23

What is wrong with your argument is this paragraph: If we imagine the chain as having many small segments, then the potential energy of each segment is $E_p=mgh$. As the number of small segments approaches infinity, their masses equalize because the difference in mass between any two segments goes to $0$ as the number of segments goes to infinity. ...


22

To put the accepted answer in mathematical terms, if you have a curve $y(x)$, hanging fixed at $x_0$ and $x_L$ at an height $h=y(x_0)=y(x_L)$, of total length $L$ and mass $M$ then then linear mass density is going to be $\lambda = M/L$. The length of the curve is given by the integral of the arc-length $$L=\int_{x_0}^{x_L} \sqrt{ 1+\left({dy \over dx}\right)...


7

The Navier-Stokes equation describes the motion of some infinitesimal volume of the fluid. That is we divide the fluid up into tiny volumes $dV$ and the equation tells us how these tiny volumes move. The overall motion of the fluid comes from combining the motions of all these tiny volumes. So the equation is really: $$ \rho \left( \dfrac{\partial v}{\...


7

There are several presentations of Zeno's paradoxes. One version is as follows (taken from Wikipedia): Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, ...


6

how long would it take for something really far away, say 300,000,000 meters, to feel the tension from the pull of a rope? When you first pull a rope you produce a mechanical wave in the rope. This wave travels at a characteristic speed called the speed of sound. The speed of sound differs for different materials. The speed of sound in a steel cable is ...


6

One objection to discrete spacetime for me is that it requires some degree of anisotropy. For example, if the space is formed by a grid with spacements at the Planck scale, some directions should be more compact than others. It is what happens in crystal lattices, and affect its macroscopic properties.


5

The tension is due to the weight of portion of the rope below the point $x$ so as you go up there’s a greater portion of the rope and thus a greater portion of the mass of the rope to pull the little portion of the rope at $x$. Note that since the tension is “local” (it changes at every $x$ rather than being constant throughout), you also need the “local” ...


4

The steps in the derivation are as follows: If we have point masses $m_i$ each moving with velocity $v_i$ then the total momentum of the system is $\vec P = \sum m_i \vec v_i$. We can show that $\frac {d \vec P}{dt} = \vec F$ where $\vec F$ is the net external force on the system (we show this by applying Newton's second law to the individual point masses ...


4

While answers posted so far have correct mathematical descriptions, I will look at what you are requesting for thinking about what a point on the rope experiences and how it can come to rest after the wave passes. We will look at a Gaussian pulse traveling down a rope, as shown below Now, according to the wave equation (which can be derived from thinking ...


4

A simplified explanation is as follows. When two molecules are some distance apart, there are some attractive forces between them, which dominate over the repulsive forces. As they keep coming closer, the repulsive forces keep getting stronger until at a critical distance, the forces are balanced. The string when lying on the ground, is in equilibrium, ...


4

This limit of a sum is by definition the integral of the chain curve. There's no such thing as "the integral of the chain curve". Curves don't have integrals. For an integral, you need three things: an integrand, measure function, and a set over which those are defined (in basic integrals, these are the function you're integrating, the ...


3

Overall tension is a body reaction force to some stimulating force (weight,pulling,external, etc) and is a specific case of a more general body stress vector field, like : And yes, tension increases proportionally to the force applied. Materials break when pressure applied to them exceeds/reaches ultimate tensile strength, measured in $\text{Pa}$ : In ...


3

Tension force like the normal force is just an aspect of the electromagnetic forces acting between molecules. 1 : You can't push something with a rope and if you try to do that the rope will buckle . Why ? For this to understand , lets take an example of magnets. When you bring like poles closer and closer , you experience greater and greater repulsive ...


3

Atoms form molecules, and molecules are arranged to make structures such as strings and things. There are generally-attractive electromagnetic forces between atoms which bond the atoms into molecules. These are generally strong. Chemical action, at a minimum, is needed destroy molecules by changing the electrical potential energy of the system. When you make ...


3

The way I understand this is that a model for a body with extension is that of a collection of $N$ point particles. Suppose that these have masses $m_1,\dots,m_N$ and follow trajectories $\vec{r}_1,\dots,\vec{r}_N$. Then we define the center of mass as an "imaginary" particle, whose mass is the total mass of the body $m=\sum_{i=1}^Nm_i$ and whose ...


3

Tension provides a restoring force, which is necessary to have oscillations (like a spring or gravity for pendulums). Otherwise any attempt to excite waves will only produce an inelastic deformation. However, if the gravity is present, there will be tension created due to the non-zero mass of the string which might be sufficient to have waves.


3

As far as I know, in continuum mechanics a body is discribed by a fixed orientable Riemannian manifold $B$ together with a (time-dependent) embedding in space, e.g. the $\mathbb{R}^3$: $$X_t:B\rightarrow\mathbb{R}^3.$$ You could then define a free Lagrangian as a integral over the model body $B$: $$L(X)=\int_B \frac{1}{2} \rho \langle\partial_t X, \partial_t ...


3

Your first equation determines the wave speed based on the properties of the medium the wave is propagating through. Your equation is specifically for a wave on a string. The equation makes sense qualitatively. If the string tension is greater, then the wave will propagate faster because there is a larger restoring force in the system. If the density of the ...


3

The simplest formulation for almost all solid mechanics modelling is Lagrangian, not Eulerian. You are considering the motion of an element of the structure which has constant mass, and therefore mass conservation is "baked in" to the model. You don't need an equation to represent it. The continuity equation would determine the local density ...


3

The speed of an impulse in the material is by definition the speed of sound. If you push or pull one end of the rod with a faster speed, you would create a shock wave in the material similar to the one created by a supersonic jet in the atmosphere: This would change the properties and integrity of the solid material and eventually destroy it. Even if your ...


3

It could be due to the torque which is directly proportional to the height of the tower. In fact when you lay the $n_{th}$ on the top it is impossible for your fingers to give it initial velocities equal to zero (due to vibrations caused by blood pressure) along the x-y directions. So we would have an impulse $I = mv=F dt$. So higher the tower higher the ...


3

I assume you're asking about compression or elongation of this structure along the primary helical axis. Let's first look at the standard helical spring model, which translates axial spring deflection into torsion of the bulk material: For simple torsion, the relationship between the twist angle $\theta$ and the torque $T$ for a circular cross section of ...


3

You replace your rope by a chain of $N$ rope segments (what your source calls the molecules, however that could be misleading, so stick with rope segments) with mass $m$. Each of them has the weight $F = mg$. Let's label them such that the lowest one (in a coordinate system where the vector $\textbf{g}$ points downwards) has index $1$, the second lowest ...


3

Zeno's idea of movement was different from ours. That is, as seen from his arrow paradox (I'm sure that he was very well aware of the non-reality of his thoughts). In his mental image of an arrow standing still there isn't the possibility for the arrow to go on. It has to remain forever at the position he images. The arrow hasn't got the internal quality &...


2

Any $\rho$ which solves the equation on the whole torus must also be a solution locally on every subset. In particular, it must be solution on the (non-toroidal) open $L \times L $ square. Since solutions on the torus are a subset of the solutions on the square, the question becomes: Do there exist solutions on the square which happen to match at the ...


2

The shear strain is due to the wire of the spring being under torsion (twist). The torsion arises because the force, F, due to the mass hanging from the spring acts along the (vertical) axis of the spring, which runs through the centres of the turns (or radius r, say). This force therefore exerts a torque $Fr$ on the wire of the spring. The torque is ...


2

$$\newcommand{\md}{\mathrm{d}}$$Yes, you are right. In the most general case, we must use $\md m = \mu\, \md s$ for the mass of the infinitesimally small element of the rope. In essence, what that equation is saying is that the mass of the small section of rope is the equal to the mass per unit length times the length of the small section. I don't know where ...


2

It's interesting to pose the problem as everything being level but each block having a small random Gaussian displacement from the one below. The displacements accumulate as the tower rises, so the position of the top block does a random walk away from the position of the bottom block. It seems sufficient to consider this in just one dimension. Let the ...


2

I think a better way of phrasing your question is, "When is the material derivative useful?" The material derivative $\frac{D}{Dt}$ is useful in the sense that the physical laws one finds for rigid bodies readily translate to continuum mechanical laws using the material derivative. For example, conservation of mass is written down as $$\frac{D(\rho ...


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