Question

An iron bar of length 10 m is heated from 0°C to 100°C. If the coefficient of linear thermal expansion of iron is 10 X 10^-6 / °C , the increase in the length of bar is
(a) 0.5 cm (b) 1.0 cm (c) 1.5 cm (d) 2.0 cm

PLEASE GIVE STEP BY STEP ANSWER

Answer 1

Answer :

• Increase in length of the iron bar is 1 cm.

Explanation :

Given :

• Coefficient of linear expansion, $\sf{\alpha = 10 \times 10^{-6}{}^{\circ}C^{-1}}$
• Initial length of the iron bar,$\sf{\it{l}_{1} = 10\:m}$
• Initial temperature of the iron bar,$\sf{\theta_{1} = 10^{\circ}C}$
• Final temperature of the iron bar,$\sf{\theta_{2} = 100^{\circ}C}$

To find :

• Increase of length of the iron bar,$\sf{\Delta l = ?}$

Knowledge required :

Coefficient of linear expansion :

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\boxed{\bf{\alpha = \dfrac{\Delta l}{l_{1}(\theta_{2} - \theta_{1})}}}$

Where,

• $\sf{\alpha}$ = Coefficient of linear expansion.
• $\sf{\it{l}_{1}}$ = Initial length of the iron bar.
• $\sf{\Delta l}$ = Increase length of the iron bar.
• $\sf{\theta_{1}}$ = Initial temperature of the iron bar.
• $\sf{\theta_{2}}$ = Final temperature of the iron bar.

Solution :

By using the equation for linear expansion and substituting the values in, we get :

$:\implies \bf{\alpha = \dfrac{\Delta l}{l_{1}(\theta_{2} - \theta_{1})}} \\ \\ :\implies \bf{10 \times 10^{-6} = \dfrac{\Delta l}{10(100^{\circ} - 0^{\circ})}} \\ \\ :\implies \bf{10 \times 10^{-6} = \dfrac{\Delta l}{10 \times 100^{\circ}}} \\ \\ :\implies \bf{10 \times 10^{-6} = \dfrac{\Delta l}{1000^{\circ}}} \\ \\ :\implies \bf{10 \times 10^{-6} \times 10^{3} = \Delta l} \\ \\ :\implies \bf{10^{-6 + 3 + 1} = \Delta l} \\ \\ :\implies \bf{10^{-2} = \Delta l} \\ \\ :\implies \bf{10^{-2} = \Delta l} \\ \\ \boxed{\therefore \bf{\Delta l = 10^{-2} m\:or\:1 cm}}$

Therefore,

• Final length of the iron bar,$\sf{\it{l}_{2} = 1 cm}$