(II) The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero.
(b) At what value(s) of 𝓍 is the magnitude of the force a minimum?
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(III) The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential)

U(r) = -(a/r⁶) + (b/r¹²) ,

where r is the distance between the two atoms and a and b are positive constants.

(a) At what values of r is U(r) a minimum? A maximum?

(III) The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential)

U(r) = -(a/r⁶) + (b/r¹²) ,

where r is the distance between the two atoms and a and b are positive constants.

(e) Let F be the force one atom exerts on the other. For what values of r is F > 0 , F < 0 , F = 0?

The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or F(r) = F₀ [2(σ/r)¹³ - (σ/r)⁷], where r is the separation between the two atoms, and σ and F₀ are constants. For an oxygen molecule (which is diatomic) F₀ = 9.60 x 10⁻¹¹ N and σ = 3.50 x 100⁻¹¹ m .

(a) Integrate the equation for F(r) to determine the potential energy U(r) of the oxygen molecule.

(II) The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero.

(a) In which interval(s) of 𝓍 is the force on the particle to the right?

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(II) The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the x axis under the influence of a conservative force. Note that the total energy E > U(x), so that the particle’s speed is never zero.

(c) At what value of 𝓍 is the magnitude of the force a maximum?

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A particle with a mass of 0.1kg moves according to the Potential Energy graph shown. What minimum speed does the particle need at Point A to reach Point B?