STEP I 1998 question 11 solution

TSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP I 1998 question 11 solution

---

Hank's Gold Mine has a very long vertical shaft of height l. A light chain of length l passes over a small smooth light fixed pulley at the top of the shaft. To one end of the chain is attached a bucket A of negligible mass and to the other a bucket B of mass m. The system is used to raise ore from the mine as follows. When bucket A is at the top it is filled with mass 2m of water and bucket B is filled with mass λm of ore, where 0 < λ < 1. The buckets are then released, so that bucket A descends and bucket B ascends. When bucket B reaches the top both buckets are emptied and released, so that bucket B descends and bucket A ascends. The time to fill and empty the buckets is negligible.

1. Find the time taken from the moment bucket A is released at the top until the first time it reaches the top again.


2. This process goes on for a very long time. Show that, if the greatest amount of ore is to be raised in that time, then λ must satisfy the condition f'(λ) = 0 where
   

Part 1
We start with A at the top and B at the bottom. To find the time taken for A to fall we first need to find the acceleration of the buckets. As the pulley is smooth the tension T is the same all along the chain. A has mass 2m and B has mass m(1+λ), and we know A accelerates downwards because 0 < λ < 1. Thus we get simultaneous equations, one for each bucket:



and



Adding these two equations eventually gives



Putting this into the equation of motion



and setting s = l, u = 0 and solving for time gives



Notice the labelling of t as this is just the first part of the motion.
Doing the same thing for the second part of the motion (notice that as A has no mass, the acceleration is simply g), we get



and so




Part 2
This part is very easy once you figure out exactly what you are looking for. You want to maximise the rate of ore raised and so the function you want to maximise is :



In differentiating this function to find the maximum we equate the derivative to zero and so the becomes irrelevant, so we can get rid of it without changing the answer. At the maximum, f'(λ) = 0 where

---