A chain lying on the ground is 10 m long and its mass is 90 kg. How much work (in J) is required to raise one end of the chain to a height of 4 m? (Use 9.8 m/s2 for g.)

Question

A chain lying on the ground is 10 m long and its mass is 90 kg. How much work (in J) is required to raise one end of the chain to a height of 4 m? (Use 9.8 m/s² for g.)

Answer 1

We want to lift one end of the chain from the ground to a height of 4 m. As we lift, the weight incrementally increases. Let y be the height of the link. After lifting, part of the chain will still be puddled on the ground.

We know that Work = weight×displacement. Weight is mass×gravity or mg.

First we need to find the density of the chain per meter. The entire chain has a mass of 90 kg, so the density, ρ = 90/10 = 9 kg/m. The incremental work:

w_i = ρgyΔy

Where y is the height of a section of the chain donated as Δy. As Δy approaches zero, it becomes dy.

So, to evaluate the work to raise one end of the chain to 4 meters, add up all the chain sections by integrating.

W=∫₀⁴ ρgydy = ρg[y²/2 ]₀⁴

You can finish the calculation by using ρ=9 and g = 9.8