Understanding the Spring Force and Work Done: A Comprehensive Analysis

Introduction to Hooke's Law

Hooke's Law describes the relationship between the force acting on a spring and the amount it is stretched or compressed. Mathematically, it is expressed as F kx, where F is the force (in pounds), x is the displacement (in inches), and k is the spring constant (in pounds per inch).

Calculating the Spring Constant

Let's consider the given scenario: a force of 3 pounds compresses a 15-inch spring a total of 3 inches. Using Hooke's Law, we can determine the spring constant. According to Hooke's Law: F kx

Given: F 3 pounds, x 3 inches

Therefore, k F / x 3 pounds / 3 inches 1 pound/inch

Determining the Required Force for Compression

Now, we need to find the force required to compress the spring 10 inches. Using the spring constant k 1 pound/inch, the force F required to compress the spring 10 inches is: F kx 1 pound/inch * 10 inches 10 pounds

Calculating the Work Done in Compression

The work done (W) in compressing a spring is given by the formula W 0.5kx^2. This formula can be derived from the integral of the force over the distance of compression. Using the spring constant k 1 pound/inch and the compression distance x 10 inches: W 0.5 * 1 pound/inch * (10 inches)^2 0.5 * 100 inch-pounds 50 inch-pounds

Converting Inch-Pounds to Foot-Pounds

To express the work done in foot-pounds (ft-lbs), we need to convert inch-pounds to foot-pounds. Since 1 foot equals 12 inches, 1 inch-pound equals 1/12 foot-pounds. Therefore, 50 inch-pounds 50 / 12 foot-pounds ≈ 4.1667 ft-lbs

Conclusion and Further Reading

In conclusion, the work done in compressing a 15-inch spring with a spring constant of 1 pound/inch by 10 inches is approximately 4.1667 ft-lbs. This result is obtained using Hooke's Law and the formula for the work done on a spring. For further reading and verification, you can refer to your textbook or resources on Hooke's Law and the work formula for springs.

Glossary of Key Terms

Spring Force: The force exerted by a spring when it is stretched or compressed. Hooke's Law: The principle that states the force required to extend or compress a spring is directly proportional to the distance of the extension or compression. Work: The energy transfer that occurs when a force causes an object to move.