Mechanics and Energy Storage of Springs and Amusement Park Physics

Energy can be stored in various fields, such as gravitational potential energy.
Energy is capable of being transferred between different forms, such as the conversion from potential energy to kinetic energy.
The most significant concept in energy transfer is the Law of Conservation: energy is never lost during a transfer; it is always conserved.
As a fundamental physical law, energy is neither created nor destroyed; it merely changes from one form to another.

Amusement parks are considered excellent environments for physicists because physical principles are observable in nearly every attraction.
A specific example is the slingshot-style ride, which consists of a cage that typically holds two people.
Historically, older versions of this ride utilized massive bungee cords to provide energy. However, this was found to be dangerous, with instances of bungee cords breaking captured in viral videos.
Modern, safer versions of the slingshot ride use steel cables connected to hundreds of smaller springs rather than bungee cords. The energy for the launch is provided by these springs.

The fundamental behavior of a spring involves a restorative property: when stretched, it pulls back toward its original state; when compressed, it pushes back toward its original state.
The magnitude of the restorative force increases in proportion to the magnitude of the displacement (stretching or compression).
Springs are foundational in physics and appear in various complex systems, including molecular bonds and quantum mechanics.

To understand "what is going on" with a spring from a physicist's perspective involves finding a mathematical equation to describe and predict its behavior.
Experimental setup for modeling springs:
- One end of a spring is attached to a wall.
- The other end is attached to a force meter (a scale with a hook and dial).
- Force ( $F$ ) is plotted on the Y-axis in Newtons ( $\text{N}$ ).
- Displacement ( $x$ ) is plotted on the X-axis in meters ( $\text{m}$ ).
Displacement ( $x$ ) is defined as the distance the spring is stretched or compressed from its rest position ( $x = 0$ ).
- Positive displacement ( $+x$ ) indicates stretching; the spring exerts a negative force ( $-F$ ) pulling back toward zero.
- Negative displacement ( $-x$ ) indicates compression; the spring exerts a positive force ( $+F$ ) pushing back toward zero.
The resulting graph of force versus displacement is linear.
This linear relationship was discovered in the 1600s by English scientist Robert Hooke and is known as Hooke's Law.
The Spring Force Equation is defined as:

$F = -k \times x$

In this equation, $k$ represents the slope of the line, termed the spring constant.

The spring constant ( $k$ ) measures the stiffness of a spring.
Units for the spring constant are Newtons per meter ( $\text{N/m}$ ).
The value of $k$ varies based on the size, composition, and construction of the spring.
Example Calculation (Cookie Pen Spring):
- The force ( $F$ ) required to max out the spring was $2\,\text{ounces}$ , which equals approximately $0.55\,\text{N}$ .
- The displacement ( $x$ ) was $2\,\text{mm}$ , which is $0.002\,\text{m}$ . (Treated as $-0.002\,\text{m}$ as it is compression).
- Applying Hooke's Law: $0.55 = -k \times (-0.002)$ .
- $k = 275\,\text{N/m}$ .
Comparison of Constants:
- Small pen spring: $275\,\text{N/m}$ .
- Stronger springs require more force for the same displacement and have higher constants.
- Car wheel coils: Approximately $50,000\,\text{N/m}$ .

Springs store potential energy when they are compressed or stretched, which is released as kinetic energy when they are let go.
Work ( $W$ ) is generally defined as force multiplied by displacement: $W = F \times \Delta x$ .
Unlike lifting an object where force is constant, the force exerted by a spring changes as it is stretched.
To calculate the energy, the "average force trick" is used:
- Initial force at relaxation ( $x = 0$ ) is $0$ .
- Force at displacement ( $x$ ) is $k \times x$ .
- The average force is calculated as $\frac{1}{2} \times k \times x$ .
The total work done (and thus the stored energy) is the average force multiplied by the distance stretched:

$W = \left(\frac{1}{2} \times k \times x\right) \times x = \frac{1}{2} \times k \times x^2$

This value corresponds to the area under the curve of a Force vs. Displacement graph.
Notation for spring potential energy varies; it can be represented as $E$ , $U$ , $PE_s$ , or $E_s$ .

Modern versions of the ride may use up to $720$ springs attached in parallel.
Mechanism: The springs move a steel cable through a series of pulleys. For every $1.25\,\text{m}$ the springs compress or stretch, the ride launches humans to an altitude of $80\,\text{m}$ .
Variables provided:
- Mass of humans and steel ball ( $m$ ): $400\,\text{kg}$ .
- Launch Height ( $h$ ): $80\,\text{m}$ .
- Acceleration due to gravity ( $g$ ): $10\,\text{m/s}^2$ .
- Spring displacement ( $x$ ): $1.25\,\text{m}$ .
- Number of springs: Approximately $700$ (used for the simplified final calculation).
Step-by-step calculation to find the spring constant ( $k$ ):
- 1. Calculate the gravitational potential energy at the peak:

$PE_g = m \times g \times h = 400\,\text{kg} \times 10\,\text{m/s}^2 \times 80\,\text{m} = 320,000\,\text{J}$

*   2. Determine energy per individual spring:

$\frac{320,000\,\text{J}}{700\,\text{springs}} \approx 460\,\text{J/spring}$

*   3. Use the spring potential energy equation to solve for  $k$ :

$460 = \frac{1}{2} \times k \times (1.25)^2$

*   Doing the math reveals that  $k$  is approximately  $580\,\text{N/m}$ .

Unexpected Springs: Objects like an archer's bowstring, metal tweezers, or a chameleon's tongue (which acts like series of connected springs when launched) can be modeled using spring math.
Spring Division: A theoretical problem is posed regarding what happens to the spring constant ( $k$ ) if a physical spring is cut in half.
Energy Leakage: In real-world systems, like a falling bottle or a launched ride, energy may "leak" into the surrounding air via air resistance. This energy is not lost but is converted into heat; total energy remains conserved.
Roller Coaster Thrill: A debate exists regarding whether sitting in the front or the back of a roller coaster provides a better thrill, focusing on how a train with potential energy at the top of a hill maximizes speed as the cars descend.

Is it better to sit in the front or the back of a roller coaster?

The speaker notes that roller coasters operate on a $100\%$ conservation of energy principle. While potential energy is highest at the top of the first hill, the question of which seat maximizes the "thrill" or speed depends on the mechanics of the train cars as they transition from potential to kinetic energy throughout the descent.