Question

A spring has a spring constant of \(80 . \mathrm{N} / \mathrm{m}\). How much potential energy does it store when stretched by \(1.0 \mathrm{~cm} ?\) a) \(4.0 \cdot 10^{-3} \mathrm{~J}\) c) 80 J e) 0.8 J b) 0.40 J d) \(800 \mathrm{~J}\)

Short answer

Answer: a) \(4.0 \cdot 10^{-3} \mathrm{~J}\)

Step-by-step solution

Convert the displacement to meters

We are given the displacement in centimeters, so we need to convert it to meters. To do this, we simply divide the displacement by 100: \(x = 1.0\ \text{cm} \times \dfrac{1\ \text{m}}{100\ \text{cm}} = 0.01\ \text{m}\)

Calculate the potential energy

Now, we can plug the values for the spring constant (k) and displacement (x) into the potential energy formula: \(PE = \dfrac{1}{2}kx^2\) \(PE = \dfrac{1}{2}(80\ \text{N/m})(0.01\ \text{m})^2\)

Evaluate

To find the potential energy stored in the spring, evaluate the expression: \(PE = \dfrac{1}{2}(80)(0.0001)\) \(PE = 40 \times 0.0001\) \(PE = 0.004\ \text{J}\)

Match the result with the answer choices

We found that the potential energy stored in the spring is \(0.004\ \text{J}\). This is equal to \(4.0 \times 10^{-3}\ \text{J}\). Therefore, the correct answer is: a) \(4.0 \cdot 10^{-3} \mathrm{~J}\)

Key concepts

Hooke's Law

Understanding how springs behave under various loads is essential in various applications, from vehicle suspensions to the mechanisms inside a clock. Central to this is Hooke's Law, which states that the force needed to extend or compress a spring by some distance is proportional to that distance. Mathematically, this is represented as:
\[\begin{equation} F = -kx \right)\], where

• \right) the proportionality constant of the spring and reflects how stiff or flexible the spring is. A higher spring constant indicates a stiffer spring, requiring more force to stretch or compress it by a given amount.
• \right) displacement from the spring's equilibrium position, either by stretching or compressing it.

Notice that the force is negative, signifying that the spring's force acts in the opposite direction to the displacement, hence the term 'restoring force'—the spring seeks to return to its original length. Investigating the relationship between force and displacement can lead to a deeper understanding of the underlying physics of spring systems.

Elastic Potential Energy

When a spring is stretched or compressed, it doesn't just exert a force; it also stores energy. This is known as elastic potential energy (EPE). For an ideal spring (one that obeys Hooke's Law and doesn't reach its elastic limit), the potential energy is calculated by the formula:
\[\begin{equation} PE = \frac{1}{2} k x^2 \right)\], where

• \right) is the amount that the spring has been stretched or compressed from its equilibrium position.

Elastic potential energy is a form of mechanical energy that is recoverable; the spring can return to its original shape and, in doing so, can do work on other objects, converting the stored potential energy into other forms, such as kinetic energy. Understanding this energy storage and conversion is fundamental for designing and analyzing systems that involve springs, like shock absorbers or toys that pop up.

Spring Constant

A key characteristic of a spring is its spring constant, denoted as k. This value is a measure of the stiffness of the spring and is determined by both the material properties of the spring and its geometry (coil diameter, number of coils, wire diameter, etc.). The spring constant units are Newtons per meter (N/m), and it directly affects the amount of force needed to cause a certain displacement. A higher spring constant means you need to apply more force to stretch or compress the spring by a set distance.
k plays a critical role in Hooke's Law and calculating elastic potential energy. By experimentally determining the spring constant, one can predict how a spring will react to different forces. Consistent testing and calculation of k ensure that spring mechanisms function safely and efficiently in their respective applications.

Energy Conversion

Springs illustrate the principle of energy conversion very effectively. When a spring is at rest, there is no energy stored in it. Once it's displaced, it stores energy in the form of elastic potential energy, as mentioned earlier. This stored energy can be converted into other forms when the spring is released. For example, the potential energy might be converted into kinetic energy if the spring is used to launch an object. Likewise, if a weight hanging on a vertical spring bounces up and down, potential energy and kinetic energy are continually being converted into one another.
Understanding the dynamics of energy conversion helps not only in comprehending spring mechanics but also in grasping broader physical concepts such as the conservation of energy, which states that in a closed system, energy can neither be created nor destroyed, only transformed from one form to another.