Problem 3 - Thomas Calculus

Question

Spring constant It took 1800 \(\mathrm{J}\) of work to stretch a spring from its natural length of 2 \(\mathrm{m}\) to a length of 5 \(\mathrm{m} .\) Find the spring's force constant.

Step-by-Step Solution

Answer

The spring's force constant is 400 N/m.

Step 1: Understand Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its natural length, given by the equation \( F = kx \), where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement.

Step 2: Calculate Displacement

The displacement \( x \) is the change in the spring's length from its natural length. Here, the spring is stretched from 2 meters to 5 meters, so \( x = 5 - 2 = 3 \) meters.

Step 3: Use Work-Energy Principle for Springs

The work done on the spring is calculated using the work-energy principle for springs, which is given by the equation \( W = \frac{1}{2}kx^2 \). Here, \( W = 1800 \) J and \( x = 3 \) meters.

Step 4: Solve for the Spring Constant

Plug the known values into the work-energy formula: \( 1800 = \frac{1}{2}k(3)^2 \). Simplifying gives \( 1800 = \frac{1}{2}k \times 9 \).

Step 5: Calculate the Value of k

To find \( k \), rearrange the equation to isolate \( k \): \( 1800 = 4.5k\). Solve for \( k \) by dividing both sides by 4.5: \( k = \frac{1800}{4.5} = 400 \) N/m.

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that defines how a spring behaves when it is either compressed or stretched. This law is epitomized by the equation:\[ F = kx \]where:

• \( F \) represents the force exerted by the spring.
• \( k \) stands for the spring constant, a measure of the stiffness of the spring.
• \( x \) is the displacement, or how far the spring has been stretched or compressed from its natural length.

In simpler words, Hooke's Law establishes that the force needed to compress or extend a spring is directly proportional to the distance of displacement. Therefore, the spring constant \( k \) plays a pivotal role, as it tells us how much force is needed for a given displacement.In practical terms, if a spring has a high spring constant, it means that more force is required to stretch or compress it, indicating a stiffer spring. Conversely, a lower spring constant reflects a spring that is more easily compressible or extendable with less force.

Work-Energy Principle

The work-energy principle in the context of springs provides a powerful way to relate the work done on a spring to its displacement. According to this principle, the work done on a spring is stored as potential energy within the spring. The formula central to this concept when dealing with springs is:\[ W = \frac{1}{2} k x^2 \]where:

• \( W \) is the work done in joules.
• \( k \) is the spring constant.
• \( x \) is the displacement.

This formula demonstrates that the work done on the spring is proportional to the square of the displacement. The factor of \( \frac{1}{2} \) arises because the force exerted by the spring is not constant but varies linearly with distance (following Hooke's Law).Understanding this principle is key when calculating the energy involved in compressing or stretching a spring, thereby allowing one to derive the spring constant if the work done and displacement are known, as in the problem given.

Displacement

Displacement in the context of springs refers to how much the spring is elongated or compressed from its original, natural length. It is a critical factor in both Hooke's Law and the work-energy principle.In the example provided, the displacement \( x \) can be calculated by taking the difference between the new length of the spring and its original length:\[ x = \text{new length} - \text{natural length} \]So, if a spring originally measures 2 meters and is stretched to 5 meters, the displacement is:\[ x = 5 - 2 = 3 \text{ meters} \]Knowing the displacement is essential because it directly influences both the force exerted by the spring and the work done on the spring. It serves as a link between the physical elongation/compression of the spring and the mathematical expressions used to describe the spring's behavior.