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Suppose your friend claims to have discovered a mysterious force in nature that acts on all particles in some region of space. He tells you
Question
Suppose your friend claims to have discovered a mysterious force in nature that acts on all particles in some region of space. He tells you that the force is always pointed away a definite point in space, which we can call the force center. The magnitude of the force turns out to be proportional to B/r3, where r is the distance from the force center to any other point. Your friend says that it has been determined that the constant of proportionality has been determined to be B= 2 (in units to be determined later), so that the magnitude of the force on a particle (in newtons) can be written as 2r^3, when the particle is at a distance r from the force center.
Required:
Write an expression of potential energy.
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Physics
4 years
2021-08-16T19:45:50+00:00
2021-08-16T19:45:50+00:00 1 Answers
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Answers ( )
Answer:
U = 1 / r²
Explanation:
In this exercise they do not ask for potential energy giving the expression of force, since these two quantities are related
F = – dU / dr
this derivative is a gradient, that is, a directional derivative, so we must have
dU = – F. dr
the esxresion for strength is
F = B / r³
let’s replace
∫ dU = – ∫ B / r³ dr
in this case the force and the displacement are parallel, therefore the scalar product is reduced to the algebraic product
let’s evaluate the integrals
U – Uo = -B (- / 2r² + 1 / 2r₀²)
To complete the calculation we must fix the energy at a point, in general the most common choice is to make the potential energy zero (Uo = 0) for when the distance is infinite (r = ∞)
U = B / 2r²
we substitute the value of B = 2
U = 1 / r²