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An ATWOOD's Machine

M2 is shown to be bigger, but it isn't

M2 is shown to be bigger, but it isn't

Shown above is an ATWOOD's Machine. The question is: Given m1 = 6 KG and m2 = 4 KG, what is the acceleration of the system?

The first step is to consider the acceleration at each point separately. Let's start at m1. We know that the net force = the mass times acceleration (Fnet = ma), so we can plug in what we already know. The net force is the force of gravity subtracted by the tension from the other mass (it pulls M1 back up). So it would look like this:

Fg - T = ma
6(9.8) - T = 6a
T = 58.8 - 6a

We can do the same thing for the other block, now we've already established the tug-of-war going on between gravity and tension. Since m1 > m2, we can assume that tension is losing to gravity on m1's side, and tension is winning on the m2 side. So the equation becomes:

T - Fg = ma
T - 4(9.8) = 4a
T = 39.2 + 4a

Now, by setting them equal to each other, you can find the tension.

T = 39.2 + 4a
T = 58.8 - 6a
58.8 - 6a = 39.2 + 4a
10a = 19.6
a = 1.96 m/s^2

Now, which way is it going? This is very simple to check. Looking back at the beginning, we assumed the Fg was greater than Tension, and if that was the case, the acceleration would be positive. The acceleration was positive, meaning Fg won, and the 6 KG mass is accelerating down at 1.96 m/s^2.

David Witten

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