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calculus

a 50 m long chain hangs vertically from a cylinder attached to a winch. assume there is no friction in the system and that the chain has a density of 5 kg/m. how much work is required to wind the entire chain onto the cylinder using the winch

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  1. Calculus question:
    Let x=length remaining below winch.
    and
    ρ=mass density
    Work done winding up a length of dx
    is equivalent to bringing the length dx up by a distance of x, i.e.
    dW=ρg xdx
    Integrate
    W=ρg∫xdx from 0 to L
    =ρg L²/2

    Physics question:
    Centroid of chain, X = -L/2
    mass of chain, M = ρL
    Total work done
    =MgX
    =ρgL(0-L/2)
    =ρgL²/2

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