10. Conservation of Energy

A particle that can move along the x-axis is part of a system with potential energy U(x) = A/x² − B/x where A and B are positive constants. Where are the particle's equilibrium positions?

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. At what value of 𝓍 is the magnitude of the force a maximum?

A system consists of interacting objects A and B, each exerting a constant 3.0 N pull on the other. What is ∆U for the system if A moves 1.0 m toward B while B moves 2.0 m toward A?

Determine the escape velocity from the Sun for an object 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.

Determine the escape velocity from the Sun for an object at the Sun’s surface ( r = 7.0 x 10⁵ km , M = 2.0 x 10³⁰ kg).

In FIGURE EX10.28, what is the maximum speed a 200 g particle could have at x = 2.0 m and never reach x = 6.0 m?

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

FIGURE EX10.25 is the potential-energy diagram for a 20 g particle that is released from rest at x = 1.0 m. What is the particle's maximum speed? At what position does it have this speed?

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?

A clever engineer designs a 'sprong' that obeys the force law F_x=−q(x−x_eq)³ , where x_eq is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, we'll let x_eq = 0 m .Then F_x = −qx³. Find an expression for the potential energy of a stretched or compressed sprong.