Question

In the situation given, all surfaces are frictionless. If $F=\frac{Mg}{2}$, the ratio of accelerations of blocks $A$ and $B$ is $\frac{n}{\sqrt{5}}$. Find the value of $n$. (Both the blocks have mass $M$)

Answer 1

Step 1: Given Data

Mass of the blocks $=M$

Given that force $F=\frac{Mg}{2}$

Let the acceleration of block $A$ be $a_1$.

Let the acceleration of block $B$ be $a_2$.

$\frac{a_1}{a_2}=\frac{n}{\sqrt{5}}$

Step 2: Acceleration of Block $A$

The force applied is acting against the motion of the block.

For block $A$ the equation of motion can be written as,

$F-M{a}_1=M{a}_1$

$\frac{Mg}{2}=2M{a}_1$

$a_1=\frac{g}{4}$

Step 3: Acceleration of Block $B$

Similarly, force applied is against the weight of the block.

For block $B$ the equation of motion can be written as,

$F–Mg=M{a}_2$

$\Rightarrow \frac{Mg}{2}-Mg=M{a}_2$

$\Rightarrow -\frac{Mg}{2}=M{a}_2$

$\Rightarrow a_2=-\frac{g}{2}$

Hence, $a_2$ is in a downward direction.

Step 4: Ratio of the Accelerations

Therefore, the ratio of the accelerations are,

$\frac{a_1}{a_2}=\frac{n}{\sqrt{5}}$

$\Rightarrow \frac{{\displaystyle \raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}{-{\displaystyle \raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=\frac{n}{\sqrt{5}}$

$\Rightarrow \frac{1}{2}=\frac{n}{\sqrt{5}}$

$\Rightarrow n=\frac{\sqrt{5}}{2}$

Hence, the value of $n$ is $\frac{\sqrt{5}}{2}$.