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Acceleration in 2D

Two weights ...

Question

Two weights $w_{1}$ and $w_{2}$ are suspended from the ends of a light string passing over a smooth fixed pulley. If the pulley is pulled up at an acceleration g, what will be the tension in the string?

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Solution

We know weight= mass $\times$ gravity;thus weight pf object 1 $= \frac{M a s s (M_{1})}{G r a v i t y (g)}$ and
weight of object 2 = $= \frac{M a s s (M_{2})}{G r a v i t y (g)}$
$M_{2} g - T = M_{2} a - g$ --------(A)
$- M_{1} g + T = M_{1} a - g$ -------(B)
(A)- (B)
$2 M_{2} g - 2 M_{1} g = M_{2} a + M_{1} a a = \frac{2 g (M_{2} - M_{1})}{(M_{1} + M_{2})}$ -------(C)
Now tension in string is
T ${= M}_{1} g + M_{1} a + M_{1} g = 2 M_{1} g + M_{1} a$
Reducing the value of mass of obejtc 1 we get,
$= \frac{W_{1}}{g} {2 g + \frac{2 g (M_{2} - M_{1})}{(M_{1} + M_{2})}} = W_{1} {\frac{2 (M_{2} + M_{1}) + 2 (M_{2} - M_{1})}{(M_{1} + M_{1})}} = W_{1} {\frac{4 \frac{W_{2}}{g}}{\frac{W_{1}}{g} + \frac{W_{2}}{g}}} = \frac{4 W_{1} W_{2}}{W_{1} + W_{2}}$

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