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I found this problem online while practicing and the answer is supposed to be D. I tried using the answer to solve for $T$ and I got $T=\frac{gm_1m_2}{m_1+m_2}$. The numerator is what's really confusing me, because I can't think of any situation where you would multiply the two masses together. My answer was $\frac{T-gm_1}{g}$.

My rationale was that the downward force of $m_2$ is $gm_2$, but $m_1$ is also exerting an upward force of $gm_1$. More rigorously, I found the net acceleration of the entire system to be $$\frac{gm_2}{m_1+m_2}$$ I then subtracted that from gravity to find the counteracting acceleration caused by $m_1$. Then I multiplied that by the mass of the system to find the counteracting force exerted by $m_1$:

$$(g-\frac{gm_2}{m_1+m_2})(m_1+m_2)$$Then I added that force to $gm_2$ to find the tension of the cord:

$$gm_2+(g-\frac{gm_2}{m_1+m_2})(m_1+m_2)=\\gm_2+g(m_1+m_2)-gm_2=\\g(m_1+m_2)$$

So we have $T=g(m_1+m_2)$

I've been spending days on this and can't find the mistake. Thank you in advance for any feedback.

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closed as off-topic by user191954, David Z Oct 27 '18 at 4:59

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I will tell you what to do. First find the net acceleration of $m_2$ using the free body diagram. The upward force is $T$ and downward force is $m_2g$. $m_1$ does not exert force $m_1g$ on $m_2$. Force on $m_1$ is only T. Find the acceleration of $m_1$. Then equate the values of acceleration and solve for $m_2$

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