Understanding Energy in Simple Harmonic Motion

Energy in simple harmonic motion (SHM) is a fundamental aspect of oscillatory systems in physics. SHM describes the periodic back-and-forth motion of a particle, where energy continuously shifts between kinetic and potential forms while the total mechanical energy remains constant throughout the motion.

Potential Energy in Simple Harmonic Motion

In SHM, potential energy arises due to the displacement of the particle from its equilibrium position. The restoring force can be represented as $F = -k x$, where $k$ is the force constant and $x$ is the displacement. The potential energy at displacement $x$ is given by $U = \dfrac{1}{2} k x^2$.

Since the relation $k = m \omega^2$ holds for SHM, the potential energy can also be written as $U = \dfrac{1}{2} m \omega^2 x^2$, where $m$ is the mass and $\omega$ is the angular frequency.

The potential energy is maximum at the extreme positions of the oscillation, where displacement is equal to the amplitude $A$. At $x = A$, $U_{max} = \dfrac{1}{2} m \omega^2 A^2$.

Kinetic Energy in Simple Harmonic Motion

Kinetic energy in SHM is due to the velocity of the particle as it moves through its path. The velocity at any displacement $x$ is described by $v = \omega \sqrt{A^2 - x^2}$, where $A$ is the amplitude. The kinetic energy is then given by $K = \dfrac{1}{2} m v^2 = \dfrac{1}{2} m \omega^2 (A^2 - x^2)$.

At the mean position $(x = 0)$, the kinetic energy reaches its maximum value $K_{max} = \dfrac{1}{2} m \omega^2 A^2$. At the extreme positions $(x = \pm A)$, the kinetic energy becomes zero as the particle momentarily comes to rest.

Total Mechanical Energy in Simple Harmonic Motion

The total mechanical energy $E$ of a particle in SHM is the sum of its kinetic and potential energies at any instant. Substituting the expressions:

$E = K + U = \dfrac{1}{2} m \omega^2 (A^2 - x^2) + \dfrac{1}{2} m \omega^2 x^2 = \dfrac{1}{2} m \omega^2 A^2$

This result shows that the total energy in SHM is independent of displacement $x$, and remains constant throughout the motion, provided no dissipative forces are present.

In terms of frequency $f$, using $\omega = 2\pi f$, the total energy can be written as $E = 2\pi^2 m f^2 A^2$.

To study the mathematical formulation of oscillations, see Simple Harmonic Motion.

Variation of Energy with Displacement

During SHM, kinetic and potential energies continuously convert into one another as the particle oscillates. At equilibrium position $(x = 0)$, the entire energy is kinetic, while at extreme positions $(x = \pm A)$, the entire energy is potential.

At any intermediate position, both kinetic and potential energies are non-zero, and their sum remains constant. The energy variations as functions of displacement are given by:

• Kinetic energy decreases as displacement increases
• Potential energy increases with displacement from equilibrium
• Total energy remains constant for all values of displacement

Graphical Representation of Energies in SHM

The graphs of kinetic and potential energies against displacement are parabolic in nature. The total energy graph appears as a straight line parallel to the displacement axis, indicating that it does not vary with position. The potential energy curve opens upwards, while the kinetic energy curve opens downwards with respect to displacement.

Study additional topics on periodic motion and energy transformations in related chapters such as Oscillations And Waves.

Frequency of Energy Variation in SHM

The kinetic and potential energies in SHM vary between zero and their maximum values during the motion. Both energies complete two full cycles in the time the displacement completes one. Hence, the frequency of variation of kinetic or potential energy is twice the frequency of the oscillation.

If the displacement is $x = A \cos(\omega t)$, the kinetic energy is $K = \dfrac{1}{2} m \omega^2 A^2 \sin^2(\omega t)$. Since $\sin^2(\omega t)$ completes two cycles in the period $T$, the energy varies with frequency $2f$.

Key Points on Energy in SHM

• Total mechanical energy depends on mass, frequency, and amplitude
• Potential energy is maximum at extreme displacements
• Kinetic energy is maximum at the mean position
• Total energy remains constant if there is no dissipation
• Frequency of variation of energy is double that of displacement

Illustrative Example: Energy Calculations in SHM

Consider a particle of mass $0.2\, \mathrm{kg}$ executing SHM with amplitude $0.05\, \mathrm{m}$ and frequency $1\, \mathrm{Hz}$. Find the total energy and maximum kinetic energy.

The value calculated gives the total energy of the system, which is equal to the maximum kinetic and maximum potential energy in SHM.

Review further applications of work and energy in oscillatory systems in Work Energy And Power.

Practical Examples of Energy in SHM

• Oscillation of a spring-mass system
• Pendulum motion near equilibrium
• Vibrations of a guitar string
• Molecules undergoing vibrational energy exchange

Study the application of SHM to mechanical systems in the Spring Mass System chapter.

Summary of Energy Relationships in SHM

In simple harmonic motion, the energy transformations between kinetic and potential forms demonstrate the conservative nature of the force involved. The total mechanical energy is solely determined by the mass, angular frequency, and amplitude of oscillation, and is independent of time or displacement.

For a conceptual understanding of thermal motions and kinetic concepts, refer to Kinetic Theory Of Gases and other foundational physics topics.