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Tension in a String

In the figure...

Question

In the figure shown, find the tension in the string connecting the blocks of mass $2 k g$ and $3 k g$ . Take $g = 10 m / s^{2}$

10 N

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12.4 N

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13.6 N

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15.2 N

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Solution

The correct option is C $13.6 N$
Considering $2 k g$ and $3 k g$ masses as a system, we have from the FBDs as

$f_{1} = μ_{1} (2 g) = 0.3 \times 20 = 6 N$
and, $f_{2} = μ_{2} (3 g) = 0.2 \times 30 = 6 N$
Also, equation of force can be written as
$5 g - T_{1} = 5 a$ ..........(i)
$T_{1} - (f_{1} + f_{2}) = 5 a$ ......(ii)
Thus, from eq (i) and (ii) we get
$5 g - (f_{1} + f_{2}) = 10 a$
$a = \frac{50 - (6 + 6)}{10} = \frac{38}{10} = 3.8 m / s^{2}$

Now, from the FBD of $2 k g$ mass we get

$T - f_{1} = 2 a$
$T = f_{1} + 2 a = 6 + 2 (3.8)$
$\Rightarrow T = 13.6 N$

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