A 3-m chain with linear mass density p(x)=?kg/m lies on the ground. Calculate the work required to lift the chain until it's fully extended.
My question is that, is the work that lift the chain from bottom equal to the work that lift the chain from top?
My understanding is that if the density is a constant, then the works are equal.
For example, if the p(x)=3.
The work is below
If the density is a variable, for example, $p(x)=2x(4-x)$, then the works are not equal.
[Math] Get the work required to lift a chain
calculus
Best Answer
The work is dependent on the density distribution. Let's assume two extreme cases. We have one chain of length $L$, and all the mass $M$ is concentrated on one end of the chain (let's call this the heavy side). If you lift the chain from the heavy side until the light side just barely touches the ground, the work done is $MgL$. If you lift from the other side until the heavy side just barely reaches the ground, the work done is $0$.