Why does tension across a rod rotating horizontally vary when it is rotating about one end?
Consider any point of the rod at a distance x from center of rotation - The tension exerted at that point is responsible for the circular motion of the rest of the rod ( l - x ) beyond that point .
At different points along the rod , the mass of l-x rotated varies and hence , the tension required for that mass' ( l - x ) circular motion changes too , being max at the point closest to the axis of rotation .
I hope you can understand this .
Oh you mean, each part needs to have the same acceleration despite having different masses. So tension has to be different. Is that correct?
Let mass of rod be m . Let it's total length be l . Consider a point at a distance x from the center of the rod . If you now consider the whole mass of length l-x to be one system , then the tension acting on that would be :
Mass of system * distance of Com of system from axis of rotation * w∧2 .
=(M* (l-x) / l )*( x + (l-x)/2 )*w∧2 .
Read this patiently and you will see that tension varies with x .
The acceleration along a rotating rod isn't constant.
What prevents you from considering parts of equal mass?
So the tension at any point..does that act as the centripetal force for the entire rod behind that point..or does it act as the centripetal force just for that point?
For the entire part behind it .
But if the tension is at x, then for the rod behind it of length l-x, as centripetal acceleration is ω2x, how can we assume that x is same for all points in the rod behind the point at which tension was calculated?
Please refer to the post of mine in which I calculate Tension . Centripetal acceleration is not (w∧2)*x .
What you want to do is to focus on a small segment of the rod between x and x + dx. The mass of this segment is mdx/L, where L is the length of the rod and m is its mass. If you do a force balance on this segment of the rod, you obtain:
Note that the tension T decreases with radial position x.
Yes. So what you are saying is that every point has different centripetal acceleration. So how can tension at one point play the role of centripetal force for all points behind it? Isn't it supposed to be just for that point only?
While true, it's not the key to variable tension. You should first try to understand a rod accelerated linearly by pulling one end. Here, the acceleration is constant along the rod, but the tension still varies.
What else would provide the centripetal force to the rest of the rod beyond that point?
I don't quite follow what you are saying here. I know you are aware that a geometric point has zero mass, so no net force is required to accelerate it. If you focus on a segment of the rod between x and x + dx, it does have mass, and the net force acting on it is T(x+dx) - T(x). So the tension is changing along the rod to provide a net force on every differential segment. If you add up the centripetal forces necessary to accelerate all the segments of the rod outboard of location x, you get T(x):
Hope this makes sense.
I got what I was looking for. Basically, that tension force isn't acting on the the particle at x or x + dx etc. Instead it is acting at the COM of the remaining part of the rod so the rod gets treated as a single point and hence we have the centripetal force equation for a single particle and not the other way round like: one tension force and different particles have different x so which x do we put in the centripetal force equation, which is what was confusing me initially. Makes sense now. :)
Not really, tension acts at the inner end of the remaining part of the rod.
Of course it does. But when we want to write the centripetal force equation, we say the mass is concentrated at the COM...and so we just have one x instead of many x's all along the rod.
So Andy, when you apply this algorithm to get the tension at location x, T(x), do you get the same answer as I obtained in post #13 by integrating the differential equation that takes into account the "many x's all along the rod?"
When we say Com , we mean the Com of the part on which tension acts as a centripetal force .
Chet , yes you do .
Could you please tell me how to quote somebody ?
Ah. I see it now. Thanks. Still, I would feel uncomfortable doing the problem this macroscopic way, and not by the microscopic balance.
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I initially solved it using ∫ . I got the answer to be the same as that due to centripetal acceleration of the Com , and then reasoned it out using the basic formula of (M*A)com = dm1 * w∧2 *x1 + .... and saw that it gives the same result ( Probably because of linear relationship between Fc and radius of circular motion ).
Ah - It works . Thank you .
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