According to Hooke's Law, within the elastic limit, the stress applied to a material is directly proportional to the strain produced. Since strain is defined as the ratio of extension to original length, stress is also directly proportional to the extension of the material. This linear relationship allows for the calculation of Young's modulus as the constant of proportionality.

Density is defined as mass per unit volume. Comparing the materials listed, aluminum has a density of approximately 2.70 g/cm³, while copper is 8.96 g/cm³, lead is 11.34 g/cm³, and mercury is 13.53 g/cm³. Therefore, aluminum is the lightest substance among the options provided.

Young's modulus is an intrinsic material property that depends solely on the nature of the material itself, not on external factors like the applied load or the dimensions of the wire. Therefore, doubling the suspended mass will increase the stress and strain proportionally, but the ratio of stress to strain, which defines the Young's modulus, remains constant for a given material at a constant temperature.

The elastic potential energy density is given by the formula U = 1/2 * stress * strain. According to Hooke's Law, stress = Y * strain, which implies strain = stress / Y. Substituting this into the energy density formula yields U = 1/2 * S * (S/Y) = S^2 / 2Y. Note: The provided option C is S/2Y, which appears to be a typographical representation of S^2/2Y.

Young's Modulus (Y) is calculated as (Force × Original Length) / (Area × Change in Length). Substituting the values: Y = (10,000 N × 1 m) / (5×10⁻⁵ m² × 0.001 m) = 10,000 / 5×10⁻⁸ = 2×10¹¹ N/m².

According to Hooke's Law, for small deformations within the elastic limit of a material, stress is directly proportional to strain. The ratio of stress to strain is known as the modulus of elasticity (e.g., Young's modulus), which remains a constant value for a given material.

The energy stored per unit volume in an elastic material is the work done per unit volume, which is the integral of stress with respect to strain. Since stress = Y * strain, the energy density is the integral of (Y * strain) d(strain), which results in 1/2 * Y * strain^2. Given the strain is α, the energy density is Y * α^2 / 2. This formula is fundamental in solid mechanics for calculating strain energy.

Young's modulus is defined as the ratio of stress to strain. A perfectly rigid body does not undergo any deformation regardless of the applied stress, meaning the strain is zero. Since division by zero is undefined, the limit of the ratio as strain approaches zero is infinite.

Elasticity is defined by Young's modulus, which measures the resistance of a material to elastic deformation. Steel has a higher Young's modulus than rubber, glass, or copper, meaning it requires more stress to produce a given strain. Therefore, steel is considered more elastic because it returns to its original shape more effectively under high stress compared to the other materials listed.

Water exhibits anomalous expansion; when it freezes into ice, its molecular structure forms a hexagonal lattice that occupies more volume than liquid water. Since density is mass divided by volume, an increase in volume for the same mass results in a decrease in density. The provided answer 'C' is factually incorrect as density decreases.