Is Physics Broken?! The Spring Paradox is an interesting one to solve.

Hey guys, so in this video I’m actually answering a question posed by one of you in a comment on one of my previous videos. Ahitagni Das, thank you for asking this question because it’s a really interesting one to answer.

The idea is as follows: Let’s say we have a spring sitting vertically on the ground, at its natural length. Then we place a block on top of the spring, which will cause the spring to compress by a certain amount. We want to try and calculate how much the spring compresses by, and we can do this in two different ways.

Firstly, we can consider balancing forces. The equilibrium position will be when the downward force on the block (i.e. its weight) will be exactly counteracted by the upward force exerted by the spring. This upward force is given by Hooke’s Law, and we can use it to calculate that the equilibrium compression of the spring, x, is given by x = mg/k where m is the mass of the block, g is the gravitational field strength on Earth, and k is the spring constant.

We can also consider the Law of Conservation of Energy, and the supposedly logical thing to do is to say that the gravitational potential energy lost by the block during the compression (mgx) is equal to the elastic potential energy subsequently stored in the spring as it compresses (0.5kx^2). If we equate the loss in GPE to the gain in EPE then we find that x = 2mg/k. This is a factor of 2 different from the answer we got when considering forces!

So we get two different answers for the same scenario using two different (supposedly) logical arguments. In this video, I dig through the logic used in both cases and show where our arguments are incorrect. Don’t worry, we do manage to resolve the spring paradox, and we can breathe a sigh of relief because physics is not quite broken.

Let me know what you think of the video guys, and if you enjoy it then please leave a thumbs up. Subscribe if you haven’t already, and follow me on Instagram @parthvlogs for more concise physics content!

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Each frame is a snapshot of a different parametric line. The formula is changed slightly between frames to create a smoothly changing video. Idea came from reading Chapter 3 of Frank Farris’s book “Creating Symmetry”. All the math necessary to replicate this is covered there. Programmed and frames generated using MATLAB.

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