# 2-d equations of motion for a Slinky going down stairs?

• benorin
In summary, The conversation discusses the problem of finding the 2-d equations of motion for a Slinky going down a flight of stairs, with the assumption that the path of the slinky is planar. The speaker mentions their fascination with this problem in lower division physics and their lack of expertise in physics compared to math. They also mention some equations that could be useful in solving the problem and ask for some guidance. The other person expresses doubt about the motion resulting in a tweaked cycloid and discusses the complexity of analyzing slinky motion. They also provide links to two papers that have studied the behavior of a slinky on stairs, one with a rigorous treatment and one with a simplified model using Lagrangian formalism.
benorin
Homework Helper
This problem fascinated me in lower division physics. Find the 2-d equations of motion for a Slinky going down a flight of stairs (assuming the path of the slinky is planar; eg only going up and down and front and back, no side to side). I do confess that whilst I do love physics I’m not terribly good at it, better with math. It’s been 20 years since physics but I still remember ##F=-kx ## seems a good place to start, or was it ##F=\tfrac{d\vec{p}}{dt}##? We’ll need some initial displacement of the slinky too. I’ll need some clues as to how to proceed... we’re going to wind up with a tweaked cycloid I bet?

benorin said:
we’re going to wind up with a tweaked cycloid I bet?
I certainly would not expect that to be the case. Think about it. The center of mass, which is what you have to be looking for the motion of, is going to be static for noticeable periods while the top of the slinky moves forward and then again briefly as the top catches up on the next step down.

Slinkies can actually get super complicated, math-wise.
This paper has the most rigorous treatment that I know of, and includes discussions on the shapes formed by a slinky under various boundary conditions: https://arxiv.org/pdf/1403.6809.pdf

This paper from the American Journal of physics uses a simplified model of the slinky and a Lagrangian formalism to look at its stair-hopping behavior: https://aapt.scitation.org/doi/full/10.1119/1.3225921

vanhees71 and anorlunda

## 1. What is a 2-d equation of motion for a Slinky going down stairs?

The 2-d equation of motion for a Slinky going down stairs is a mathematical representation of the motion of the Slinky as it moves down a set of stairs. It takes into account both the vertical and horizontal components of the Slinky's motion.

## 2. How is the 2-d equation of motion for a Slinky derived?

The 2-d equation of motion for a Slinky is derived using Newton's laws of motion and the principles of kinematics. It involves breaking down the motion into its individual components and using equations such as velocity, acceleration, and distance to describe the Slinky's movement.

## 3. What factors affect the 2-d motion of a Slinky going down stairs?

The 2-d motion of a Slinky going down stairs is affected by several factors, including the mass of the Slinky, the angle of the stairs, the force of gravity, and any external forces acting on the Slinky.

## 4. Can the 2-d equation of motion be used to predict the behavior of a Slinky going down stairs?

Yes, the 2-d equation of motion can be used to predict the behavior of a Slinky going down stairs. However, it is important to note that the equation is based on certain assumptions and may not accurately reflect real-world conditions.

## 5. Are there any limitations to the 2-d equation of motion for a Slinky going down stairs?

Yes, there are limitations to the 2-d equation of motion for a Slinky going down stairs. It assumes that the Slinky is a perfect spring with no internal friction or resistance, and that the stairs are a smooth and even surface. In reality, these factors may affect the Slinky's motion and cause deviations from the predicted behavior.

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