Two equations for centripetal acceleration

[AARMA]

My question is how can Ac (centripetal acceleration) be both inversely related to radius and directly related in two equations:
1) Ac = V²/r
2) Ac = 4pi²r/T²
In the first equation Ac is inversely proportional to radius in the second one Ac is directly proportional to radius. Why and how is that so?

 

[Dadface]

It is inversely proportional to r for constant v and directly proportional to r for constant T.

 

[AARMA]

Dadface
Can you please elaborate on that explanation.. can you go more in depth, I really want to understand.

 

[D H]

Yet another expression is a=rω², from which your second expression arises, AARMA. Another expression related to this expression for acceleration is that for velocity, v=rω. Thus v²=r²ω². Dividing by r yields rω², the equation for acceleration.

 

[Vagn]

Do you know the equation for the arc length, when you know the radius and the angle tended by the arc?

S = r \theta

If you take the time derviative of this, you get v = r \omega
When you substitute this into the the equation for centripetal acceleration then you get
a = \omega^2 r

When \omega is constant, then you can use the definition \omega = \frac{2 \pi }{T}
which gives you the equation you wanted when you substitute in.

 

[Dadface]

Hello AARMA,in each of the two equations you presented there are two possible variables r being one of the variables and featuring in both equations.In the first equation the other possible variable is v and A is inversely proportional to r but only if v does not vary and is constant.In the second equation T is the second variable and A is directly proportional to r but only if T is constant.There are other ways to express the equation as shown by DH.

 

[Dale]

Hi AARMA, you might find it instructive to find the relationship between v and T. If that relationship depends on r then it makes sense that the power of r would change in the two expressions. In fact, once you have the relationship between v and T you should be able to substitute it into one of the above expressions and get the other.

 

[AARMA]

Thank you for the help. This is what I wrote for my extra credit on a test and got only +1

The reason why centripetal acceleration (Ac) is inversely related to radius (r) in the equation:
Ac = V²/r and directly related to the radius in the equation: Ac = 4pi²r/T² is because in the first equation if you keep the velocity constant the bigger the radius the slower the object is going to move and the smaller the centripetal acceleration is going to be because that object is going to change its direction at a slower rate.
On the other hand, in the second equation if the period (T) is kept constant then the bigger the radius the bigger the velocity is going to get and thus the bigger the centripetal acceleration.
I couldn't use the angular velocity explanation because we learned nothing of that concept so I just tried explaining it in simple terms. Can you tell me what could have I added to this explanation to make it better?