Why is one Tension Negative and the Other Positive - Atwood Machines

[Blakeasd]

Why is one Tension Negative and the Other Positive -- Atwood Machines

Hello,

Please consider the following:

I am trying to understand how to solve for acceleration when dealing with Atwood machines. I keep getting hung up on one part: the direction of tension. In the image above, why is tension in the sum of the forces for m₂ negative when if I were to draw a force diagram it is pointing up. Furthermore, why is gravity positive? Tension is pulling the masses up and gravity is pulling them down. A force diagram's directions are not different between m₁ and m₂, so why are they different when summing the equations.

Many Thanks.

 

[dauto]

Obviously the author is using coordinates where the positive axis point down for the mass 2. for the mass 1 they are using a different coordinate axis where the positive side points upward as usual. This is common choice designed to keep the acceleration positive for both masses. If the axis had been chosen with the positive end pointing up (as you expected), the equation would be $$T - m_2 g = - m_2 a.$$ That equation is identical to the one found by the author except that the whole equation was multiplied by -1. That, of course, makes no difference at all for the solution.

 

[Blakeasd]

I see, thanks. But why is T - m₂g = to a negative m₂a ?

What is the reasoning behind m₂a being negative. If when dealing with the first mass ma is positive; what makes the difference here?

 

[dauto]

I see, thanks. But why is T - m₂g = to a negative m₂a ?

What is the reasoning behind m₂a being negative. If when dealing with the first mass ma is positive; what makes the difference here?

The difference is that the mass 2 is accelerating downwards which has a negative component in the vertical direction (If we chose upwards as positive). The acceleration, just as the forces, may have negative components. That's easy to forget. That's why it is common to chose axis such that the acceleration is positive avoiding having to deal with negative accelerations.

 

[Blakeasd]

The difference is that the mass 2 is accelerating downwards which has a negative component in the vertical direction (If we chose upwards as positive).

Yes, but isn't mass 1 accelerating downwards as well. It also is being affected by gravity. Why then is it equal to positive ma and not negative ma. I expect it to be negative, but that is not the case. Am I missing something obvious?

Thanks!

 

[Doc Al]

Yes, but isn't mass 1 accelerating downwards as well.

They can't both be accelerating downward. Mass 2 accelerates down while mass 1 accelerates up.

 

[Blakeasd]

They can't both be accelerating downward. Mass 2 accelerates down while mass 1 accelerates up.

I see!

Thank you.

 

[dauto]

Yes, but isn't mass 1 accelerating downwards as well. It also is being affected by gravity. Why then is it equal to positive ma and not negative ma. I expect it to be negative, but that is not the case. Am I missing something obvious?

Thanks!

Yes, you're missing the obvious fact that if one mass goes down than the other mass must go up

 

[AlephZero]

There are two different ways to attack this sort of problem.

The picture in the OP uses the principle that "we know which way everything moves, so let's draw a picture where all the quantities are positive."

The other way says, "We don't know what happens, because we haven't solved the problem yet. So let's measure everything positive in the same direction (e.g. upwards), and if some things turn out negative, that's not a big deal." Using this method, because the string is a fixed length, if m1 has acceleration a, m2 has acceleration -a.

You can use either method, but the mixing them up usually leads to mistakes.