Question

Two thin and light rods $A$ and $B$ are attached to a block of mass $m$ with the help of a massless string as shown in the figure. Then identify the correct statement, if the angle between the rods is acute.

• A. Both $A$ and $B$ experience compression
• B. Both $A$ and $B$ experience extension
• C. $A$ experiences extension and $B$ experiences compression
• D. $B$ experiences extension and $A$ experiences compression

Answer 1

The correct option is C $A$ experiences extension and $B$ experiences compression

The tension in rods and string connecting the block is shown in figure given below.


F.B.D of block,


Let tension in the rods $A$ and $B$ be $T_A$ and $T_B$ respectively.
Applying equilibrium condition on block,
$T=mg....\left(i\right)$
Applying equilibrium condition for the point at junction of both rods.
For vertical equilibrium,
$T_A\sin \theta =T=mg...\left(ii\right)$
For horizontal equilibrium,
$\sum F_x=0$
$T_B+T_A\cos \theta =0$
$\Rightarrow T_B=-T_A\cos \theta ...\left(ii\right)$
From Eq.$\left(i\right)$,
$\Rightarrow T_B=-\frac{mg}{\sin \theta }\times \cos \theta$
$\Rightarrow T_B=-mg\cot \theta$
Since $\theta$ is acute angle i.e $\cot \theta$ will be positive and $mg$ is itself a positive quantity, therefore $T_B$ comes to be negative.

$\Rightarrow$Hence negative sign for $T_B$(tension in rod $B$), indicates that rod $B$ experiences compression. However the sign of tension in rod $A$ is $+ve$, so it will experience extension.