Two light wires $A$ and $B$ shown in the figure are made of the same material and have radii $r_{A}$ and $r_{B}$ respectively. The block between them has a mass $m$ . When the force $F$ is $m g / 3$ , one of the wires breaks.

A

breaks if

r_{A} = r_{B}

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A

breaks if

r_{A} < 2 r_{B}

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Either

A

B

may break if

r_{A} = 2 r_{B}

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The lengths of

A

and

B

must be known to predict which wire will break.

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Solution

The correct options are
A $A$ breaks if $r_{A} < 2 r_{B}$
B $A$ breaks if $r_{A} = r_{B}$
C Either $A$ or $B$ may break if $r_{A} = 2 r_{B}$
Tension in wire A is $T_{A} = 4 m g / 3$
Tension in wire B is $T_{B} = m g / 3$
Now stress in wire A is $S_{A} = \frac{T_{A}}{π r_{A}^{2}}$
Stress in wire B is $S_{B} = \frac{T_{B}}{π r_{B}^{2}}$
$S_{A} = \frac{4 m g / 3}{π r_{A}^{2}}$
$S_{B} = \frac{m g / 3}{π r_{B}^{2}}$
Now wire A will break if $S_{A} > S_{B}$
which is true when $r_{A} < 2 r_{B}$
when $r_{A} = 2 r_{B}$
$S_{A} = S_{B}$
Therefore, either of the wire can break.

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