Suppose your friend claims to have discovered a mysterious force in nature that acts on all particles in some region of space. H

Question

3 months ago

11

Suppose your friend claims to have discovered a mysterious force in nature that acts on all particles in some region of space. H

e 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.

Physics

1 answer:

ValentinkaMS [3.4K] · Answer 1 · 2026-03-10T19:06:09+03:00

Answer:

U = 1 / r²

Explanation:

In this problem, the task does not require calculating potential energy via the force equation since these two variables are interconnected.

F = - dU / dr

This derivative represents a gradient, meaning it indicates direction, leading us to write

dU = - F. dr

The formula for force becomes

F = B / r³

Now, let’s apply this in the integral:

∫ dU = - ∫ B / r³ dr

Here, the force aligns with the displacement, simplifying the scalar product to the product of magnitudes.

Now, we can solve the integrals:

U - Uo = -B (- / 2r² + 1 / 2r₀²)

To finalize the calculations, a reference point for energy must be designated; commonly, potential energy is set to zero (Uo = 0) at infinity (r = ∞).

U = B / 2r²

Substituting B = 2, we arrive at:

U = 1 / r²