In a double pulley system, if the mass of the object is 10 kg and the object is accelerating upwards at 2 m/s$^2$, what is the tension in the ropes?
I figured the weight of the box is 100N and the upward force is 20 N. Since these forces are in opposite direction I would think to subtract them and get 80N and divide it by the two ropes. 40N
However my textbook says
$$2T =m(a+g)\\
2T = 10kg (2m/s^2 + 10m/s^2)\\
2T = 10kg \cdot 12m/s^2\\
2T= 120N \\
T = 60N $$
Basically its saying that the 2 forces are added together. This does not make intuitive sense to me, even when looking at a diagram of a double pulley.
Can someone explain this to me.
Best Answer
This is kind of hard to describe without a diagram but anyway i will give it a shot. I am assuming you mean something like the fourth photo on this page: http://www.oldschool.com.sg/index.php/module/PublicAccess/action/Wrapper/sid/9595afb87c8cf767f034c3ae53e74bae/coll_id/4745/recs_ppg/5/desc/wrap-function/pg_id/3
If you consider the simpler case where the mass is held stationary. Due to the two contact points in the ceiling and the two ropes on the left we have halfed the force needed to hold the mass still.
To then account for the acceleration is just a simple application of newtons laws.
If you consider the work down to move the mass is always the same regardless of the pulley system, we reduce the amount of force needed with pulleys, but we need to pull the rope through a greater distance.