity also increases

as angular velocity = (initial angular velocity) +(angular acceleration×time)

and

velocity =r×(angular velocity)

and

centripetal acceleration=(velocity)²/r

To address your question about the effects of constant tangential acceleration on a particle in uniform circular motion, let's break down the concepts involved and see how they interact with each other.

Understanding Circular Motion

In uniform circular motion, a particle moves along a circular path at a constant speed. While the speed remains constant, the direction of the velocity vector changes continuously, which means there is always a centripetal acceleration directed towards the center of the circle. This centripetal acceleration can be calculated using the formula:

• Centripetal Acceleration (a_c) = v²/r

where v is the linear speed and r is the radius of the circular path.

Introducing Tangential Acceleration

Now, if we introduce a constant tangential acceleration (let's call it a_t), this means that the particle is no longer in uniform circular motion. Instead, the particle will experience an increase in its speed as it moves along the circular path. The tangential acceleration affects the linear speed of the particle, which in turn influences the centripetal acceleration.

Effects on Angular Velocity

As you correctly pointed out, the angular acceleration (α) can be related to tangential acceleration through the radius:

• Angular Acceleration (α) = a_t/r

With this angular acceleration, the angular velocity (ω) will change over time:

• Angular Velocity (ω) = Initial Angular Velocity (ω₀) + α × t

As the angular velocity increases, the linear velocity (v) also increases since:

• Linear Velocity (v) = r × ω

Impact on Centripetal Acceleration

Now, let's examine how this increase in linear velocity affects centripetal acceleration. Since centripetal acceleration is dependent on the square of the velocity:

• Centripetal Acceleration (a_c) = v²/r

As the linear velocity increases due to the tangential acceleration, the centripetal acceleration will also increase. This is because the velocity is squared in the formula, meaning even a small increase in velocity results in a larger increase in centripetal acceleration.

Conclusion

In summary, when you apply a constant tangential acceleration to a particle in circular motion, it will indeed lead to an increase in both the angular velocity and the linear velocity. Consequently, the centripetal acceleration will also increase as a result of the increased linear velocity. This relationship highlights the interconnected nature of linear and angular motion in circular dynamics.