I went to my home town during my vacations, and found that it was easier to lift a bucket of water with pulley as compared to that without pulley. Can anyone explain it mathematically assuming that pulley and rope is massless and frictionless?

• It's a question about resolution of forces. When you pull things at an angle, you are only pulling a component of the net force compared to working directly against gravity. Have a look at this explainthatstuff.com/pulleys.html – xCodeZone Oct 13 '16 at 5:26
• @xCodeZone: When you pull things at an angle, you are only pulling a component of the net force compared to working directly against gravity. No, that isn't true. Whatever the angle the rope 'transmits' the full force of gravity. By your reasoning if you pulled horizontally, you wouldn't have to pull at all! Your link is about multiple pulleys, I don't think that's what the OP has in mind. – Gert Oct 13 '16 at 5:46
• You very likely experienced the mechanical advantage of a pulley system, which is explained with diagrams here.. Succinctly, if there are $n$ sections of rope supporting a load $W$, the rope tension is $W/n$. – chandra Oct 13 '16 at 7:30
• It's worth noting that there are two ways to use a single pulley for this job. One is to attach it to the bucket and get a mechanical advantage (which does not require that the pulley be either massless or frictionless, but does put a limit of how much friction can be present for it still to be a win) or it can be a fixed overhead pulley used to change the direction of work needed to one better suited to the way humans are built. The math for the second case is non-trivial because it depends on details of human anatomy. – dmckee Oct 13 '16 at 17:08

The only thing that changes is the direction of the force needed. Without a pulley, you need to exert an upward force $F=mg$ on the bucket to lift it.
Using a ceiling mounted pulley you need to exert a downward force $F=mg$ on the rope to lift the bucket.