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I think $U$ must be the potential energy to bring the last particle in from infinity. Thus $U/5$ is the average pairwise potential between this last bead and each of its neighbors. By symmetry though, each bead has this same average pairwise potential with the other beads. The total potential of the configuration is half the sum of the pairwise potential of ...


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There follows my try to decompose the solution into a minimal amount of calculation and apart from that only geometrical considerations. The centripetal acceleration is $a_c=\frac{v^2}R$. It is directed towards the center. We define the $z$ coordinate as starting at the top and pointing vertically downwards (see the following Figure). The conservation ...


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OP considers an equations of motion of the form $$\tag{1}\dot{\bf x}~=~{\bf B}({\bf x}),$$ where the vector field ${\bf B}$ is of the form$^1$ $$\tag{2} {\bf B}~=~{\bf \nabla}\times {\bf A}.$$ In other words, ${\bf B}$ is divergence-free $$\tag{3} {\bf \nabla}\cdot {\bf B}~=~0.$$ Eq.(3) is locally eqivalent to eq. (2), cf. Poincare's Lemma. Let ...


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A mass moving in a circle has centripetal acceleration $v^2/r$ directed toward the center of the circle. You can get $v$ from potential energy. The mass here has two forces on it. Gravity is constant and down. The reaction force of the surface (assuming no friction) is normal to the surface. When the sum of these two forces becomes less than centripetal ...


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The quote is taken from just above eq. (1.32) in Ref. 1: [...] If the internal forces are also conservative, then the mutual forces between the $i$th and $j$th particles, ${\bf F}_{ij}$ and ${\bf F}_{ji}$, can be obtained from a potential function $V_{ij}$. To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance ...


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The strong law of action and reaction says that the forces that two bodies exert on each other have the same magnitude, opposite direction and act along the line joining the particles. When you want that last bit to be true and you want to write the force on particle $i$ as $-\nabla_i V_{ij}$, then the potential has to be a function of the relative distance. ...


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I have no background in this matter, but I think some basic intuition is in order. Sliding down: Suppose you have chain of rubber balls connected by elastic springs. Hold up the chain and let it dangle. Notice that the energy of the system is exactly as above, with $E(R_n)=mgZ_n$ where $Z_n$ is the height of ball $n$. What will it look like? Numbering ...


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The dot product is missing. The integral must also be multiplied by a Cosine of the angle between the vectors. $dl$ and $dr$ are the same thing. It's just an infinitesimally small distance on the direction of the field.


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The energy stored in the spring is equal to the work done in compressing it. The force needed to compress the spring to the maximum is only a small part of this calculation. Suppose the spring is completely relaxed, and you apply a small force compressing the spring slightly. The energy stored is the product of the force and the distance, measured in ...


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I am assuming that by 'winch's force' you mean the maximum force that can be generated by the torque produced in the winch. If this force is not higher than the load, then no it cannot pull it. The maximum potential energy the winch can hold depends on how much turns it is given, the properties of the spring are totally irrelevant here. w = mgh where m is ...


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When using "your aproach" you are assuming that the mass is going to move exactly into the equilibrium position and stay still there. Instead, the mass will descend with some velocity till that point and overpass it, and then bounce back and forth like a harmonic oscilator. In real life the oscillation will vanish due to friction after some time, and the ...



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