Question

A part of a guard rail along a highway has been tested by pulling sideways. A force of \(10 \mathrm{kN}\) leads to a lateral displacement of \(1 \mathrm{~cm}\). What will be the additional displacement is the force is increased to \(20 \mathrm{kN}\), more or less than \(1 \mathrm{~cm}\) ?

Short answer

#tag_title#Step 2: Solve for the elastic constant#tag_content#To solve for the elastic constant, we simply rearrange the equation and plug in the given values: $$ k = \frac{10 \mathrm{kN}}{1 \mathrm{~cm}} $$ Converting the units so that force is in newtons (N) and displacement is in meters (m), we get: $$ k = \frac{10,000 \mathrm{N}}{0.01 \mathrm{~m}} = 1,000,000 \mathrm{N/m} $$ So, the elastic constant of the guard rail is 1,000,000 N/m. #tag_title#Step 3: Calculate the new displacement when the force is increased to 20 kN#tag_content#Now that we have the elastic constant, we can use it to find the new displacement when the force is increased to 20 kN. Applying Hooke's Law again: $$ F = kD $$ We now have a force of 20 kN, which is equal to 20,000 N. Rearranging the equation to solve for the new displacement, we get: $$ D = \frac{F}{k} $$ Plugging in the force and elastic constant values, we have: $$ D = \frac{20,000 \mathrm{N}}{1,000,000 \mathrm{N/m}} = 0.02 \mathrm{~m} $$ Converting this value back to centimeters, we get: $$ D = 0.02 \mathrm{~m} \times \frac{100 \mathrm{~cm}}{1 \mathrm{~m}} = 2 \mathrm{~cm} $$ So, when the force is increased to 20 kN, the guardrail will be displaced by 2 cm.

Step-by-step solution

Find the elastic constant

We can utilize Hooke's Law to find the elastic constant of this guard rail. Hooke's Law states that the force exerted on an object is directly proportional to its deformation (displacement), mathematically: $$ F = kD $$ Where: \(F\) - Force exerted on the object (in this case, the guard rail) \(k\) - Elastic constant of the object \(D\) - Deformation (displacement) of the object Using the given information, we can find the elastic constant of the guard rail: $$ 10 \mathrm{kN} = k(1 \mathrm{~cm}) $$

Key concepts

Elastic Constant

The elastic constant, often denoted by the symbol 'k', is a fundamental property of materials that measures their stiffness. It plays a crucial role in Hooke's Law, which is a principle that enables us to understand how a material deforms under the influence of a force.

The relationship is quite straightforward: the stiffer or more rigid a material is, the higher its elastic constant will be. When we apply a force to an object, like the guard rail in the textbook exercise, the elastic constant helps us determine the extent of the deformation it will experience. If 'k' is large, the object won't deform much; if it's small, the object will be more elastic and experience more significant deformation.

To calculate the elastic constant using Hooke's Law, we simply rearrange the equation to solve for 'k', which gives us: \[ k = \frac{F}{D} \] where 'F' represents the force applied to the object and 'D' is the resulting deformation. Using the provided example, with a force of 10 kN causing a deformation of 1 cm, we find that:\[ k = \frac{10 \, \mathrm{kN}}{1 \, \mathrm{cm}} \] Remember, the correct units for 'k' depend on the units used for force and displacement. In this case, 'k' would have the units of kN/cm, representing the force per unit deformation.

Lateral Displacement

Lateral displacement refers to the movement of an object sideways or laterally when it is subjected to a force not aligned with its primary axis. This is a key concept in the study of mechanics and material science, especially when analyzing the behavior of structures like guard rails, beams, or even buildings under lateral forces such as wind or an impact.

In the classroom example, when the guard rail is pulled sideways with a force of 10 kN, it experiences a lateral displacement of 1 cm. This displacement is the measure of how much the guard rail has shifted from its original position. It's crucial to understand that, in a real-world scenario, excessive lateral displacement could compromise the structural integrity and safety of a design.

Engineers and designers frequently calculate the lateral displacement to ensure that structures can withstand applied forces without failing or bending beyond acceptable limits. They use Hooke's Law as a fundamental tool to predict this displacement and design accordingly to ensure resilience and durability.

Force-Deformation Relationship

Understanding the force-deformation relationship is essential for predicting how materials and structures will behave under load. Hooke's Law provides a simple yet powerful way of establishing this relationship for elastic materials, stating that the force applied to an object is linearly proportional to its deformation, as long as the material remains within its elastic limit.

The equation \[ F = kD \] encapsulates this relationship: 'F' is the applied force, 'k' is the elastic constant, and 'D' is the deformation. This equation holds true for small deformations where the material behaves elastically - meaning it returns to its original shape after the force is removed.

In the context of the guard rail's textbook problem, using Hooke's Law enables us to predict the additional displacement when the force is increased. For example, if the force is doubled to 20 kN, and assuming the guard rail remains within its elastic limit, the displacement will also double, meaning an additional 1 cm of displacement for a total of 2 cm. This linear behavior is very useful in engineering and design, as it allows us to foresee how objects and materials will react to changes in applied forces.