Question

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

• A. A breaks if ${\text{r}}_{\text{A}}={\text{r}}_{\text{B}}$
• B. A breaks if ${\text{r}}_{\text{A}}<{\text{r}}_{\text{B}}$
• C. Either A or B may break if ${\text{r}}_{\text{A}}=2{\text{r}}_{\text{B}}$
• D. The lengths of A and B must be known to predict which wire will break

Answer 1

Answer: (A), (B), (C)

The correct options are
A

A breaks if ${\text{r}}_{\text{A}}={\text{r}}_{\text{B}}$

B

A breaks if ${\text{r}}_{\text{A}}<{\text{r}}_{\text{B}}$

C

Either A or B may break if ${\text{r}}_{\text{A}}=2{\text{r}}_{\text{B}}$

Stress on wire A,

${\sigma }_{\text{A}}=\frac{F_A}{\pi r_A^2}=\frac{mg+\frac{mg}{3}}{\pi r_A^2}$

$=\frac{4\text{mg}}{3\pi r_A^2}$

Stress on wire B,

${\sigma }_{\text{B}}=\frac{F_B}{\pi r_B^2}=\frac{\frac{mg}{3}}{\pi r_B^2}$

$=\frac{\text{mg}}{3\pi r_B^2}$

If $r_A\le r_B,{\sigma }_A>{\sigma }_B$ and A will break. Therefore options (a) and (b) are correct. If $r_A=2r_B$, then ${\sigma }_A={\sigma }_B$. Therefore either A or B may break.