In the arrangement shown, inclined plane is smooth, strings and pulleys are massless. If T2T1=xy Where x and y are smallest possible integers find x+y

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In the arrangement shown, inclined plane is smooth, strings and pulleys are massless. If $\frac{T_{2}}{T_{1}} = \frac{x}{y}$ Where $x a n d y$ are smallest possible integers find $x + y$

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answer is 13.

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Detailed Solution

5 kg : along the plane: 50 $\sin 30^{0} - T_{1} = 5 a$ ……….(i)
3 kg : along the plane: 30 $\sin 30^{0} + T_{1} - T_{2} = 3 a$ …………(ii)
2 kg: $↑ : T_{2} - 20 = 2 a$ ………………………………………………(iii)
Adding Eqs. (i), (ii) and (iii), we get
$20 = 10 a \Rightarrow a = 2 m s^{- 2}$
$T_{1} = 15 N$ and $T_{2} = 24 N$
$∴ \frac{T_{1}}{T_{2}} = \frac{15}{24} = \frac{5}{8}$