Question:
Two bars $A$ and $B$ of circular cross-section and of same volume and made of the same material are subjected to tension. If the diameter of $A$ is half that of $B$ and if the force applied to both the rods is the same and it is in the elastic limit, the ratio of extension of $A$ to that of $B$ will be
Two bars $A$ and $B$ of circular cross-section and of same volume and made of the same material are subjected to tension. If the diameter of $A$ is half that of $B$ and if the force applied to both the rods is the same and it is in the elastic limit, the ratio of extension of $A$ to that of $B$ will be
Updated On: Jun 7, 2022
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The Correct Option is A
Solution and Explanation
Young modulus, $Y = \frac{FV}{A^2 \Delta l} $
$Y, F$ and $V$ are constants
$\therefore \:\:\:\:\:\: \Delta l \propto \frac{1}{A^2}$
$ \therefore \:\:\: \frac{\Delta l_A}{\Delta l_B} = \frac{D^4_B}{D^4_A} = \frac{D^4_B}{\left( \frac{1}{2} D_B \right)^4} = 16$
$Y, F$ and $V$ are constants
$\therefore \:\:\:\:\:\: \Delta l \propto \frac{1}{A^2}$
$ \therefore \:\:\: \frac{\Delta l_A}{\Delta l_B} = \frac{D^4_B}{D^4_A} = \frac{D^4_B}{\left( \frac{1}{2} D_B \right)^4} = 16$
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Concepts Used:
Mechanical Properties of Solids
Mechanical properties of solids intricate the characteristics such as the resistance to deformation and their strength. Strength is the ability of an object to resist the applied stress, to what extent can it bear the stress.
Therefore, some of the mechanical properties of solids involve:
- Elasticity: When an object is stretched, it changes its shape and when we leave, it retrieves its shape. Or we can say it is the property of retrieving the original shape once the external force is removed. For example Spring
- Plasticity: When an object changes its shape and never attains its original shape even when an external force is removed. It is the permanent deformation property. For example Plastic materials.
- Ductility: When an object is been pulled in thin sheets, wires or plates, it will be assumed that it has ductile properties. It is the property of drawing into thin wires/sheets/plates. For example Gold or Silver
- Strength: The ability to hold out applied stress without failure. Many types of objects have higher strength than others.