Problem 234 - Calculus Volume 1

Question

A 12 -in. spring is stretched to 15 in. by a force of \(75 \mathrm{lb}\). What is the spring constant?

Step-by-Step Solution

Answer

The spring constant is 25 lb/in.

Step 1: Understand Hooke's Law

The exercise involves a spring, so we need to use Hooke's Law. Hooke's Law is given by the equation \( F = k \cdot x \), where \( F \) is the force applied to the spring, \( k \) is the spring constant, and \( x \) is the amount of stretch (or compression) from the spring's original length.

Step 2: Identify Given Values

We know the original length of the spring is 12 inches, and it is stretched to 15 inches. The force applied is 75 pounds. This means the spring is stretched by \( x = 15 - 12 = 3 \) inches.

Step 3: Convert Units if Necessary

Ensure that all units are compatible. Since the force is in pounds and the stretch is in inches, there is no need for unit conversion here. We can directly substitute these values into Hooke's Law equation.

Step 4: Rearrange Hooke's Law to Solve for the Spring Constant

Rearrange the formula \( F = k \cdot x \) to solve for the spring constant: \( k = \frac{F}{x} \).

Step 5: Substitute Values

Substitute the known values into the equation: \( k = \frac{75}{3} \).

Step 6: Calculate the Spring Constant

Perform the division: \( k = 25 \). Thus, the spring constant is 25 pounds per inch.

Key Concepts

Understanding the Spring Constant

The spring constant, often denoted as \( k \), is a crucial concept when analyzing the behavior of springs. It describes the stiffness of a spring.

• A higher spring constant indicates that a spring is stiffer and requires more force to stretch or compress it by a certain amount.
• A lower spring constant means the spring is more flexible.

In the context of this exercise, the spring constant is determined by observing how much force is needed to cause a specific stretch in the spring. In simpler terms, it tells us how resilient the spring is to changes in shape.

Force and Displacement in Springs

In Hooke's Law, force \( F \) and displacement \( x \) have a direct relationship. Force is measured in pounds (lb) or newtons (N), while displacement refers to how much the spring is stretched or compressed from its original length, often measured in meters (m) or inches (in).When solving problems involving springs:

• Identify the initial and final lengths of the spring to calculate displacement: \( x = \text{final length} - \text{original length} \).
• Ensure that force is exerted in a manner to stretch or compress the spring.

By measuring these two variables, you can understand how energy transfers through the spring and how much work is done in the system.

Unit Conversion Importance

Unit consistency is crucial when applying physical formulas like Hooke's Law. Inconsistent units can lead to incorrect calculations and results. For this particular exercise:

• Check that all measurements are in compatible units before performing calculations. Here, no conversion was needed because both the force and the displacement used pounds and inches respectively.
• When dealing with mixed units, such as converting inches to meters or pounds to newtons, use known conversion factors. For instance, 1 inch is 0.0254 meters.

Being vigilant about unit conversion ensures that formulas are applied correctly and results are accurate.

Effective Physics Problem Solving

Approaching physics problems systematically can simplify complex problems. Here's a structured approach: 1. **Understand the Problem:** Identify the physical laws that apply. Here, Hooke's Law was crucial. 2. **Extract Information:** Gather all known values and what needs to be found. Translate word problems into variables. 3. **Check Units:** Ensure all measurements are in compatible units to prevent calculation errors. 4. **Apply Mathematical Models:** Use relevant formulas to solve for the unknowns. Rearrange equations if necessary to isolate the desired variable. 5. **Perform Calculations:** Carefully substitute values, keeping track of each step. 6. **Verify Results:** Double-check calculations and consider if results make sense contextually. This step-by-step method can be adapted to tackle a wide range of physics exercises, ensuring clarity and precision.