A composite bar has two segments of equal length $L$ each. Both segment are made of same material but cross sectional area of segment OB is twice that of OA. The bar is kept on a smooth table with the joint at the origin of the co-ordinate system attached to the table. The temperature of system is $θ^{°} C$ Now the temperature is increased by $∆ θ^{°} C$ . The junction of the rods will shift from its initial position by $\frac{L α ∆ θ}{n}$ Then the value of n is

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answer is 6.

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Detailed Solution

Mass of OA= m and Mass of OB = 2m
Let the joint shift by x to left.
The COM of the composite rod will not move.
The COM of segment OA will move to left by $x + \frac{L α Δ θ}{2}$
The COM of segment BO will move to right by $\frac{L α Δ θ}{2} - x$
For COM of the composite rod to remain unmoved, we must have
$m (x + \frac{L α Δ θ}{2}) = 2 m (\frac{L α Δ θ}{2} - x)$
$\Rightarrow 3 x = \frac{L α Δ θ}{2} \Rightarrow x = \frac{L α Δ θ}{6}$
$∴$ x co-ordinate of joi9nt is $- \frac{L α Δ θ}{6}$