Pulleys & Forces Homework: Struggling to Understand

[FeelTheFire]

Homework Statement

Hey guys I'm really having problems understanding this despite looking at the solution, which is fully laid out here: http://iweb.tntech.edu/murdock/books/v1chap4.pdf

I know that m2 moves twice the distance that m1 moves. I have tried looking at each rope as a length that doesn't change made up of intervals that do change. I keep hitting a wall trying to come to the conclusion that 2a1 = a2.

I am doing pretty good understanding net forces, but this absolutely throws me for a loop. Any insight would be terrific.

 

[Legaldose]

The first thing you should always do in a mechanics problem is to draw a free body diagram for each mass. Your insight may show itself there.

 

[gneill]

Imagine that there are measurement rules positioned beside each mass for determining their positions and arranged so that at some given instant they both read zero (so they're synchronized). When the m1 rule reads some value, the other rule reads twice that. So the readings are related: x₂ = 2x₁. Differentiate both sides to find the velocity relationship. Differentiate again to find the acceleration relationship.

The thing is, once a displacement relationship is fixed it fixed the velocity and acceleration relationships with the same ratio. This applies anytime there are related rates.

 

[FeelTheFire]

The first thing you should always do in a mechanics problem is to draw a free body diagram for each mass. Your insight may show itself there.

Thanks for the reply. I have drawn a FBD for both pullies and both masses. Attached is as far as I got with what they tell me. *Edit* The m2g in the lower right corner should read m1g. My mistake.Gneill,
Thanks for the reply. The problem is I can't even get to that point. I only know that m2 moves twice as far as m1 because I looked at the solution. I can't, in mathematical terms, come to the conclusion that it displaces twice as far. I tried calling each length of rope it's own distance, with each changing but their sums always adding to the same number since the total rope length doesn't change. The problem I ran into was I kept getting circular answers, for instance Rope 1 length = Rope 1 length.

 

[gneill]

You just need to make more diagrams until you can picture what's happening. Suppose a pulley is supported by a loop of rope, one end tied to the ceiling and the other passes though a hole in the ceiling. Now imagine lifting the pulley out of its rope cradle by some unit distance. How much rope "slack" would you have to pull through the hole in the ceiling to get the rope seated against the pulley again?

You should be able to see that there are two units of slack that must be gathered in in order to re-seat the pulley. So the pulley moves 1 unit, the rope moves 2 units.

This is the same situation as with your pulley setup. The wheel of the pulley is displaced by half the amount of the rope end moves, or conversely, the rope moves twice as far as the pulley moves.