Problem 23 - Algebra and Trigonometry

Question

Hooke's Law 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 cm and a force of 40 N is required to maintain the spring stretched to a length of 15 cm, find the spring constant. (c) What force is needed to keep the spring stretched to a length of 14 cm?

Step-by-Step Solution

Answer

(a) \( F = kx \). (b) \( k = 8\) N/cm. (c) 32 N.

Step 1: Understanding Hooke's Law Equation

Hooke's Law states that the force \( F \) needed to stretch or compress a spring by a distance \( x \) is directly proportional to that distance. This can be expressed as an equation: \( F = kx \), where \( k \) is the spring constant, and \( x \) is the displacement of the spring from its natural length.

Step 2: Determine Displacement of the Spring

For parts (b) and (c), you need to calculate \( x \), the displacement from the natural length. The natural length of the spring is 10 cm, so if the spring is stretched to 15 cm, the displacement \( x = 15 - 10 = 5 \) cm. Similarly, for a length of 14 cm, \( x = 14 - 10 = 4 \) cm.

Step 3: Find the Spring Constant

Using the information given in part (b), we know that \( F = 40 \) N and \( x = 5 \) cm. Substitute these values into Hooke's Law: \( 40 = k \, \times 5 \). Solving for \( k \) gives \( k = \frac{40}{5} = 8 \). The spring constant \( k \) is 8 N/cm.

Step 4: Calculate Force for New Displacement

For part (c), use the spring constant \( k = 8 \) N/cm. The displacement \( x \) is 4 cm. Substitute these into Hooke's Law equation: \( F = 8 \, \times 4 \). This gives \( F = 32 \) N.

Key Concepts

Spring Constant

The spring constant, often denoted as \( k \), is a fundamental concept in Hooke's Law. It represents the stiffness of a spring. A higher \( k \) value means a stiffer spring, which requires more force to stretch or compress it by a certain amount.
Understanding the spring constant is crucial because it helps us quantify the behavior of different springs under force. It is unique to each spring and depends on factors like the material of the spring and its coil thickness.
In the given exercise, we use the spring constant to relate the force applied and the displacement of the spring. By using Hooke's Law formula \( F = kx \), we can determine \( k \) when we know the force and displacement.

• For instance, with a force of 40 N needed to stretch the spring by 5 cm, as seen in the example, we calculate \( k \) as \( k = \frac{40}{5} = 8 \) N/cm.
• This means each centimeter of stretch requires 8 N of force, showcasing how the spring constant helps control and predict spring behavior.

Essentially, understanding the spring constant enables you to determine how a spring will react to various forces, which is particularly useful in fields like engineering and physics.

Force Calculation

When using Hooke's Law, one of the primary tasks is calculating the force required to reach a particular spring displacement. Hooke's Law is beautifully simple, comprising the formula \( F = kx \), where \( F \) represents the force applied to the spring.
The force is directly proportional to the displacement \( x \), multiplied by the spring constant \( k \). This direct proportionality means that if you double the displacement, the force will also double, holding the spring constant steady. Let's explore how this calculation unfolds.
To calculate the force:

• Identify the spring constant \( k \). In our exercise, the spring constant was found to be 8 N/cm.
• Determine the displacement \( x \). For instance, stretching the spring to 14 cm gives a displacement of 4 cm (since the natural length is 10 cm).
• Substitute these values into the equation: \( F = 8 \times 4 \).

This results in \( F = 32 \) N, meaning 32 Newtons of force are necessary to maintain the 4 cm stretch of the spring.
This simple method provides a practical way to calculate force requirements, aiding in tasks spanning from everyday objects like retractable pens to complex machinery.

Displacement of Spring

Displacement refers to the change in length of a spring from its original, or natural, length when a force is applied. This is a crucial component of Hooke's Law, represented as \( x \) in the equation \( F = kx \).
In our exercise, determining displacement helps in solving for both spring constant and required forces. To find the displacement:

• Identify the natural length of the spring. For example, the natural length is 10 cm.
• Measure the new stretched length, such as 15 cm.
• Compute the displacement \( x \) by subtracting the original length from the stretched length: \( x = 15 - 10 = 5 \) cm.

The displacement is effectively the "stretch" or "compression" of the spring. When force is applied to a spring, knowing this value is key.
By examining this component, students can visualize how much a spring deviates from its original form and use this understanding to predict spring behavior and design practical solutions, enhancing fundamental physics and engineering knowledge.