# Calculus

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(i) A 20m chain with a mass-density of 3kg/m (coiled on the ground). How
much work is performed lifting the chain so that it is fully extended (and
one end touches the ground)?

(ii) How much work is performed to lift 1/4 of the chain?

• Calculus -

This sounds like a physics problem:
Work done = increase in potential energy (PE)
Let
PE at ground = 0
After lifting, the centre of gravity of the chain is at h=10 m above ground, so
potential energy
=mgh
=(3 kg/m * 20m)*9.8 m/s²*10m
= 5880 joules

Using calculus,

Let
ρ = length density = 3 kg/m
x=distance of top of chain from ground in metres
then
length of chain off ground = x
mass of chain off ground = ρx
weight of chain off ground = ρxg
lifting the chain through a distnce dx does work of
ρxg*dx
Total work done through a length of 0 to L metres
=∫ρx;gdx [0,L]
=14.7L²
For L=20m,
Work done = 14.7(20²)=5880 joules
as before.

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