A uniform heavy plank of length L is resting on two rods of identical length and area of cross-section, made up of different materials. It is observed that, to keep the planks horizontal, rod A has to be at the end of the plank and rod B has to be at a distance $\frac{L}{4}$ from centre of the plank as shown in figure.

In another scenario, the rods are kept between two rigid walls such that the separation between the walls is equal to the natural length of the rod. Now the temperature of both the rods is gradually increased in the same manner, and it is observed that thermal stress developed in rod A is always thrice the thermal stress developed in rod B. Ratio of coefficient of linear expansion of rod A to coefficient of linear expansion of rod B is. (Answer upto two digits after the decimal point)

Open in App

Solution

Let $T_{A} and T_{B}$ be tension developed in both the rods respectively.

By balancing vertical forces we get
$T_{A} + T_{B} = m g$

By balancing torque we get
$T_{A} \frac{l}{2} = T_{B} \frac{l}{4}$

So $T_{A} = m g / 3$ and $T_{B} = 2 m g / 3$

We know $Y_{A} (\frac{Δ l}{l}) = \frac{T_{A}}{A}$ and

$Y_{B} (\frac{Δ l}{l}) = \frac{T_{B}}{A}$

$⟹ \frac{Y_{B}}{Y_{A}} = \frac{T_{B}}{T_{A}} = 2$

Thermal stress in A is $Y_{A} (\frac{Δ l}{l}) = Y_{A} (\frac{Δ l}{l}) α_{A} Δ t$

Thermal stress in B is $Y_{B} (\frac{Δ l}{l}) = Y_{B} (\frac{Δ l}{l}) α_{B} Δ t$

Given thermal stress developed in A is thrice thermal stress developed in B. So

$⟹ Y_{A} α_{A} Δ t = 3 Y_{B} α_{B} Δ t$

$⟹ \frac{α_{A}}{α_{B}} = 3 \frac{Y_{B}}{Y_{A}} = 6$

Suggest Corrections

Similar questions

Q. A metal rod is fixed rigidly at two ends so as to prevent its thermal expansion. If

L, α

and

Y

respectively denote the length of the rod, coefficient of linear thermal expansion and Young's modulus of its material, then for an increase in temperature of the rod by

Δ T

, the longitudinal stress developed in the rod is:

Q. Two rods of different materials having coefficients of thermal expansion and Young's moduli

Y_{1}, Y_{2}

, respectively are fixed between two rigid massive walls. The rods are heated such that undergo the same increase in temperature. There is no bending of the rods. If

α_{1} : α_{2} = 2 : 3

, the thermal stresses developed in the two rods are equal provided

Y_{1} : Y_{2}

is equal to:

Q. Two rods identical in geometry but of different materials having coefficient of thermal expansion

α_{1} a n d α_{2}

and Youngs moduli

Y_{1} a n d Y_{2}

respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of the rods If

α_{1} : α_{2}

= 2 : 6, the thermal stresses developed in the two rods are equal provided

Y_{1} : Y_{2}

is equal to

Two rods of different materials, having coefficients of linear expansion $α_{1}$ and $α_{2}$ and Young's moduli $Y_{1}$ and $Y_{2}$ respectively, are fixed between two rigid walls. The rods are heated to the same temperature. There is no bending of the rods. If $α_{1} : α_{2} = 2 : 3$ , the thermal stress developed in the two rods will be equal provided $Y_{1} : Y_{2}$ is equal to

Q. Two rods of different materials having coefficients of linear expansion

α_{1}, α_{2}

and Young's modulii

Y_{1}

and

Y_{2}

respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. The rods are heated such that they undergo the same increases in temperature. There is no bending of rods. If

α_{1} : α_{2} = 2 : 3

. the thermal stress developed in the two rods are equal provided

Y_{1} : Y_{2}

is equal to-

Join BYJU'S Learning Program