Concept #1: Relationship Between Force and Potential Energy (In 1-Dimension)

Concept #2: Partial Derivatives

Concept #3: General Relationship Between Force and Potential Energy

A potential energy function is given by
U = Ax3 + Bx2 - Cx + D
Find the stable equilibrium position for A = 3.45 J/m 3, B = 1.05 J/m 2, C = 1.7 J/m, and D = 4.2 J.

The following graph represents a hypothetical potential energy curve for a particle of mass m.If the particle is released from rest at position r 0, its speed ||→v|| at position 2r0 is most nearly

The plot shown represents the potential energy of a particle that can only move along the x-axis, as a function of the coordinate x. Which of the following statements is correct:[a] In B and D the force is zero;[b] The force in B is positive;[c] A and C are stable equilibrium positions;[d] The force in D is positive;[e] A and C are unstable equilibrium positions.

The potential energy between two neutral atoms is called the Lennard-Jones potential, and is given by the equation
U(x) = A/r12 - B/r6
where A and B are positive constants.
a) What is the force due to this potential energy?
b) For what range of r is the force attractive, and for what range of r is the force repulsive? Note that if the force is positive, it is repulsive, and if the force is negative, it is attractive.

A single conservative force acts on a 5.00-kg particle within a system due to its interaction with the rest of the system. The equation Fx = 2x + 4 describes the force, where Fx is in newtons and x is in meters. As the particle moves along the x axis from x = 1.00 m to x = 5.00 m, calculate(a) the work done by this force on the particle(b) the change in the potential energy of the system(c) the kinetic energy the particle has at x = 5.00 m if its speed is 3.00 m/s at x = 1.00 m

A particle that can move along the x-axis experiences an interaction force
Fx=(3x2-5x) N where x is in m. Find an expression for the system's potential energy.

The following image is the potential-energy diagram for a 500 g particle that is released from rest at A.(a) What is the particles speed at B?(b) What is the particles speed at C?(c) What is the particles speed at D?