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)

and increase in tangential acceleration means increase in 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)

so velocity also increase with increase in angular velocity

and

centripetal acceleration=(velocity)²/r

and centripetal acceleration also increase with increase in velocity

To address your question about the effects of constant tangential acceleration on a particle in uniform circular motion, let's break it down step by step. The key concepts here are tangential acceleration, centripetal acceleration, and how they interact in circular motion.

Understanding Circular Motion

In uniform circular motion, a particle moves along a circular path at a constant speed. However, even though the speed is constant, the direction of the velocity vector is continuously changing, which means there is always a centripetal acceleration directed towards the center of the circle. This centripetal acceleration is given by the formula:

• Centripetal Acceleration (a_c) = v²/r

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

Introducing Tangential Acceleration

Now, when we introduce a constant tangential acceleration (a_t), we are essentially changing the speed of the particle as it moves along the circular path. Tangential acceleration affects the magnitude of the velocity but not its direction. The relationship between tangential acceleration and angular acceleration (α) is given by:

• α = a_t/r

This means that if you have a constant tangential acceleration, the angular acceleration will also be constant, leading to an increase in angular velocity over time.

Effects on Velocity and Centripetal Acceleration

As the angular velocity increases due to the constant tangential acceleration, the linear velocity (v) of the particle also increases, as described by:

• v = r × ω

where ω is the angular velocity. Since the linear velocity is increasing, we can substitute this into the centripetal acceleration formula:

• a_c = v²/r

As v increases, the centripetal acceleration will also increase because it is proportional to the square of the velocity.

Conclusion on Centripetal Acceleration

So, to answer your question directly: yes, the centripetal acceleration will increase when you apply a constant tangential acceleration to a particle in circular motion. This is because the tangential acceleration leads to an increase in the particle's speed, which in turn increases the centripetal acceleration due to the relationship defined by the formula a_c = v²/r.

Visualizing the Concept

Think of it like this: if you’re riding a merry-go-round and you start pushing off with your legs (applying tangential acceleration), you not only go faster but also feel a stronger pull towards the center as you speed up. This is the essence of how tangential acceleration affects centripetal acceleration in circular motion.