A liquid having a coefficient of volume expansion γ is filled in a cylindrical vessel made out of glass having coefficient of linear expansion α = γ 5 . At room temperature, the level of liquid in the vessel is l 0 and when the temperature is increased by ∆ T , the level of liquid in the vessel becomes l ≈ l 0 1 + n α ∆ T . What is the value of n ? [Given, α ∆ T ≪ 1 ]

Question

A liquid having a coefficient of volume expansion γ is filled in a cylindrical vessel made out of glass having coefficient of linear expansion α = γ 5 . At room temperature, the level of liquid in the vessel is l 0 and when the temperature is increased by ∆ T , the level of liquid in the vessel becomes l ≈ l 0 1 + n α ∆ T . What is the value of n ? [Given, α ∆ T ≪ 1 ],

MARKS App · Accepted Answer

Let us assume that the volume of the liquid in the vessel is V 0 and the area of cross-section of the vessel is A 0 , then l 0 = V 0 A 0 Similarly, l = V A = V 0 1 + γ ∆ T A 0 1 + 2 α ∆ T ⇒ l = l 0 1 + γ ∆ T 1 + 2 α ∆ T - 1 Using binomial approximation, we get l = l 0 1 + γ ∆ T 1 - 2 α ∆ T ⇒ l = l 0 1 + γ - 2 α ∆ T - 2 α γ ∆ T 2 Neglecting 2 α γ ∆ T 2 , since it is insignificant as compared to γ - 2 α ∆ T , we get l ≈ l 0 1 + γ - 2 α ∆ T ⇒ l ≈ l 0 1 + 5 α - 2 α ∆ T ⇒ l ≈ l 0 1 + 3 α ∆ T So, n = 3

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