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Chapter 5: Problem 5

For any spring obeying Hooke's Law, show that the work done in stretching a spring a distance \(d\) is given by \(W=\frac{1}{2} k d^{2}\).

Short Answer

Expert verified
The work done is given by \( W = \frac{1}{2} k d^{2} \) for a spring obeying Hooke's Law.

Step by step solution

01

Understanding Hooke's Law

Hooke's Law states that the force required to stretch or compress a spring by a distance \( x \) from its equilibrium position is proportional to \( x \). Mathematically, it's given by \( F = kx \), where \( k \) is the spring constant.
02

Relating Work to Force and Displacement

The work \( W \) done by a force in moving an object through a displacement is given by the integral of force with respect to displacement. For a variable force like in a spring, it's \( W = \int F(x) \, dx \).
03

Determining the Expression for Work Done

Substituting Hooke's Law (\( F = kx \)) into the work integral gives \( W = \int kx \, dx \). The limits of integration are from 0 to \( d \), as the displacement starts from 0 and goes to \( d \).
04

Solving the Integral

To find the work done, evaluate the integral \( W = \int_{0}^{d} kx \, dx = k \int_{0}^{d} x \, dx \). This integral is \( k \left[ \frac{x^2}{2} \right]_{0}^{d} \).
05

Evaluating the Result

Substituting the limits, we find \( W = k \left( \frac{d^2}{2} - \frac{0^2}{2} \right) = \frac{1}{2} kd^2 \). Thus, the work done in stretching the spring by a distance \( d \) is \( W = \frac{1}{2} kd^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spring Constant
In the world of physics, the spring constant is a key characteristic that describes a spring's ability to resist deformation. It is denoted by the letter \( k \) and is a measure of the stiffness of the spring. A higher spring constant means a stiffer spring, requiring more force to stretch or compress it.
  • Hooke's Law formula that includes the spring constant is \( F = kx \), where \( F \) is the force applied to the spring and \( x \) is the displacement from the spring鈥檚 equilibrium position.
  • The spring constant is crucial for predicting how much force is needed for a certain displacement, important in various applications from vehicle suspensions to measuring devices.
  • Spring constants vary depending on material and construction, allowing designers to tailor springs for specific needs.
Understanding the spring constant is foundational in mechanics, providing insight into spring behavior and helping solve problems related to Hooke鈥檚 Law.
Work Done
Work done is a fundamental concept in physics that refers to the energy transferred to or from a system by a force acting over a distance. In the context of a spring, work is done when a force stretches or compresses it.
  • For a spring following Hooke's Law, the work done \( W \) is given by \( W = \frac{1}{2} kd^2 \). This equation shows that work depends on the spring constant \( k \) and the square of the distance \( d \) the spring is stretched.
  • The equation reveals that the work done is directly proportional to the spring constant and quadratically proportional to the distance.
  • This formula results from integrating the force over the distance from 0 to \( d \), reflecting the way force varies with displacement in a spring.
Understanding how to calculate work helps in analyzing energy transformations in mechanical systems, critical for engineering and physics applications.
Integral Calculus
Integral calculus is a branch of mathematics concerned with calculating areas under curves, finding volumes, and solving problems involving accumulation. It is essential for deriving and understanding physical laws involving changing quantities.
  • In the exercise, integral calculus is used to determine the work done by integrating the force function \( F(x) = kx \) over the distance \( x \) from 0 to \( d \).
  • This involves evaluating the integral \( W = \int_{0}^{d} kx \, dx \), which equals \( k \left[ \frac{x^2}{2} \right]_{0}^{d} = \frac{1}{2} kd^2 \).
  • The process of integration helps in finding work done by a variable force, capturing how small changes accumulate over a range.
  • This mathematical tool is pivotal in physics and engineering, allowing for the calculation of quantities that are not constant but change with respect to time or space.
Mastering integral calculus provides profound insights into analyzing dynamic systems and solving real-world problems.

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