Problem 29 - Precalculus Mathematics for Calculus

Question

Hooke's Law states that the force needed to keep a spring stretched \(x\) units beyond its natural length is directly proportional to \(x\). Here the constant of proportionality is called the spring constant. (a) Write Hooke's Law as an equation. (b) If a spring has a natural length of \(10 \mathrm{cm}\) and a force of \(40 \mathrm{N}\) is required to maintain the spring stretched to a length of \(15 \mathrm{cm},\) find the spring constant. (c) What force is needed to keep the spring stretched to a length of \(14 \mathrm{cm} ?\)

Step-by-Step Solution

Answer

(a) \( F = kx \); (b) \( k = 800 \text{ N/m} \); (c) \( F = 32 \text{ N} \).

Step 1: Understanding Hooke's Law

Hooke's Law states that the force \( F \) needed to stretch or compress a spring by a distance \( x \), from the spring’s natural length, is proportional to that distance. This is mathematically expressed as \( F = kx \), where \( k \) is the spring constant.

Step 2: Determine the Spring Constant

Given that the natural length of the spring is 10 cm, when a force of 40 N extends it to 15 cm, we first find the displacement, \( x = 15 \text{ cm} - 10 \text{ cm} = 5 \text{ cm} \) (or 0.05 m). Using Hooke's Law, \( F = kx \), we set up the equation: \( 40 = k imes 0.05 \). Solving for \( k \), we get \( k = \frac{40}{0.05} = 800 \, \text{N/m} \).

Step 3: Calculate the Force for a New Stretch

To find the force needed to keep the spring stretched to 14 cm, first calculate the new displacement: \( x = 14 \text{ cm} - 10 \text{ cm} = 4 \text{ cm} \) (or 0.04 m). Using Hooke's Law, \( F = kx = 800 \times 0.04 = 32 \, \text{N} \).

Key Concepts

Spring Constant

The spring constant, often denoted as \( k \), is a crucial component in understanding how springs behave when subject to forces. It quantifies the stiffness of a spring, indicating how much force is needed to stretch or compress the spring by a given distance. A higher spring constant means that the spring is stiffer and requires more force to change its length.
The spring constant is unique to each spring and is determined experimentally.

• Measured in newtons per meter (N/m).
• Calculated using Hooke's Law.
• Represents the ratio of force exerted to the displacement caused by that force.

Understanding the spring constant helps in predicting how a spring will behave under different amounts of force and is essential for applications where precise control over spring deformation is required.

Proportionality

Proportionality is a fundamental concept in Hooke's Law. It relates to the idea that one quantity changes in a consistent way with another. In the context of Hooke's Law, the force ( \( F \)), required to stretch a spring is directly proportional to the distance ( \( x \)) the spring is stretched from its natural length.
Mathematically, this is expressed as \( F = kx \), where \( k \) is the spring constant. This equation signifies that as the displacement increases, the amount of force required increases proportionally.
Some key points about proportionality in Hooke's Law:

• As the distance x doubles, the force F also doubles.
• If the force applied is halved, the displacement is also halved.
• This linear relationship ensures predictability of spring behavior.

By understanding proportionality, students can better grasp how changing the distance impacts the force needed and vice versa.

Force Calculation

Force calculation using Hooke's Law involves determining the amount of force needed to stretch or compress a spring a certain amount. The core formula is \( F = kx \), where \( F \) is the force in newtons, \( k \) is the spring constant in N/m, and \( x \) is the displacement in meters.
To calculate force:

1. Identify the displacement \( x \) from the spring's natural length.
2. Make sure to convert any units if necessary (e.g., cm to meters).
3. Multiply the spring constant \( k \) by the displacement \( x \) to find the force \( F \).

For example, if \( k = 800 \, \text{N/m} \) and the displacement \( x \) is 0.04 m, the force is \( F = 800 \times 0.04 = 32 \, \text{N} \). This process helps in various applications, from mechanical engineering to everyday objects like mattresses.

Natural Length of Spring

The natural length of a spring is its original length when no external force is applied. It is the reference point from which all stretches or compressions are measured. In problems involving Hooke's Law, the natural length is critical to determining the displacement (\( x \)).
To understand the natural length:

• Find the spring's length under no external force.
• Subtract this length from any extended or compressed length to find displacement.
• This displacement is what drives the calculations for force using Hooke's Law.

In exercises, knowing the natural length allows for accurate measurement of how far the spring has been stretched or compressed. For example, if a spring's natural length is 10 cm, and it is stretched to 15 cm, the displacement \( x \) is 5 cm or 0.05 m. This baseline is essential for accurate application of the law.