Centripetal acceleration and tangential acceleration

[Maxo]

When having a circular acceleration motion, we have both tangential acceleration and centripetal acceleration.

The tangential acceleration is a_T=r*α where α=1/2*(ω_f-ω₀). So we can see tha a_T is dependant on both the initial angular velocity ω₀ and the final ω_f).

For centripetal acceleration, we instead have a_C=r*ω_f².

My question is, how come the centripetal acceleration is only dependant on the final angular velocity, and not the initial?
Is there a physical explanation for this?

 

[dauto]

First: α isn't 1/2*(ω_f-ω₀). The correct equation is α = (ω_f-ω₀)/(t_f-t₀).

Second: The equation above is the expression for the AVERAGE angular acceleration. If the circular motion is UNIFORMLY accelerated than you can use the average angular acceleration to calculate the tangential speed since angular acceleration is constant, otherwise you have to use the instantaneous angular acceleration - say α_f and the expression for tangetial acceleration becomes a_Tf = r α_f

 

[Maxo]

Ok thanks for the correction. But I still wonder why the centripetal acceleration is only dependant on the final angular velocity, and not the initial angular velocity. Is there a physical explanation for this?

 

[namanjain]

centripetal acceleration is caused to continue circular motion so at every point. so at that point it has to accelerate the mass toward center so as to change it direction.
now for at the point it has angular velocity omega then centripetal acceleration will vary with that only

 

[PJC01]

I can explain the difference between tangential and centripetal acceleration and why they are dependent on different variables.

First, let's define what these two types of acceleration are. Tangential acceleration is the rate of change of tangential velocity, which is the velocity along the circular path. This means that tangential acceleration is responsible for changing the speed of an object moving in a circular path. On the other hand, centripetal acceleration is the acceleration towards the center of the circular path. It is responsible for keeping the object moving in a circular path by constantly changing its direction.

Now, let's look at the equations for tangential and centripetal acceleration. As you mentioned, tangential acceleration is given by aT = r*α, where r is the radius of the circular path and α is the angular acceleration. This equation tells us that tangential acceleration is dependent on both the radius of the circle and the angular acceleration.

On the other hand, centripetal acceleration is given by aC = r*ωf^2, where ωf is the final angular velocity. This equation shows us that centripetal acceleration is only dependent on the final angular velocity and not the initial. This is because, in a circular motion, the final angular velocity is the same as the initial velocity, since the object is constantly moving at a constant speed along the circular path.

To understand why this is the case, we need to look at the forces acting on the object. In circular motion, there are two main forces at play - tangential force and centripetal force. The tangential force is responsible for changing the speed of the object, while the centripetal force is responsible for changing its direction.

When an object starts moving in a circular path, there is no tangential force acting on it, and the only force is the centripetal force. This force is what causes the object to move in a circular path. As the object moves, the tangential force increases, and the centripetal force decreases until they reach a balance, and the object moves at a constant speed along the circular path. This is why the final and initial angular velocities are the same, and the centripetal acceleration is only dependent on the final velocity.

In conclusion, the difference in the dependence of tangential and centripetal acceleration on initial and final angular velocities can be explained by the forces acting on the object in circular motion. While tang