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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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 a1+a2=na3, then find n.

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 $n$ .