A steel ring of radius r and cross-sectional area A is...

A steel ring of radius $r$ and cross-sectional area $A$ is fitted on a wooden disc of radius $R(R>r)$. If Young's modulus be $E$, then the force with which the steel ring is expanded is (a) $A E \frac{R}{-}$ (b) $A E \frac{(R-r)}{r}$ (c) $\frac{E}{A}\left(\frac{R-r}{A}\right)$ (d) $\frac{E r}{A R}$

A steel ring of radius $r$ and cross-sectional area $A$ is fitted on a wooden disc of radius $R(R>r)$. If Young's modulus be $E$, then the force with which the steel ring is expanded is
(a) $A E \frac{R}{-}$
(b) $A E \frac{(R-r)}{r}$
(c) $\frac{E}{A}\left(\frac{R-r}{A}\right)$
(d) $\frac{E r}{A R}$

Key Concepts

Young's Modulus

Young's modulus is a fundamental property of materials that measures the stiffness of an elastic material. It relates the stress (force per unit area) experienced by a material to the strain (relative deformation) induced in it by that stress, assuming the deformation is within the elastic limit.

Hooke's Law

Hooke's law is a principle in elasticity stating that, for small deformations, the stress in a material is directly proportional to the strain. This linear relationship, expressed as stress = Young's modulus × strain, underpins much of the analysis in elastic deformation problems.

Stress and Strain in Elastic Materials

In the context of elasticity, stress is the force applied over a unit area and strain is the resulting deformation expressed as the ratio of the change in dimension to the original dimension. The relationship between these two quantities, mediated by Young's modulus, allows engineers to determine the forces required to achieve a given deformation.

Circumferential Strain in Circular Structures

For circular structures such as rings, circumferential or hoop strain is defined as the change in circumference divided by the original circumference. This strain results from an expansion or contraction of the ring’s radius and is critical when analyzing forces resulting from interference fits or other expansion scenarios.

Interference Fit

An interference fit is a design technique where two components are assembled with a deliberate dimensional mismatch, requiring one part to deform elastically in order to be fitted. The resulting stresses and strains, calculated using properties such as Young's modulus and the geometric relationship of the parts, determine the force required to achieve the assembly.

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