Problem 966 - JEE Physics

Question

Assertion and Reason: Read the assertion and reason carefully to mark the correct option out of the option given below (A) If both assertion and reason are true and reason is the correct explanation of the assertion. (B) If both assertion and reason are true but reason is not the correct explanation of the assertion. (C) If assertion is true but reason is false. (D) If assertion and reason both are false. Assertion: Young's modulus for a perfectly plastic body is zero. Reason: For a perfectly plastic body restoring force is zero. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) d

Step-by-Step Solution

Answer

The correct option is (A) because both the assertion and reason are true, and the reason correctly explains the assertion. Young's modulus for a perfectly plastic body is indeed zero, as there is no restoring force after deformation, which leads to no relationship between stress and strain.

Step 1: Understand the Young's modulus

Young's modulus (Y) is a measure of the stiffness of a material, quantifying the relationship between stress (force per unit area) and strain (deformation) for an elastic material. It is defined by the formula: \[ Y = \frac{stress}{strain} = \frac{F/A}{\Delta L/L} \] where: - Y is the Young's modulus - F is the force applied to the material - A is the cross-sectional area of the material - ΔL is the change in length of the material - L is the original length of the material Step 2: Understand the behavior of a perfectly plastic body

Step 2: Understand the behavior of a perfectly plastic body

A perfectly plastic body is a hypothetical material that undergoes permanent deformation under stress without requiring any additional force. In other words, once the yield strength is reached, the material deforms at a constant stress without increasing the force necessary to sustain this deformation. For a perfectly plastic body, there is no restoring force that allows the material to return to its original shape. Step 3: Evaluate the assertion

Step 3: Evaluate the assertion

The assertion states that the Young's modulus for a perfectly plastic body is zero. This is true because the perfectly plastic body experiences no restoring force after deformation, and therefore there is no relationship between stress and strain, resulting in a Young's modulus of zero. Step 4: Evaluate the reason

Step 4: Evaluate the reason

The reason states that for a perfectly plastic body, the restoring force is zero. This is also true, as we have previously described the behavior of a perfectly plastic body and established that it has no restoring force, allowing it to return to its original shape after deformation. Step 5: Choose the correct option

Step 5: Choose the correct option

Since both the assertion and reason are true and the reason explains the assertion (the restoring force being zero is the cause of the Young's modulus being zero), the correct option is (A).

Key Concepts

Perfectly Plastic Body

A perfectly plastic body is a fascinating concept in physics and materials science. Imagine a material that can undergo a change in shape or form without bouncing back to its original state. This is a perfectly plastic body. When stress is applied to this kind of material, it deforms permanently once the stress exceeds a certain threshold, known as the yield point. The unique thing about a perfectly plastic body is that it doesn't need more force to keep deforming once the yield point is reached.

• It behaves in a way where the stress level remains constant after yielding.
• There is no additional increase in force needed for further deformation.

Understanding this is crucial for applications where permanent deformation is desired, and no return to the original shape is necessary, like in metal forming processes.

Restoring Force

The restoring force is the force that allows materials to return to their original shape after being deformed. In elastic materials, this force kicks in once the stress is removed. However, for perfectly plastic bodies, the story is a bit different. There is no restoring force present.

• This is because, once deformed, these materials do not return to their original shape.
• They exhibit permanent deformation, unlike elastic bodies which "bounce back".

The lack of a restoring force in a perfectly plastic body is precisely why its Young's modulus is zero. With no force pushing the body back to its undeformed state, there is no relationship maintained between stress and strain once the yield point is surpassed.

Stress and Strain

Stress and strain are fundamental concepts when discussing material deformation. Stress refers to the force applied over an area, while strain measures how much a material deforms in response to that stress. The formula connecting them in the context of Young's modulus is \[Y = \frac{stress}{strain} = \frac{F/A}{\Delta L/L}\]

• Where \(F\) is the applied force, \(A\) is the area, \(\Delta L\) is the change in length, and \(L\) is the original length.

For most materials, stress and strain have a linear relationship up to a certain point. However, in perfectly plastic bodies, after the yield strength, strain occurs without additional increases in stress, disrupting this relationship and resulting in a Young's modulus of zero.

Elasticity in Physics

Elasticity in physics describes the ability of a material to recuperate its shape after being distorted. When you pull on a rubber band and then let go, it springs back. That's elasticity at work.

• Elastic materials possess a restoring force that returns them to their original shape once the stress is removed.
• This behavior is neatly quantified by Young's modulus, which measures how stiff or flexible a material is.

However, when it comes to perfectly plastic bodies, elasticity is absent. They do not exhibit a returning force; instead, they remain permanently deformed once the yield point is exceeded. This lack of elasticity leads to their Young's modulus being zero, as there is no elasticity to quantify.