15.4:
Energy in Simple Harmonic Motion
Consider a block of mass m attached to a spring with force constant k on a frictionless surface at equilibrium.
When the block is displaced, the work done by the spring force along the displacement from its initial to final position equals the amount of potential energy stored in the spring relative to its displacement. This is called elastic potential energy.
When released, the block undergoes simple harmonic motion. The energy required for its back-and-forth movement is called translational kinetic energy, and is directly proportional to the square of its velocity and its mass.
During oscillations, both energies continuously interchange and are represented by sinusoidal waveforms. The elastic potential energy is at maximum at the maximum displacement, while the translational kinetic energy is at maximum at the equilibrium position.
At other positions, the block has different kinetic and potential energy values, and their sum equals the system's total energy.
So, the total energy in the system remains constant and conserved, as it oscillates between translational kinetic energy and elastic potential energy.
15.4:
Energy in Simple Harmonic Motion
To determine the energy of a simple harmonic oscillator, consider all the forms of energy it can have during its simple harmonic motion. According to Hooke's Law, the energy stored during the compression/stretching of a string in a simple harmonic oscillator is potential energy. As the simple harmonic oscillator has no dissipative forces, it also possesses kinetic energy. In the presence of conservative forces, both energies can interconvert during oscillation, but the total energy remains constant. The total energy for a simple harmonic oscillator is equal to the sum of the potential and kinetic energy and is proportional to the square of the amplitude. It can be expressed in the following form:
The magnitude of the velocity in a simple harmonic motion is obtained by rearranging and solving the equations of the total energy.
Manipulating this expression algebraically gives the following:
where
Notice that the maximum velocity depends on three factors and is proportional to the amplitude. If the displacement is maximal, the velocity will also be maximal. Additionally, the maximum velocity is greater for stiffer systems because they exert greater force for the same displacement. This observation can be seen in the expression for the maximum velocity. The maximum velocity is proportional to the square root of the force constant. Finally, the maximum velocity is smaller for objects with larger masses since the maximum velocity is inversely proportional to the square root of the mass.
Suggested Reading
- OpenStax. (2020). College Physics [Web version]. (Pg. No. 680-682) https://openstax.org/details/books/college-physics