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How could I estimate the force I apply when I squeeze a new toothpaste tube? I want to either be able to calculate it without having to perform an experiment, or refer to some scientific journal with a similar calculation or rough estimate.

I was thinking that I could use Hooke's law and calculate the force exerted when a similar force to that of gently squeezing the tube is exerted on a spring with a spring constant similar to the resistivity of the toothpaste tube...I could make numbers up (as the value of the force I am looking for can very much be an estimate), but I have no clue what an estimate k value and x value would be to calculate the force using Hooke's law.

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I can give you an idea. The force you are exerting per area on the toothpaste cover is due to the viscosity of the toothpaste. I don't think there is a paper published on toothpaste. But there is several good experiments on the viscosity of fluids. Renold's Number and several others can help.

Hook's law is only applicable on those physical materials which regain back their original shape. As the toothpaste does not regain back the original position, you cannot apply that. And in no way the toothpaste with the paste act like a spring.

You can conduct this experiment which would be better provided that you are ready to sacrifice many toothpaste as waste.

1)Take the toothpaste and put series of weights on it. you may have a weighing machine the amount of toothpaste that comes out weigh it.

2)once you have done that you can assert that the weight put on the paste is directly praportional to the weight of the paste that comes out. $$W=kw_t$$ here if my assumption is correct then you can plot a graph of the weights vs the amount of paste. by this you may get a rough estimate of the force.

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Toothpaste is a Non-Newtonian Fluid, hence you cannot treat it using the Hooks Law.

Is a fluid of the class Bingham plastic, that means that applying a stress on the toothpaste tube, there is a threshold before which the fluid don't flows.

Shear stress is given by $\sigma= \frac{Force}{Area}$. Since the hole of the tube has a fixed area, you'll have to apply a force $F_0$ to start the toothpaste flowing out of the container (see figure 1 in the link). In other words you have to provide an initial Yeld stress.

Unfortunately every fluid has its rehogram, and the rehogram depends from the composition, so it is very difficult to estimate the force to squeeze a toothpaste tube

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For one, it depends whether the tube is full or not and also where the force is applied. As a very rough estimate my guess is that when it's full or near full it takes about $1/5 * 9.8 N \approx 2N$ (same force as putting a 200 g object over the tube to squeeze the toothpaste).

Now when the tube is almost empty, I personally have to press much stronger (around 2 to 3 *9.8N so about 30 N at most, same as if a mass of 3 kg was over the tube at a particular spot, where my fingers are pressing).

To sum up, since you asked for a rough estimate, with common sense I get a force of between 2 N and 30 N when using fingers (the force is applied on an area equal to that of fingers pressing the tube), depending on how full the tube is.

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