Question

A uniform plank of young's modulus \(\mathrm{Y}\) is moved over a smooth horizontal surface by a constant horizontal force \(\mathrm{F}\), The area of cross-section of the plank is \(\mathrm{A}\). What is the compressive strain on its plank in the direction by the force ? (A) (F / AY) (B) \((2 \mathrm{~F} / \mathrm{AY})\) (C) \((1 / 2)(\mathrm{F} / \mathrm{AY})\) (D) (3F / AY)

Short answer

The compressive strain on the plank in the direction of the force is (A) \(\frac{F}{AY}\).

Step-by-step solution

Write down the given information

We are given the following: - Young's Modulus (Y) - A constant horizontal force (F) - Cross sectional area of the plank (A) We need to find the compressive strain on the plank in the direction of the force.

Recall the definition of strain and the formula for Young's Modulus

Strain is the change in length per unit length. The formula for strain is: \[ Strain=\frac{\Delta L}{L} \] Young's modulus (Y) is defined as the ratio of stress to strain. The formula for Young's modulus is: \[ Y = \frac{Stress}{Strain} \]

Replace stress in the Young's Modulus formula with force and area

Stress is defined as the force per unit area. So, we can replace stress in the Young's modulus formula with force(F) and area(A). \[ Y = \frac{F / A}{Strain} \]

Solve the equation for strain

Rearrange the equation to solve for strain. \[ Strain = \frac{F}{AY} \]

Identify the correct answer

Now we can compare the obtained result to the given choices: (A) (F / AY) (B) \((2F / AY)\) (C) \((1 / 2)(F / AY)\) (D) (3F / AY) So, the correct answer is (A) (F / AY).

Key concepts

Compressive Strain

Compressive strain is a crucial concept in understanding how materials respond to forces. It describes how a material such as our uniform plank changes in length under the influence of a compressive force, that is, a force that acts to compress or shorten the material. In mathematical terms, compressive strain is expressed as the change in length divided by the original length, or \( \frac{\Delta L}{L} \).

This concept is important in many real-world applications, such as understanding how structural beams in buildings might react to heavy loads. By determining the compressive strain, engineers can ensure that materials do not deform or compress to a damaging extent under load. Compressive strain is linked directly to stress, as stress is the cause of this strain when a force is applied concurrent to the material's surface area.

Uniform Plank

A uniform plank refers to a piece of material that is consistent and even throughout its structure. This uniformity means that it has the same density, shape, and material properties across its entire length.

A uniform plank is often preferred in construction and mechanical applications because it provides predictable and consistent behavior under stress. The symmetry ensures the plank will not bend or compress inconsistently, which is crucial for maintaining structural integrity. For our problem, the uniformity of the plank simplifies the calculations of stress and strain since we assume consistent properties across the entire plank length.

Stress and Strain

Stress and strain are interrelated concepts that describe how materials deform under various forces. Stress is defined as the force applied per unit area, calculated by \( \frac{F}{A} \), where \( F \) is the force and \( A \) is the cross-sectional area.

Strain, as mentioned earlier, is the ratio of change in length to the original length, given by \( \frac{\Delta L}{L} \). Thus, while stress describes the pressure on the object's material, strain quantifies how that material changes shape as a result of this stress.

Understanding the interplay between stress and strain is fundamental to materials science. It allows us to predict material behavior and ensure that structures can withstand required forces without undergoing failure or unacceptable deformations.

Cross-Sectional Area

Cross-sectional area is a crucial geometrical factor in determining how a material will react to force. It is the size of the surface that is being acted upon by a force, calculated as the area of a section that is perpendicular to the force direction.

In our case, with the force \( F \) applied horizontally, the cross-sectional area would be the area of one end of the uniform plank. The stress a material experiences is directly related to its cross-sectional area; a smaller area means higher stress for the same force.

• For example, a narrower plank will experience greater stress than a wider one when pushed with the same force, potentially leading to more significant deformation.

This helps engineers and designers choose appropriate material shapes and sizes for strength and durability in various applications.