Two rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature will be (α1, α2= linear expansion coefficient A1, A2= Area of rods Y1, Y2= Young modulus)

Question

Two rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature will be (α1, α2= linear expansion coefficient A1, A2= Area of rods Y1, Y2= Young modulus) (A) (A1/A2)=(α1 Y1/α2 Y2) (B) (A1/A2)=(L1 α1 Y1/L2 α2 Y2) (C) (A1/A2)=(L2 α2 Y2/L1 α1 Y1) (D) (A1/A2)=(α2 Y2/α1 Y1)

Tardigrade Admin · Accepted Answer

Y=( text Stress / text Strain )=(T / A/Δ l / l) T=(Y ⋅ Δ l/l) A=Y ⋅ A α Δ T In both the rods tension will be same so T1=T2, Hence Y1 A1 α1 Δ T =Y2 A2 α2 Δ T (A1/A2)=(Y2 α2/Y1 α1)

Q. Two rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature will be ( $α_{1}, α_{2} =$ linear expansion coefficient
$A_{1}, A_{2} =$ Area of rods
$Y_{1}, Y_{2} =$ Young modulus)

$\frac{A _{1}}{A _{2}} = \frac{α _{1} Y _{1}}{α _{2} Y _{2}}$

$\frac{A _{1}}{A _{2}} = \frac{L _{1} α _{1} Y _{1}}{L _{2} α _{2} Y _{2}}$

$\frac{A _{1}}{A _{2}} = \frac{L _{2} α _{2} Y _{2}}{L _{1} α _{1} Y _{1}}$

$\frac{A _{1}}{A _{2}} = \frac{α _{2} Y _{2}}{α _{1} Y _{1}}$

$Y = \frac{Stress}{Strain} = \frac{T / A}{Δ l / l}$
$T = \frac{Y \cdot Δ l}{l} A = Y \cdot A α Δ T$
In both the rods tension will be same so
$T_{1} = T_{2}$ , Hence $Y_{1} A_{1} α_{1} Δ T$
$= Y_{2} A_{2} α_{2} Δ T$
$\frac{A _{1}}{A _{2}} = \frac{Y _{2} α _{2}}{Y _{1} α _{1}}$

Q. Two rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature will be (α1​,α2​= linear expansion coefficient A1​,A2​= Area of rods Y1​,Y2​= Young modulus)

A2​A1​​=α2​Y2​α1​Y1​​

A2​A1​​=L2​α2​Y2​L1​α1​Y1​​

A2​A1​​=L1​α1​Y1​L2​α2​Y2​​

A2​A1​​=α1​Y1​α2​Y2​​