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Answer 1

QUESTION:

A rod, of length L at room temperature and uniform area of cross section A, is made of a metal having coefficient of linear expansion α/°C. It is observed that an external compressive force F, is applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by ΔT K. Young's modulus, Y, for this metal is

ANSWER:

$\textsf{Option 3) \dfrac{F}{A\alpha \Delta T} }$

FORMULA:

$\textsf{\bf{Young's Modulus = \dfrac{Stress}{Strain} }}\\ \\\textsf{\bf{Thermal Strain}} = \textsf{\bf{ \dfrac{ \Delta\: l}{l} }} = \alpha\Delta t\\ \\\textsf{\bf{Strain = \dfrac{Force}{Area} }}$

EXPLANATION:

$Young's \: Modulus=\dfrac{ \dfrac{F}{A} }{\dfrac{ \Delta l}{l} } \\ \\Substitute \: \: \dfrac{ \Delta l}{l} = A \alpha\Delta t \\ \\ Young's \: Modulus = \frac{F}{A \alpha\Delta t}$

MORE INFORMATION:

• Young's modulus is a measure of the ability of a material to resist the changes while applying longitudinal compression.

• It is also called af modulus of elasticity.

• It is also defined as the ratio of longitudinal stress to the longitudinal strain.

• S.I unit of Y = N/m².

• Dimension = $[M\:\:L^{-1}\:\:T^{-2}]$

• S.I unit of Stress = N/m².

• Strain is a dimensionless quantity as it is the ratio of change in length to length.

Answer 2

Refer to the attached image.