Please Help!
A chain 63 meters long whose mass is 27 kilograms is hanging over the edge of a tall building and does not touch the ground. How much work is required to lift the top 11 meters of the chain to the top of the building? Use that the acceleration due to gravity is 9.8 meters per second squared. Hint: Don't forget that when you lift the top 11 meters of the cable you are also lifting the bottom 52 meters of the cable, just not all the way to the top.
Best Answer
Because energy is conserved in this problem (no friction), you can either integrate the force with respect to distance, or else calculate the change in potential energy. Both are correct; the latter is often easier.
For the force approach, note that when $x$ meters of chain have been pulled up, there are $63-x$ meters remaining, with a mass of $27(1-x/63)=27-(3/7)x$ kilograms, and so you're exerting a force of $(27-(3/7)x)\times(9.8$ kg-m/s$^2$) along the direction of motion. Integrating this gives $$ \frac{W}{9.8J}=\int_{x=0}^{11}F(x)dx=\int_{x=0}^{11}\left(27-\frac{3x}{7}\right)dx=\left(27x-\frac{3x^2}{14}\right)\Bigg\vert_{x=0}^{11}=\frac{3795}{14}, $$ or $W=2656.5$ kg-m$^2$/s$^2$.
I'll leave the potential energy approach as an exercise.