# Rope on an inclined plane problem

My book says the answer is (a)zero but i don't understand how it came zero. What will the acceleration if horizontal level of the two ends of the rope are different?

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work out the acceleration of the 2 parts as if you cut the rope on the tip of the wedge – ratchet freak Feb 3 '14 at 10:53

There's a few ways to think about this but I'll try the one that I hope will be clearest from a conceptual point of view.

Imagine that the rope has uniform mass distribution and that the middle of the rope has been placed on the tip of the wedge. This means that there are equal masses of rope on either side of the wedge. Now imagine the two extremes.

1) $\alpha = \beta = 0$:

Clearly there will be no acceleration here, the string is just laying flat on the surface.

2) $\alpha = \beta = 90^{\circ}$:

Both sides of the string are now just hanging vertically, and they both have the same mass, so there cannot be any difference in acceleration between the two sides.

Now, these example have one very important hidden characteristic. The ends of the string fall to the same vertical position.

We can imagine, an example where the two angles ($\alpha$ and $\beta$) are not equal, with the middle of the string still placed on the tip of the wedge.

Let's say, for example, that $\beta = 90^{\circ}$ and $\alpha < \beta$. In this case, the acceleration felt by the right side of the string will just be $g$, and and the acceleration felt by the other end (the acceleration down the plane) will be $gsin(\alpha)$. In this case there WILL be difference in the acceleration felt by the two sides of the string (and therefore a greater force on one side, because the masses of the two sides are equal) (in the example, the right side will feel greater acceleration and therefore the string will fall to the right side of the wedge).

BUT what did we change from our original scenario? With both sides having strings of length $L/2$, but different angles of inclination, one of the ends of the string (the left end) is now HIGHER off the surface than the other! Shifting the string over to the left so that the two ends of the sting are at the same height off the ground would increase the mass on the side that has less acceleration which means that the force on that side ($F = ma$) will increase, and the force on the other side would decrease...

Now, I won't give the whole thing away, but there's a fairly simple way (using only the information we've used in these examples) to prove that if the two ends of the sting are at the same vertical height off the surface, the net acceleration on the string will be zero. But I'll leave that to you.

Hope this helped!

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