Calc Leaky bucket problem

Question

A bucket is filled with sand so that the total mass (sand and bucket) is 15 kgs. The bucket is to be lifted to the top of a 10 meter tall building by a rope of negligible weight. However, the bucket has a hole in it, and leaks 0.1 kgs of sand each meter it is lifted.

Find the work done lifting the bucket to the top of the building.

Answer 1

The problem can be visualized as the sum of a large number of small lifts. The first lift is for a mass of 15 kg being lifted Δx, The second lift is for a slightly lower mass also being lifted Δx, the last lift is for a mass of

15 - 10(.1) = 14 kg being lifted Δx. The work associated the first lift is 15 g Δx etc.

This goes over to the integral ∫ g ( 15 - .1 x ) dx with limits 0 to 10

This works out to be 145 g