The school day is slowly approaching its end; you’re bored in class and are watching the time slowly tick by all while constantly clicking your pen in anticipation of that final bell. At that moment, your sole focus may be going home, however, subconsciously you are demonstrating Hooke’s law every time you click your pen. Maybe there is a payoff to being diligent in class after all? Hooke’s law is the primary mechanism behind retractable pens. Retractable pens consist of springs, attached to an ink cartridge, and a plastic tube that is fixed in place. Together, they form an internal mechanism that works using Hooke’s law. Every time we click our pens down, we unlock the ink cartridge allowing us to write, and when we are done and click the pen again, we enable the ink cartridge to lock once again. Therefore, let us use this example as a starting point in understanding Hooke's law and introduce definitions and examples that will help expand your knowledge on this topic.
Fig. 1: Demonstrating Hooke's law using a pen.
Hooke's Law Definition
Hooke's law describes the linear relationship between force and displacement. The paradigmatic example of Hooke's law is that of a mass attached to an ideal spring. Intuitively, you'd expect that pulling on the mass from its rest position and letting it go will result in the spring pulling the mass back to its original resting position. Alternatively, pushing the mass will compress the spring which, upon being let go, will push the mass back to its original resting position.
A key aspect of Hooke's law is that it assumes that the spring under consideration is ideal.
An ideal spring has negligible mass and exerts a force proportional to the change in its length as measured from its relaxed length.
With this concept in mind, we arrive at the statement of Hooke's law as follows:
Hooke's law states that the force required to extend or compress an ideal spring by a certain distance is proportional to that distance.
Another name for Hooke's law is Hooke's law of elasticity. Elasticity is an object's ability to stretch or compress.
Hooke's Law Formula
The mathematical formula of Hooke's law is
$$F= -kx$$
where \(F\) is the force applied measured in newtons \(\mathrm{N}\), \(k\) is the spring constant measured in \(\frac{\mathrm{N}}{\mathrm{m}}\), and \(x\) is the displacement measured in meters \(\mathrm{m}\). The negative sign indicates the restoring force, the force which acts on the object to bring it back to its equilibrium position, is in the opposite direction of the displacement. However, one must note that this formula applies only to ideal springs.
The spring constant refers to the stiffness of the spring and will vary depending on the spring. A larger spring constant requires a larger force to stretch the spring a certain distance.
Hooke's Law Graph
When examining springs, we apply Hooke's law. For many springs, the stretch or compression of the spring is proportional to the force applied and a graph of the force applied to a spring vs. the displacement of the spring shows a region that is linear. This is the region over which the assumption that the spring is ideal holds and, therefore, Hooke’s law applies. However, if we apply a force beyond this regime to the spring, it will reach the limit of proportionality. This means that the spring becomes permanently deformed, as it cannot return to its normal shape, and eventually breaks.
Fig. 2: A graph depicting Hooke's law.
Beyond the limit of proportionality, the relation between the force applied to the mass and its displacement from its original position ceases to be linear and becomes non-linear instead.
Hooke's Law Experiment
We can conduct an experiment to investigate and prove the validity of Hooke's law consisting of a vertically hanging spring, different masses, and a meter stick. To this end, we must first calculate the weight of the mass
\[W=mg \]
to determine the force it exerts on the spring. We then calculate the spring's displacement by measuring the position of the spring before and after we attach the mass to it. The difference in the position of the spring is its displacement. To keep this information organized, we should build a table like the one below.
| Mass (kg) | Force (N) | Starting Position (m) | Final Position (m) | Displacement (m) |
| | | | |
| | | | |
Note that, since we're assuming that the spring is ideal, we do not include its mass in the table of measurements. Using different masses for the weight attached to the spring, we must repeat this process until we have gathered enough data points. Once we are finished, we can use the information to construct a graph of force vs. displacement to test if Hooke's law is correct. If the graph depicts a straight line of constant slope, it indicates a linear relationship between force and displacement.
Importance of Hooke’s Law
Hooke’s law is important because it describes the linear relationship between force and displacement. Consequently, it has played a role in many different aspects ranging from the invention of modern devices to the creation of new scientific disciplines. Some common inventions include retractable pens/pencils and spring scales seen in grocery stores. Note that the spring scale measures the compression or expansion of the spring and translates it to weight. The development of seismology, the study of earthquakes, was a result of Hooke's law. Engineers even rely on Hooke's law to aid them in their ability to calculate the stress or strain on bridges and skyscrapers.
Hooke’s law is named after the English experimental physicist, Robert Hooke, as he discovered the relationship between force and displacement. However, did you know that besides this discovery he also contributed to other scientific disciplines such as biology and astronomy? His contribution to biology came when he first discovered and coined the concept of “cells” through his observations of living plant matter. In the field of astronomy, he is credited as one of the first scientists to build a working Gregorian telescope.
Hooke's Law Examples
To solve problems related to Hooke's law, one can apply the equation
\[ F= -kx \]
to different situations. As we have defined Hooke's law and discussed the linear relationship between force and displacement, let us work through some examples to further our understanding. Note that before solving a problem, we must always remember these simple steps:
- Read the problem and identify all variables given within the problem.
- Determine what the problem is asking and what formulas are needed.
- Draw a picture if necessary to provide a visual aid.
- Apply the necessary formulas and solve the problem.
Example 1
Let us apply Hooke's law to calculate the spring constant of a system.
A spring is stretched \( 0.83\,\mathrm{m} \) by a \( 1.5\,\mathrm{kg} \) mass. Calculate the spring constant of the system. Note that the mass extending the spring is in equilibrium between gravity and tension, and the force of tension counteracting the pull of gravity is supplied by the spring.
Fig. 3: Using an ideal spring mass to calculate the spring constant. StudySmarter Original
After reading the problem, we are given the mass and displacement.
Therefore, before applying Hooke's law to determine the spring constant, we must first calculate force as follows:
$$\begin{align}F &= mg\\ F &= \left(1.5\,\mathrm{kg}\right)\left(9.8\,\mathrm{{\frac{m}{s}^2}}\right)\\F&= 14.7\,\mathrm{N}.\\\end{align}$$
Now using Hooke's law, our calculations will proceed as follows:
$$\begin{align}F&= -kx\\ 14.7\,\mathrm{N} &= -k \left(0.83\,\mathrm{m}\right)\\ \frac{14.7\,\mathrm{N}}{ 0.83\,\mathrm{m}} &= -k\\-k & = 17.7\,\mathrm{{\tfrac{N}{m}}}\\k & = -17.7\,\mathrm{{\tfrac{N}{m}}}.\\\end{align}$$
The spring constant of the system is \( k = -17.7\,\mathrm{{\frac{N}{m}}}. \)
Example 2
Let us apply Hooke's law to determine the displacement of a system.
A force of \( 12\,\mathrm{N} \) is applied to a spring, with a spring constant of \( 23\,\mathrm{{\frac{N}{m}}}. \) What is the displacement of the spring?
Fig. 4: Using an ideal spring to calculate displacement. StudySmarter Original
After reading the problem, we are given the force and the spring constant.
Therefore, applying Hooke's law to determine displacement, our calculations are as follows:
$$\begin{align}F&= -kx\\ 12\,\mathrm{N} & = -23\,\mathrm{{\tfrac{N}{m}}} \left(x\right)\\ \frac{12\,\mathrm{N}}{ -23\,\mathrm{\frac{N}{m}}} &= x\\x & = -0.52\,\mathrm{m}.\\\\\end{align}$$
The displacement of the spring is \( x =-0.52\,\mathrm{m}. \)
Hooke's Law - Key takeaways
- Hooke's law is a principle within physics that states that the force required to extend or compress an ideal spring by a certain distance is proportional to that distance.
- The mathematical formula of Hooke's law is \( F= -kx. \)
- The graph of Hooke's law shows that the linear relationship between force and displacement only applies until the spring reaches the limit of proportionality.
- Experiments demonstrate and prove the linear relationship between force and displacement., ie Hooke's law.
- From the invention of modern devices to creating scientific disciplines and aiding engineers in the construction of complex systems, Hooke’s law plays an important role.
References
- Figure 1: Demonstrating Hooke's law using a pen. (https://pixabay.com/photos/arm-hand-write-planner-planning-1284248/) by Pixabay. Licensed by CC0 1.0 Universal.
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