Calculus Computing Work using Hooke's Law for Springs Easy Example

TL;DR

Work done in compressing a spring can be calculated using Hooke's law and the definite integral.

Transcript

a force of 10 pounds compresses a 12-inch spring a total of 2 inches how much work is done in compressing the spring 7 inches from its natural length let's go ahead and work this out i'll start by drawing a picture of the spring just so you see what's going on so this is our 12-inch spring and here it's at equilibrium so the force acting on it is e... Read More

Key Insights

• 👮 Hooke's law describes the relationship between the force and displacement of a spring.
• 💦 The work done in compressing a spring can be found by using the definite integral of the force function.
• 🇧🇫 The formula for Hooke's law is F = kx, where k is the spring constant.
• 🤌 A 12-inch spring compressed by a force of 10 pounds results in a total compression of 2 inches.
• 💁 Using the given information, the value of the spring constant (k) is found to be 5.
• 🤌 The work done in compressing the spring 7 inches from its natural length is calculated to be 122.5 inch pounds.
• 💦 The work done represents the energy transferred to the spring during compression.

Questions & Answers

Q: How is the force required to compress or stretch a spring related to the displacement?

According to Hooke's law, the force required to compress or stretch a spring is directly proportional to the displacement. This means that as the displacement increases, the force required also increases.

Q: What is the formula for Hooke's law?

Hooke's law can be expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement from the spring's natural length.

Q: How is the work done in compressing a spring calculated?

The work done in compressing a spring is calculated using the definite integral of the force function with respect to the displacement. This integral represents the area under the force-displacement curve.

Q: What are the units of the calculated work?

The units of the calculated work done in compressing the spring are inch pounds, which indicate the amount of work done per unit displacement.

Summary & Key Takeaways

• A 12-inch spring is compressed by a force of 10 pounds, resulting in a total compression of 2 inches.

• Hooke's law states that the force required to compress or stretch a spring is proportional to the displacement.

• Using Hooke's law and the definite integral, the work done in compressing the spring 7 inches from its natural length is found to be 122.5 inch pounds.