Calculus- What is the work required?

Question

A cable that weight 1.5kg/meter and is attached to a bucket that weight 75kg. Initially there are 500kg of grain in the bucket and as the bucket is raised 2kg of grain leaks out of the hole in the bucket for every meter the bucket is raised. The bucket is 200meters below a bridge. How much work is required to raise the bucket to the top of the bridge?

Answer 1

Recall that the work done against uniform gravity when lifting is W = ∫ F dh = ∫ mg dh where the potential energy at ground level is taken to be 0.

 

Note how the mass of the bucket at height h is

 

m(h) = (75 + 500 - 2h)  kg = (575 - 2h) kg.

 

So the work done is

 

W = [9.81 ∫₀²⁰⁰ (575-2h) dh ] J = 9.81[[575h - h²]₀²⁰⁰ ] J

 

= 9.81[575·200 - 200²] J = 7.36 × 10⁵ J