Solve T(y)-T(y+dy)=ug(dy) | Easier Method?

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To solve the equation T(y) - T(y+dy) = ug(dy), one can interpret it as a differential equation where dT = -ug dy represents the change in tension. The simplest approach involves considering the mass of the chain segment below a given point y. By integrating this differential equation, one can derive an expression for T(y). The discussion emphasizes the importance of understanding the relationship between tension and the mass per unit length of the chain under gravity. Ultimately, integrating the equation provides a straightforward method to find T(y).
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Homework Statement
Find an expression for T(y) which is the tension at the point y along the chain.

Chain is length L, take the top of the chain hanging vertical to be 0, and the length from top to bottom be L. T(y) tension at point y down the chain. T(y+dy) tension at the point y+dy, which is below y. It has mass per unit length u. The section from T(y) down to T(y+dy) supports the weight ug(dy), which has mass u(dy) as g is just gravity. There is also a force 'f' supporting the chain from falling.

Using this information, find an expression for T(y).
Relevant Equations
T(y)-T(y+dy)=ug(dy)
T(y)-T(y+dy)=ug(dy) is what I have got. How would I solve this? Or is there a simpler method.
 
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jayjackson said:
Homework Statement:: Find an expression for T(y) which is the tension at the point y along the chain.

Chain is length L, take the top of the chain hanging vertical to be 0, and the length from top to bottom be L. T(y) tension at point y down the chain. T(y+dy) tension at the point y+dy, which is below y. It has mass per unit length u. The section from T(y) down to T(y+dy) supports the weight ug(dy), which has mass u(dy) as g is just gravity. There is also a force 'f' supporting the chain from falling.

Using this information, find an expression for T(y).
Relevant Equations:: T(y)-T(y+dy)=ug(dy)

T(y)-T(y+dy)=ug(dy) is what I have got. How would I solve this? Or is there a simpler method.
You haven’t provided the full problem statement, but I gather this just a chain hanging vertically.
The simplest way would be just to consider the mass of the chain below a given point. But from the equation you got you can easily take limits to get an integral.
 
haruspex said:
You haven’t provided the full problem statement, but I gather this just a chain hanging vertically.
The simplest way would be just to consider the mass of the chain below a given point. But from the equation you got you can easily take limits to get an integral.
im unsure on how to solve the T(y)-T(y+dy) equation. How do i go about it?
 
jayjackson said:
im unsure on how to solve the T(y)-T(y+dy) equation. How do i go about it?
T(y+dy)-T(y) is the change in T, which is written dT. So your equation is dT=-ug dy.
Do you know how to integrate?
 
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