Problem 45 - Precalculus with Limits

Question

HOOKE'S LAW In Exercises 45-48, use Hooke's Law for springs, which states that the distance a spring is stretched (or compressed) varies directly as the force on the spring. A force of 265 newtons stretches a spring 0.15 meter (see figure). (a) How far will a force of 90 newtons stretch the spring? (b) What force is required to stretch the spring 0.1 meter?

Step-by-Step Solution

Answer

The force of 90 newtons will stretch the spring a distance of \( x = 90 / k \) meters. The force required to stretch the spring 0.1 meter is \( F = k \cdot 0.1 \) newtons.

Step 1: Calculate the proportionality constant

Hooke's Law can be expressed as \( F = k \cdot x \) where \( F \) is the force, \( x \) is the distance the spring is stretched, and \( k \) is a proportionality constant. Using the initial conditions, we can express \( k \) as: \( k = F / x \). Substituting for \( F = 265 \) newtons and \( x = 0.15 \) meter, we find \( k = 265 / 0.15 \).

Step 2: Find the distance a force of 90 newtons will stretch the spring

Using the proportionality constant from the previous step, we now set \( F = 90 \) newtons. We can solve for \( x \) using the formula \( x = F / k \). So \( x = 90 / k \).

Step 3: Find the force required to stretch the spring 0.1 meter

We again use the same formula, but now set \( x = 0.1 \) meter. The force is found using \( F = k \cdot x \)} So, \( F = k \cdot 0.1 \).

Step 4: Calculate results

Now we can perform the calculations using the value of \( k \) obtained in the first step.

Step 5: Interpret the results

We should now assess if the answers for each part are reasonable given the context of the problem (stretching a spring).

Key Concepts

Proportionality Constant

When we talk about Hooke's Law and springs, a key player in the equation is the proportionality constant, commonly denoted as 'k'. This constant is crucial because it links the force applied to a spring to the distance it stretches or compresses. In a sense, it describes how 'stiff' or 'flexible' a spring is.

The actual value of 'k' varies from one spring to another, depending on factors such as the material of the spring and its physical dimensions. Think of it as a measure of the spring's resistance to being stretched or compressed. In the context of the exercise, once we have a force and an associated extension, we can calculate 'k' by dividing the force (in newtons) by the extension (in meters). This relationship gives us a means to predict how the spring will behave under different forces.

Spring Constant

Understanding the Spring Constant

The spring constant, represented by the same proportionality constant 'k' in Hooke's Law, plays a pivotal role in describing the behavior of springs. It gives a quantitative measure of the stiffness of the spring: the higher the spring constant, the stiffer the spring, and conversely, the lower the value, the more easily the spring can be stretched or compressed.

Think about it as a personality trait of a spring. If you have two springs in front of you and one has a higher spring constant than the other, it will not stretch as much as the other when subjected to the same force. In our exercise example, once we've calculated the spring constant using the formula provided in the solution, it can be used to find out how the spring reacts to any other force value.

Force and Extension Relationship

Dynamics of Force and Extension

At the heart of Hooke's Law is a straightforward yet profound relationship: the force applied to a spring is directly proportional to the extension it causes. This means if you double the force, the extension doubles, assuming we are within the elastic limits of the spring. If you halve the force, the extension halves.

This relationship is crucial for engineers and designers who work with springs in mechanical systems. Knowing this allows them to predict the extension of a spring for a given force, thereby tailoring the design to their needs. In our exercise, by understanding this relationship, we can solve for unknown quantities like the extension for a new force (part a) or the force needed for a desired extension (part b), simply by rearranging the equation of Hooke's Law.