Question

Three rods of equal length f are joined to form an equilateral triangle PQR. 0 is the mid point of PQ. Distance OR remains same for small change in tem-perature. Coefficient of linear expansion for PR and RQ is same i.e. ${\alpha }_2$ but that for PQ is ${\alpha }_1$. Then

• A. ${\alpha }_2=3{\alpha }_1$
• B. ${\alpha }_1=4{\alpha }_2$
• C. ${\alpha }_1=3{\alpha }_2$
• D. ${\alpha }_2=4{\alpha }_1$

Answer 1

The correct option is B ${\alpha }_1=4{\alpha }_2$

from pythagoras theorem:

$R{O}^2=R{Q}^2-Q{O}^2$

$={\ell }^2-{\left(\frac{\ell }{2}\right)}^2$

$={\left[\ell (1+{\alpha }_{2}t)\right]}^2-{\left[\frac{\ell }{2}(1+{\alpha }_{1}t)\right]}^2$

${\ell }^2-\frac{{\ell }^2}{4}={\ell }^2(1+{\alpha }_2^2{t}^2+2{\alpha }_{2}t)-\frac{{\ell }^2}{4}(1+{\alpha }_1^2{t}^2+2{\alpha }_{2}t)$

$0={\ell }^2\left(2{\alpha }_{2}t\right)-\frac{{\ell }^2}{4}\left(2{\alpha }_{1}t\right)$

$2{\alpha }_2=2\frac{{\alpha }_1}{4},{\alpha }_1=4{\alpha }_2$