15.2 Energy in simple harmonic motion  (Page 5/8)

If you are interested in this interaction, find the force between the molecules by taking the derivative of the potential energy function. You will see immediately that the force does not resemble a Hooke’s law force $\left(F=\text{−}kx\right)$ , but if you are familiar with the binomial theorem:

the force can be approximated by a Hooke’s law force.

Velocity and energy conservation

Getting back to the system of a block and a spring in [link] , once the block is released from rest, it begins to move in the negative direction toward the equilibrium position. The potential energy decreases and the magnitude of the velocity and the kinetic energy increase. At time $t=T\text{/}4$ , the block reaches the equilibrium position $x=0.00\text{m,}$ where the force on the block and the potential energy are zero. At the equilibrium position, the block reaches a negative velocity with a magnitude equal to the maximum velocity $v=\text{−}A\omega$ . The kinetic energy is maximum and equal to $K=\frac{1}{2}m{v}^2=\frac{1}{2}m{A}^2{\omega }^2=\frac{1}{2}k{A}^2.$ At this point, the force on the block is zero, but momentum carries the block, and it continues in the negative direction toward $x=\text{−}A$ . As the block continues to move, the force on it acts in the positive direction and the magnitude of the velocity and kinetic energy decrease. The potential energy increases as the spring compresses. At time $t=T\text{/}2$ , the block reaches $x=\text{−}A$ . Here the velocity and kinetic energy are equal to zero. The force on the block is $F=+kA$ and the potential energy stored in the spring is $U=\frac{1}{2}k{A}^2$ . During the oscillations, the total energy is constant and equal to the sum of the potential energy and the kinetic energy of the system,

The equation for the energy associated with SHM can be solved to find the magnitude of the velocity at any position:

The energy in a simple harmonic oscillator is proportional to the square of the amplitude. When considering many forms of oscillations, you will find the energy proportional to the amplitude squared.

Summary

• The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = − kx , where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
• Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given by $U=\frac{1}{2}k{x}^2.$
• Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
  
  $E_{\text{Total}}=\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2=\frac{1}{2}k{A}^2=\text{constant.}$
• The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using
  
  $\left|v\right|=\sqrt{\frac{k}{m}\left(A^2-x^2\right)}.$

Conceptual questions

Problems