Question

Match the values of column II with column \(\mathrm{I}\) and mark the appropriate choice. $$ \begin{array}{|l|l|l|l|} \hline {\text { Column I }} & && {\text { Column II }} \\ \hline \text { (A) } & \begin{array}{l} \text { The shape of rubber heel } \\ \text { changes under stress } \end{array} & \text { (p) } & \begin{array}{l} \text { Young's } \\ \text { modulus of } \\ \text { elasticity is } \\ \text { involved } \end{array} \\ \hline \text { (B) } & \begin{array}{l} \text { In a suspended bridge, } \\ \text { there is a strain in the } \\ \text { ropes by the load of the } \\ \text { bridge } \end{array} & \text { (q) } & \begin{array}{l} \text { Bulk modulus } \\ \text { of elasticity is } \\ \text { involved } \end{array} \\ \hline \text { (C) } & \begin{array}{l} \text { In an automobile tyre, } \\ \text { when air is compressed, } \\ \text { the shape of tyre changes } \end{array} & \text { (r) } & \begin{array}{l} \text { Modulus of } \\ \text { rigidity is } \\ \text { involved } \end{array} \\ \hline \text { (D) } & \begin{array}{l} \text { A solid body is subjected } \\ \text { to a deforming force } \end{array} & \text { (s) } & \begin{array}{l} \text { All the moduli } \\ \text { of elasticity } \\ \text { are involved } \end{array} \\ \hline \end{array} $$ (a) \((A) \rightarrow(i),(B) \rightarrow(i i),(C) \rightarrow(\) iv),\((D) \rightarrow(\) iii) (b) (A) \(\rightarrow\) (iii), (B) \(\rightarrow\) (i), (C) \(\rightarrow\) (ii), (D) \(\rightarrow\) (iv) (c) (A) \(\rightarrow\) (ii), (B) \(\rightarrow\) (iii), (C) \(\rightarrow\) (iv), (D) \(\rightarrow\) (i) (d) \((\mathrm{A}) \rightarrow(\mathrm{i}),(\mathrm{B}) \rightarrow\) (iii), (C) \(\rightarrow\) (ii), (D) \(\rightarrow\) (iv)

Short answer

(a) (A) \rightarrow(i),(B) \rightarrow(i),(C) \rightarrow(ii),(D) \rightarrow(iii)

Step-by-step solution

Identify Types of Elasticity

Recognize that Column I describes scenarios involving deformation of materials, and Column II lists the types of modulus of elasticity that correspond to each deformation. Young's modulus (p) applies to unidirectional tension or compression, the bulk modulus (q) applies to uniform compression, the modulus of rigidity (r), also called shear modulus, applies to changes in shape at constant volume, and (s) indicates that all moduli of elasticity are involved.

Match (A) with Column II

The change in shape of a rubber heel under stress relates to a change in length and not volume, thus it involves Young's modulus of elasticity. Therefore, (A) matches with (p).

Match (B) with Column II

In a suspended bridge, the ropes experience tension due to the load, this is associated with Young's modulus. Therefore, (B) matches with (p).

Match (C) with Column II

When air is compressed inside an automobile tire, it leads to a change in volume without a change in shape, indicating that the bulk modulus of elasticity is involved. Therefore, (C) matches with (q).

Match (D) with Column II

A solid body subjected to a general deforming force can involve all types of deformation including tension, compression, and shear. This means all the moduli of elasticity are involved. Therefore, (D) matches with (s).

Choose the Correct Option

Combining all the matches: (A) matches with (p), which is (i) in the provided options. (B) matches with (p), which is (i) in the provided options. (C) matches with (q), which is (ii) in the provided options. (D) matches with (s), which is (iii) in the provided options. Therefore, the correct match is (a).

Key concepts

Young's Modulus

Young's modulus, or the modulus of elasticity, is a measure of the stiffness of a material. It's a parameter that represents how a material deforms under longitudinal stress—this includes stretching and compressing along one axis.

Mathematically, Young's modulus, often denoted by the symbol \( E \), is defined by the ratio of stress (force per unit area) to strain (proportional deformation) in the range of stress where Hooke's Law holds. This is expressed as \( E = \frac{\text{stress}}{\text{strain}} = \frac{\text{force} / \text{area}}{\text{(change in length)} / \text{(original length)}} \).

When it comes to understanding problems related to Young's modulus, such as the change in shape of a rubber heel under stress, it's crucial to think about how the material is being deformed. In this case, the rubber heel's length changes under tension, but not its volume, highlighting the involvement of Young's modulus. To improve the understanding of this concept, consider everyday examples like stretching a rubber band - the rubber band gets longer (stretch) and thinner (compression), which is a classic representation of Young's modulus at work.

Bulk Modulus

Bulk modulus is a measure of a material's resistance to uniform compression. It is a fundamental property that indicates how compressible a substance is.

The bulk modulus (\( K \)) is defined by the formula \( K = -V \frac{\text{dp}}{\text{dV}} \), where \( p \) is the pressure, \( V \) is volume, and \( \frac{\text{dp}}{\text{dV}} \) represents the rate of change of pressure with volume. A negative sign is used because an increase in pressure typically leads to a decrease in volume.

Using the automobile tire as an example from the exercise, when air is pumped into the tire, it compresses and increases in pressure. This compression without shape deformation indicates the bulk modulus is at work. Real-world applications also include the compression of fluids and the volumetric elasticity of solid materials. To aid student understanding, visualize squeezing a sealed syringe partially filled with liquid—the syringe itself doesn't change shape, but the liquid inside compresses.

Modulus of Rigidity

The modulus of rigidity, also known as the shear modulus or torsion modulus, relates to a material's ability to resist deformation under shear stress—this means forces applied tangentially or parallel to the face of a material.

Its mathematical definition is given by \( G = \frac{\text{shear stress}}{\text{shear strain}} = \frac{\text{force} / \text{area}}{\text{tangential displacement} / \text{initial length}} \). Essentially, it measures how a material deforms when subjected to shear forces, such as when a material is twisted.

This modulus explains why certain materials can withstand heavy loads without changing their volume. While modulus of rigidity wasn't directly applicable in the given exercise, understanding its role is vital for engineering applications, such as in the design of shafts in mechanical systems or the twisting of metal beams.

Deformation of Materials

Deformation of materials is the change in shape or size of an object due to an applied force or a change in temperature. There are different types of deformation, including elastic, plastic, and viscous deformation, each with their own characteristics and governed by various physical properties.

Elastic deformation is temporary and the material returns to its original shape when the force is removed—this is closely related to Young's modulus, the bulk modulus, and the modulus of rigidity. Plastic deformation is permanent and occurs when the material is stretched beyond its elastic limit. Viscous deformation is typically seen in fluids and occurs over time.

In the context of the exercise, a solid body subjected to a deforming force involves all types of deformation and therefore all the moduli of elasticity are applicable. This is an essential concept for engineers and scientists, as understanding the various types of deformation can aid in predicting material behavior under different stress conditions and designing materials and structures accordingly.