I am a bit tripped up on the following work problem --
A 30 ft long chain is hanging from one end on a hook, 25 feet above the ground; naturally, this means 5 feet of the hook are on the ground itself. The total weight of the chain is 120 lbs. How much work is required to life the other end of the chain to the same hook, so that both ends of the chain are 25 feet above the ground?
I have solved problems similar to this that didn't include the 5 trailing feet on the ground–but will this really affect the setup of the integral?
I set it up as follows :
$$\int_0^{30} 4x dx $$ ⇒
The chain on the ground starts out with zero potential energy, so the upper limit should be $25$. This gives you the starting value. At the end, the chain is doubled, so you can consider it to be two $15$ foot chains hanging from the hook. Where is the bottom?