Three rods A, B and C, having identical shape and size, are hinged together at ends to form an equilateral triangle. Rods A and B are made of same material having coefficient of linear thermal expansion a while that of material of rod C is $α_{2}$ . By how many kelvin must the system of rods be heated to increase the angle opposite to rod C by $Δ θ$ .

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A
When the system is heated, the rods expand and the triangle does not remain equilateral. Let lengths of rods A, B and C be $l_{1}$ , $l_{2}$ and $l_{3}$ respectively. Then

$c o s θ = \frac{l_{1}^{2} + l_{2}^{2} - l_{3}^{2}}{2 l_{1} l_{2}} o r 2 l_{1} l_{2} c o s θ = l_{1}^{2} + l_{2}^{2} - l_{3}^{2} \dots \dots (1) D i f f e r e n t i a t i n g e q . (1), w e g e t 2 l_{1} c o s θ d l_{2} + 2 l_{2} c o s θ d l_{1} - 2 l_{1} l_{2} s i n θ d θ = 2 l_{1} d l_{1} + 2 l_{2} d l_{2} - 2 l_{3} d l_{3} \dots \dots (2)$
Let the temperature of the system be increased by $Δ T$ . Then
$d l_{1} = l_{1} α_{1} Δ T d l_{2} = l_{2} α_{1} Δ T \dots \dots (3) a n d d l_{3} = l_{3} α_{2} Δ T I n c a s e o f e q u i l a t e r a l t r i a n g l e l_{1} = l_{2} = l_{3} = l (s a y) a n d θ = 60^{\circ} f r o m e q n . (2) a n d (3), w e g e t 2 l^{2} c o s 60 α_{1} Δ T + 2 l^{2} c o s 60 α_{1} Δ T \frac{α_{1}}{2} Δ T + \frac{α_{1}}{2} Δ T - \frac{\sqrt{3}}{2} d θ = α_{1} Δ T + α_{1} Δ T - α_{2} Δ T α_{1} Δ T - \frac{\sqrt{3}}{2} d θ = 2 α_{1} Δ T - α_{2} Δ T (α_{2} - α_{1}) Δ T = \frac{\sqrt{3}}{2} d θ Δ T = \frac{\sqrt{3} d θ}{2 (α_{2} - α_{1})}$

Suggest Corrections

Similar questions

Q. Three rods A, B and C, having identical shape and size, are hinged together at ends to form an equilateral triangle. Rods A and B are made of same material having coefficient of linear thermal expansion a while that of material of rod C is

α_{2}

. By how many kelvin must the system of rods be heated to increase the angle opposite to rod C by

Δ θ

Q. Two rods of length

L_{2}

and coefficient of linear expansion

α_{2}

are connected freely to a third rod of length

L_{1}

of coefficient of linear expansion

α_{1}

to form an isosceles triangle. The arrangement is supported on the knife edge at the midpoint of

L_{1}

which is horizontal. The apex of the isosceles triangle is to remain at a constant distance from the knife edge if

Two rods of different materials having coefficients of thermal expansion $α_{1}$ and $α_{2}$ and Young's modulii $Y_{1}$ and $Y_{2}$ are fixed between two rigid and massive walls. The rods are heated to the same temperature. If there is no bending of the rods, the thermal stresses developed in them are equal provided