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Calculate the compressional force requir...

Calculate the compressional force required to prevent the metallic rod of length l cm and cross - sectional area `A cm^(2)` when heated through `t""^(@)C`, from expanding lengthwise. Young's modulus of elasticity of the metal is E and mean coefficient of linear expansion is `alpha` per degree celsius.

Text Solution

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The change in natural length = `Delta1_(t)=1alphat`
The natural length of rod at temperature `t^(@)C` is `1+lalphat`
The decrease in natural length due to developed stress = `Deltal`
But the length of rod remains constant.
`thereforeDelta1_(t)-Delta1=0" "thereforeDelta1=Delta1_(t)=1alphat`
`thereforeE=("stress")/("strain")=(F/A)/((-DeltaL)/(l+Deltal_(t)))`
`thereforeF=(EADeltal)/(l+Deltal_(t))=(EADeltal_(t))/(l+Deltal_(t))=(-EA1alphat)/(l+lalphat)=-(EAalphat)/((1+alphat))` Here, negative sign indicates that the forces is compressive in nature.
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