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9

Indeed the problem with boundary conditions, generally speaking, is not well-posed. There are boundary conditions admitting no curves or admitting many curves, satisfying both these conditions and Euler-Lagrange equations. Examples. (1) Think of a particle constrained to stay on a smooth sphere where it can freely move. If you assign the North and the ...


4

This is the setup described in the equation: The acceleration is defined in terms os the displacement of the bow $x$ by: $$ a = 6000 \left(1 - \tfrac{4}{3}x\right) \tag{1} $$ So initially $x=0$ and when we substitute this into equation (1) we get $a = 6000 \text{ms}^{-2}$. When the arrow leaves the bow so $x=\tfrac{3}{4}$ and we get $a=0$. So far so ...


3

The oldest work on this preceeds quantum mechanics by more than 100 years. it was done by Malus in 1809 about experiments with polarized light. See http://www.mat.univie.ac.at/~neum/papers/physpapers.html#CQlightslides


3

The issue is that the underlying classical physics is determined by equations of motion (EOMs) (i.e. Newton's 2nd law), which are common for initial value problems (IVPs) and boundary value problems (BVPs). For BVPs , the EOMs can often alternatively be formulated as Euler-Lagrange (EL) equations of a stationary action principle. The latter approach does ...


2

What happens during folding is that the material undergoes plastic deformation. When a sheet of material is bent slightly that deformation is usually elastic, meaning it will return to its original shape when the deforming stress is withdrawn. But when the deformation is larger we enter the plastic zone: the material will no longer fully recover its ...


2

In the case of inorganic matter, such as metal sheets, folding/creasing produces a substantial bulk stress in the material which can modify the molecular structure in a large number of complex (and not fully understood) ways. For example, it can break bonds, cause amorphisation, and propagate dislocations. These same mechanisms are at play when you cut a ...


1

You forgot about the free body diagram for the cupboard! It is pushing on the earth too. So you can't apply a net force to the earth by pushing between the floor and the cupboard which are both fastened to the earth.


1

Whatever you are doing, I think you are making this too complicated. The normal way to solve this problem (approximately) is to ignore all the terms in the Fourier expansion, except the one at the resonant frequency of the swing. Then the solution is just the normal time-dependent solution for a damped oscillator forced on resonance.


1

The force of friction is defined as $F_f = \mu N$, where $N$ is the normal force. In the case of a flat surface free of external forces, you can use Newton's laws to determine that $N = mg$, where $m$ is the mass of the object. Notice that we have made no reference to the objects size, or area of contact. This is because in these examples we have ...


1

About second part of your question, I should say that I couldn't understand it because it may the polygon like a star has no side contacted with the ground. About first part of your question, I should say "It is not possible". “Let us say a polygon shaped object is stable on a side when the center of mass "falls" inside the base”. If you accept this phrase ...


1

Yes, the viscosity of a dilatant suspension will decrease if the viscosity of the solvent decreases. Surprisingly I struggled to find experimental data to back this up. Perhaps everyone thinks it's too obvious to be worth publishing. The best I could do is this school exeriment report. The authors timed the fall of a ball through the suspension, so lower ...



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