In the arrangement of three blocks as shown in the figure, the string is inextensible. If the directions of accelerations are as shown in the given figure, and $a_{1} + a_{2} = n a_{3}$ , then find $n$ .

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Solution

Let us assume the respective distance of each block as shown in figure. Since the length of the string is constant, therefore, $x_{1} + x_{2} + 2 x_{3} =$ constant
On differentiating twice w.r.t. time, we get
$\frac{d^{2} x_{1}}{d t^{2}} + \frac{d^{2} x_{2}}{d t^{2}} + 2 \frac{d^{2} x_{3}}{d t^{2}} = 0$
Since $x_{1}$ and $x_{2}$ are assumed to be decreasing with time, therefore,
$\frac{d^{2} x_{1}}{d t^{2}} = - a_{1}$ and $\frac{d^{2} x_{2}}{d t^{2}} = - a_{2}$
and $x_{3}$ is assumed to be increasing with time, therefore,
$\frac{d^{2} x_{3}}{d t^{2}} = + a_{3}$
Thus $- a_{1} - a_{2} + 2 a_{3} = 0$ or $a_{1} + a_{2} = 2 a_{3}$

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