Calculate the work required to lift a chain.

Question

A 4-m chain with linear mass density ρ(x) = 2x(4 − x) kg/m from x = 0 to x = 4 lies on the ground. Calculate the work required to lift the chain from its front end so that its bottom is 2 m above ground. (Assume that the chain is lifted from the end at x = 4. Assume that acceleration due to gravity is g = 9.8 m/s². Round your answer to one decimal place.) 

 J

Answer 1

I actually have a 15.2-meter, knotted rope as well a 4.4-meter chain; I took the chain outside and laid it in a straight line with the rope attached to one end of the chain. I placed that attached end at the foot of a 9.7-meter ladder standing against a tree and climbed roughly 7 meters up the ladder with the free end of the rope. With the attached chain-end directly below me, I started pulling the rope up, hand-under-hand.

I was surprised to see that the entire chain began to effortlessly move toward the point directly beneath me, practically gliding like a snake, and I could see no discernible signs of opposition or friction between the chain and the ground.

At the reference point directly below me, each link of the chain would lift almost vertically off the ground and climb at 90 degrees to the ground; there seemed no need to calculate sines and cosines of angles. When the entire chain was hanging vertically with the far end of the chain 2 meters above ground, every link of the chain having moved (4.4 + 2) meters, I felt as if the entire displacement of the chain had been vertical and through air.

In Physics, Work is described as a Force exerted or acting through a Distance. The Distance in this case would be 6 meters.

The Force (or Mass of the chain Times Gravity) is given by 9.8∫_{(from x=0 to x=4)}(8x − 2x²)dx equal to

9.8[4x² − (2/3)x³|_{(from x=0 to x=4)}] or 209.066666666 Newtons.

Finally, the Work sought is obtained by 209.066666666∫_{(from x=0 to x=6)}dx equal to

1254.4 "Newton-Dot-Meters" or Joules.