10. Conservation of Energy

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?

(II) Determine the escape velocity from the Sun for an object:

(a) at the Sun’s surface ( r = 7.0 x 10⁵ km , M = 2.0 x 10³⁰ kg);

(II) Determine the escape velocity from the Sun for an object:

(b) at the average distance of the Earth ( 1.50 x 10⁸ km . Compare (give factor for each) to the speed of the Earth in its orbit.

II) A particle is constrained to move in one dimension along the x axis and is acted upon by a force given by

F→(x) = - (k/x³) î

where k is a constant with units appropriate to the SI system. Find the potential energy function U(x) , if U is arbitrarily defined to be zero at x = 2.0m , so that U (2.0m) = 0.

(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.

(b) At what value(s) of 𝓍 is the magnitude of the force a minimum?

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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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(I) The escape velocity from planet A is double that for planet B. The two planets have the same mass. What is the ratio of their radii, r_A / r_B?