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Why are the two acceleration considered as a1 and a2 here? What is the reason for this even though the tension of both the strings are same.enter image description here

T is equal to 5gTcos 53 for left side and 5gTcos37 for the right side

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    $\begingroup$ How much lifting work is being done on m1 and m2 to raise their potential energy? Consider the extreme cases where plane 1 was horizontal and plane 2 was vertical? (I presume the sliding surfaces are frictionless.) $\endgroup$ – Transistor Jan 23 at 14:21
  • $\begingroup$ @Transistor the sliding surfaces are frictionless. In your extreme case , the mass m1 and m2 will definitely move. $\endgroup$ – srijan Sri Jan 23 at 14:22
  • $\begingroup$ @Transistor is it because the components of gravity will be different right $\endgroup$ – srijan Sri Jan 23 at 14:23
  • $\begingroup$ @Transistor any idea on how can we find acceleration for 6kg mass $\endgroup$ – srijan Sri Jan 23 at 14:24
  • $\begingroup$ I haven't done this stuff for many, many years. Start with your initial conditions. Edit your question to show what the initial value of T is. $\endgroup$ – Transistor Jan 23 at 14:25
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As posed, you have three unknowns - $a1$, $a2$, and $a3$. The tension on the cable is everywhere uniform, but the accelerations are different because the static force components affecting the tension are different.

$F=ma$ at each mass. So for mass $m1$, the tension is
$m1*(g*cos53 +a1)$. Likewise
$m2*(g*cos37 + a2)$ and
$1/2*m3*(g*cos90 +a3)$.

These are all equal. And the length of the line doesn't change.

You can assume that you start from a stationary condition (all velocities are initially zero) - that doesn't change the accelerations. You can choose any initial length that like - that doesn't change the accelerations either.

So $m1(g*cos53 + a1) = 1/2 m3(g*cos90 + a3) = m2(g*cos37 +a2) $ from equal tension.

And $a1 = 2*a3 - a2$ from constant length.

$ 5 kg (0.600g + a1) = 3 kg (1g + a3) = 5 kg ( 0.799g + a2)$
$0.01 kg + 5 kg * a1/g = 3 kg * a3/g = 0.993 kg + 5 kg * a2/g$
$0.0033 + 1.666 a1/g = a3/g = 0.331 + 1.666 a2/g$

$0.0033 + 1.666 (2a3 - a2)/g = a3/g$
$0.0033 - 1.666 a2/g = -2.332 a3/g$
$-0.0014 + 0.7144 a2/g = a3/g $

$-0.0014 + .7144 a2/g = 0.331 + 1.666 a2/g$
$-0.3324 = .9516 a2/g$

$a2 = -0.349 g$
$a3 = -0.2509 g$
$a1 = -0.1529 g$

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