The mechanical energy of a particle in a potential U(x) is E = = mv +U...

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The mechanical energy of a particle in a potential U(x) is E = = mv² +U (x), mv². and should be constant in the absence of dissipation. For a harmonic oscillator with F = − kx, U(x) = kx²/2; note that F = −dU/dx. One check on the quality of a numerical solution of a conservative system is to see if the energy is constant in the numerical solution. 2.2. Now consider motion in the Morse potential, which models the potential energy of vibrating diatomic molecules. The Morse potential function U(x) = A[1-e (x-B)/C1², (5) where A> 0 is a constant with units of energy and B > 0 and C> 0 are constants with units of length. Here, x > 0 represents the separation between the two molecules. The force asso- ciated with the Morse potential is F(x)= -(x−B)/C[1 − e¯(x−B)/C]. 2.2.1. Argue that the point x = B is a stable static equilibrium point, i.e., a point where a parti- cle initially at rest will remain for all time, and where a small disturbance will result in a force that tends to bring the particle back to B. Then, linearize the equation of motion around x = B in the limit of small displacements and show that for small displacements, the particle describes SHM with w? = 2A/(mC²). == du dx (4) 2A C e (6) The mechanical energy of a particle in a potential U(x) is E = = mv² +U (x), mv². and should be constant in the absence of dissipation. For a harmonic oscillator with F = − kx, U(x) = kx²/2; note that F = −dU/dx. One check on the quality of a numerical solution of a conservative system is to see if the energy is constant in the numerical solution. 2.2. Now consider motion in the Morse potential, which models the potential energy of vibrating diatomic molecules. The Morse potential function U(x) = A[1-e (x-B)/C1², (5) where A> 0 is a constant with units of energy and B > 0 and C> 0 are constants with units of length. Here, x > 0 represents the separation between the two molecules. The force asso- ciated with the Morse potential is F(x)= -(x−B)/C[1 − e¯(x−B)/C]. 2.2.1. Argue that the point x = B is a stable static equilibrium point, i.e., a point where a parti- cle initially at rest will remain for all time, and where a small disturbance will result in a force that tends to bring the particle back to B. Then, linearize the equation of motion around x = B in the limit of small displacements and show that for small displacements, the particle describes SHM with w? = 2A/(mC²). == du dx (4) 2A C e (6) The mechanical energy of a particle in a potential U(x) is E = = mv² +U (x), mv². and should be constant in the absence of dissipation. For a harmonic oscillator with F = − kx, U(x) = kx²/2; note that F = −dU/dx. One check on the quality of a numerical solution of a conservative system is to see if the energy is constant in the numerical solution. 2.2. Now consider motion in the Morse potential, which models the potential energy of vibrating diatomic molecules. The Morse potential function U(x) = A[1-e (x-B)/C1², (5) where A> 0 is a constant with units of energy and B > 0 and C> 0 are constants with units of length. Here, x > 0 represents the separation between the two molecules. The force asso- ciated with the Morse potential is F(x)= -(x−B)/C[1 − e¯(x−B)/C]. 2.2.1. Argue that the point x = B is a stable static equilibrium point, i.e., a point where a parti- cle initially at rest will remain for all time, and where a small disturbance will result in a force that tends to bring the particle back to B. Then, linearize the equation of motion around x = B in the limit of small displacements and show that for small displacements, the particle describes SHM with w? = 2A/(mC²). == du dx (4) 2A C e (6) The mechanical energy of a particle in a potential U(x) is E = = mv² +U (x), mv². and should be constant in the absence of dissipation. For a harmonic oscillator with F = − kx, U(x) = kx²/2; note that F = −dU/dx. One check on the quality of a numerical solution of a conservative system is to see if the energy is constant in the numerical solution. 2.2. Now consider motion in the Morse potential, which models the potential energy of vibrating diatomic molecules. The Morse potential function U(x) = A[1-e (x-B)/C1², (5) where A> 0 is a constant with units of energy and B > 0 and C> 0 are constants with units of length. Here, x > 0 represents the separation between the two molecules. The force asso- ciated with the Morse potential is F(x)= -(x−B)/C[1 − e¯(x−B)/C]. 2.2.1. Argue that the point x = B is a stable static equilibrium point, i.e., a point where a parti- cle initially at rest will remain for all time, and where a small disturbance will result in a force that tends to bring the particle back to B. Then, linearize the equation of motion around x = B in the limit of small displacements and show that for small displacements, the particle describes SHM with w? = 2A/(mC²). == du dx (4) 2A C e (6)

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