Problem 862 - JEE Physics

Question

If longitudinal strain for a wire is \(0.03\) and its poisson's ratio is \(0.5\), then what is its lateral strain ? (A) \(0.003\) (B) \(0.0075\) (C) \(0.015\) (D) \(0.4\)

Step-by-Step Solution

Answer

The lateral strain is 0.015 (C).

Step 1: Substitute the given values into the formula

We are given: Longitudinal Strain = 0.03 Poisson's Ratio = 0.5 The formula to find the lateral strain is: Lateral Strain = -Poisson's Ratio × Longitudinal Strain

Step 2: Calculate the lateral strain

Now, we will substitute the given values into the formula and find the lateral strain: Lateral Strain = -0.5 × 0.03

Step 3: Solve for the lateral strain

Multiplying the values, we get: Lateral Strain = -0.015 Since the lateral strain is negative, it means that the wire contracts laterally when it's stretched longitudinally. So, the lateral strain is -0.015 or 0.015 (in terms of magnitude). The correct answer is (C) 0.015.

Key Concepts

Longitudinal Strain

Longitudinal strain is a measure of the deformation experienced by an object in response to an applied force along its length. It is essentially the ratio of the change in length to the original length. When a force stretches or compresses an object, it undergoes a change in its dimensions. In this case, we focus on how the length of the material elongates or shrinks.

• Formula: \ Longitudinal Strain = \( \frac{\text{Change in Length}}{\text{Original Length}} \)
• Example: For a wire initially 100 cm long, if it stretches to 103 cm, the longitudinal strain is \( \frac{3}{100} = 0.03 \).

Longitudinal strain is a dimensionless quantity, meaning it has no units. Understanding this concept is crucial for solving problems involving materials experiencing tensile or compressive forces. It provides insights into how materials behave under such conditions.

Poisson's Ratio

Poisson's Ratio is a property of materials that provides a relationship between longitudinal and lateral strains. When a material is stretched, it not only changes in length but also in width (or lateral dimension). Poisson's Ratio helps in understanding this relationship, particularly how much a material expands or contracts in directions perpendicular to the force applied.

• Formula: \ Poisson's Ratio = \( - \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} \)
• Example: If a rubber band is stretched, it becomes thinner. If its longitudinal strain is 0.02 and Poisson's Ratio is 0.5, its lateral strain is calculated to be \( -0.01 \).

Poisson's Ratio typically ranges from 0 to 0.5 for most materials. A Poisson's Ratio of 0.5 is characteristic of incompressible materials that do not change in volume during deformation. Negative Poisson's Ratio, although rare, can also be observed in some novel materials, meaning they expand laterally when stretched.

Elasticity

Elasticity refers to a material's ability to return to its original shape and size after the force causing the deformation is removed. This property is vital for materials that undergo mechanical stress and strain. Materials with high elasticity, like rubber, can endure substantial deformation and revert to their initial state without permanent alteration.

• Elastic Limit: The maximum extent to which a material can be deformed and still return to its original shape once the load is removed.
• Hooke’s Law: Within the elastic limit, the deformation experienced by a material is proportional to the force applied to it, defined by the equation: \ \( F = k \times x \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement.
• Applications: Elasticity is employed in designing springs, shock absorbers, and other systems that must absorb and release energy effectively.

Elasticity is not just about flexibility; it also involves resilience and the ability to resist permanent deformation. It is a fundamental property exploited in numerous engineering and construction applications.