Question

hi guys....solve it....​

Answer 1

As we know that :-

Coefficient of linear expansion = α

Coefficient of superficial expansion = β

Coefficient of volume expansion = γ

Now as we have got these :-

$\bigstar \sf { Linear \: expansion = l_0(1 + \alpha\triangle t ) }$

$\bigstar \sf { Superficial \: expansion = A_0(1 + \beta\triangle t ) }$

$\bigstar \sf { Volume \: expansion = V_0(1 + \gamma\triangle t ) }$

First :-

Relationship between α and β

We know that

Area = length × breadth

$\rightarrow A_0 = l_0 \times b_0$

And

$\rightarrow A_t = l_t \times b_t$

Also

$\rightarrow A_t =A_0(1 + \beta\triangle t )$

Now by further solving

$\rightarrow A_t = l_t \times b_t$

$\rightarrow A_t = l_0(1+\alpha\triangle t) \times b_0 (1+\alpha\triangle t)$

$\rightarrow A_t = l_0.b_0 (1+\alpha\triangle t)^2$

$\rightarrow A_t = A_0( 1 + 2\alpha\triangle t + \alpha^2\triangle t^2)$

Now as

$\alpha^2 \to 0$

$\rightarrow A_t = A_0( 1 + 2\alpha\triangle t )$

Now

$\rightarrow A_0( 1 + 2\alpha\triangle t ) = A_0(1 + \beta\triangle t )$

$\rightarrow 1 + 2\alpha\triangle t = 1 + \beta\triangle t$

$\rightarrow 2\alpha\triangle t = \beta\triangle t$

$\rightarrow 2\alpha = \beta$

$\rightarrow \alpha = \dfrac{1}{2} \beta$

Second :-

Relationship between α and γ

We know that

Volume = length × breadth × width

$\rightarrow V_0 = l_0 \times b_0 \times h_0$

And

$\rightarrow V_t = l_t \times b_t \times h_t$

Also

$\rightarrow V_t =V_0(1 + \gamma\triangle t )$

Now by further solving

$\rightarrow V_t = l_t \times b_t \times h_t$

$\rightarrow V_t = l_0(1+\alpha\triangle t) \times b_0 (1+\alpha\triangle t) \times h_0 (1+\alpha\triangle t)$

$\rightarrow V_t = l_0.b_0.h_0 (1+\alpha\triangle t)^3$

$\rightarrow V_t = V_0( 1 + 3\alpha\triangle t + 3\alpha^2 \triangle t^2 + \apha^3 \triangle t^3)$

Now as

$\alpha^2 \to 0 \\\\ \alpha^3 \to 0$

$\rightarrow V_t = V_0( 1 + 3\alpha\triangle t )$

Now

$\rightarrow V_0( 1 + 3\alpha\triangle t ) = V_0(1 + \gamma\triangle t )$

$\rightarrow 1 + 3\alpha\triangle t = 1 + \gamma\triangle t$

$\rightarrow 3\alpha\triangle t = \gamma\triangle t$

$\rightarrow 3\alpha = \gamma$

$\rightarrow \alpha = \dfrac{1}{3} \gamma$

So from above :-

$\huge{\boxed{\sf{\rightarrow \alpha = \dfrac{1}{2} \beta = \dfrac{1}{3} \gamma}}}$

Answer 2

Sorry sis not γ/2, its γ/3

Well the answer:

Alpha (α) is coefficient of linear expansion.

Increase in length during expansion is called linear expansion.

It means object one side increases.

Hence α = 1

Beta (β) is coefficient of superficial expansion.

Increase in area during expansion is called superficial expansion.

We know area = Length × Breadth

                       = Length × Length

                       = 2Length

Hence, β = 2

Gamma (γ) is coefficient of cubical expansion.

Increase in volume during expansion is called cubical expansion.

We know, volume = Length × Breadth × Height

                              = Length × Lenth × Length

                             = 3Length

Hence, γ  = 3

So, α=β/2=γ/3 (is proved)