Question

You do a very precise experiment to test the properties of a piece of wire. You stretch the wire by applying an outward force to each end of the wire and measure its change in length. If you then precisely double this force and the wire still remains elastic but is beyond proportionality limit, the change in length of the wire. (1) will exactly double (2) will just slightly more than double (3) will just slightly less than double (4) will decrease by exactly a factor of two

Short answer

(2) will just slightly more than double

Step-by-step solution

- Understanding the Linear Elastic Region

Under Hooke's Law, the elongation of a material is proportional to the applied force, provided the material remains within its elastic limit. This region is known as the linear elastic region.

- Proportionality Limit

The proportionality limit is the maximum stress that a material can withstand while still obeying Hooke's Law. Beyond this limit, the relationship between force and elongation is no longer linear.

- Effect of Doubling the Force Beyond Proportionality Limit

Beyond the proportionality limit, the material does not obey Hooke's Law, and the elongation will increase in a nonlinear fashion when the force is doubled.

- Resulting Change in Length

Since the material is beyond the proportionality limit, doubling the force will result in an elongation that is slightly more than double the original length due to the nonlinear nature of the stress-strain relationship at this stage.

Key concepts

Hooke's Law

Hooke's Law states that, within the elastic limit of a material, the amount it stretches is directly proportional to the force applied to it. Mathematically, this is expressed as \(F = k \Delta L\), where \(F\) is the force, \(k\) is the spring constant (a measure of the stiffness of the material), and \(\Delta L\) is the change in length.

For example, if you apply a force to a spring, it will stretch. If you double the force, the stretch will also double, as long as you stay within the elastic limit of the material.

This linear relationship makes it easy to predict how a material will behave when subjected to different forces. However, different materials have different elastic limits, beyond which Hooke's Law no longer applies.

Proportionality Limit

The proportionality limit is a critical point in the stress-strain curve of a material. It marks the maximum stress that can be applied while the material still behaves elastically and follows Hooke's Law.

When stretching a wire, as long as the applied force does not exceed this limit, the wire will return to its original length once the force is removed. The relationship between stress (force per unit area) and strain (relative deformation) is linear within this region.

However, once the force surpasses the proportionality limit, the material enters a plastic deformation phase. This means the material will not return to its original shape entirely and Hooke's Law no longer applies.

Nonlinear Stress-Strain Relationship

Beyond the proportionality limit, the stress-strain relationship becomes nonlinear. The material no longer follows Hooke's Law, and its response to applied forces changes.

In the nonlinear region, doubling the force does not result in exactly doubling the elongation. Instead, the change in length becomes more complicated. The wire will elongate more than double the initial increase when doubling the force.

This behavior occurs because the material is experiencing plastic deformation and other complex internal changes. Understanding this nonlinear relationship requires a more advanced comprehension of the material's properties and is crucial for applications where materials are subjected to high stresses.