Centripetal acceleration - why?

[Cloud 9]

Just a general "why" question. Why does the centripetal acceleration increase when the radius decreases? This is not a homework question but rather something I'm trying to make sense of. I read that: "The centripetal acceleration has to continuously change the velocity vector back towards the center of the circle to keep the object moving in a circle."

So shouldn't it be that when the distance from the circle is higher (radius), the centripetal acceleration is higher to change the velocity vector back towards the center?? That isn't the case though since: as the radius decreases, the centripetal acceleration increases...why is this so?

 

[Feldoh]

Imagine a ball traveling in a circle at a constant speed.

If we change the radius of this ball the only difference is that the direction of the ball will change at a different rate. Also, remember that a change in direction is an acceleration. So a ball can be accelerating even at a constant speed IF it is changing direction.

It appeals to the mind that if the ball is on a shorter radius the direction will change faster (and hence) the acceleration will be larger than if the ball were really far away. This makes sense because the distance the ball has to travel to complete one full circle is a lot less if the radius is smaller.

 

[Cloud 9]

Thanks, that makes more sense. That sentence is just worded weirdly I guess. I'm terrible at physics anyway, give me a chemistry or organic chemistry equation and I'm all on it. lol...well thank you again, I appreciate the help!

 

[gomunkul51]

centripetal acceleration = tangential velocity squared / radius

are you familiar with that ?

 

[rwisz]

I can provide an explanation for this phenomenon. First, we need to understand what centripetal acceleration is. Centripetal acceleration is the acceleration that is directed towards the center of a circular path. It is responsible for keeping an object moving in a circular motion.

Now, let's consider the equation for centripetal acceleration: a = v^2/r, where v is the velocity of the object and r is the radius of the circular path. This equation tells us that as the radius decreases, the centripetal acceleration increases.

To understand why this is the case, we need to look at the relationship between velocity and radius. As an object moves in a circular path, its velocity is constantly changing, as it is always directed tangent to the circle. This change in velocity is what causes the object to accelerate towards the center of the circle.

When the radius is larger, the object has a longer distance to travel in a given time period, which means the velocity is lower. This results in a lower centripetal acceleration. However, when the radius decreases, the object has a shorter distance to travel in the same time period, which means the velocity is higher. This results in a higher centripetal acceleration.

In simpler terms, as the object moves in a smaller circular path, it needs to travel a shorter distance in the same amount of time, which requires a higher velocity. And since acceleration is directly proportional to velocity, a higher velocity results in a higher centripetal acceleration.

In conclusion, the reason why the centripetal acceleration increases when the radius decreases is because the object needs to travel a shorter distance in the same amount of time, which requires a higher velocity and therefore a higher centripetal acceleration.