Energy in Simple Harmonic Motion

Consider the example of a pendulum. When it is at its mean posture, it is said to be at ease. It’s in motion while it’s moving towards its extreme position, and it comes to a complete stop as soon as it gets to its extreme position. Simply put, harmonic motion is the continuous exchange of kinetic energy and potential energy between two points in space. Potential energy reaches its largest value at the greatest distance from the equilibrium point, whereas kinetic energy is at its lowest value. At the equilibrium point, the potential energy is zero, and the kinetic energy is at its highest level possible. 

Kinetic Energy (K.E.) in S.H.M

When an item is in motion, it has kinetic energy, which is the amount of energy it has. Come with me as I teach you how to calculate the kinetic energy of an item. Consider the motion of a particle with mass m down a route AB, which is characterized by simple harmonic motion. Assume that O is the mean position. Therefore, OA = OB = a.

Concerning a distance x from the mean location, the instantaneous velocity of the particle executing S.H.M. is given by 

v= ±ω a2 – x2

  v2 = 2 ( a2  – x2)

∴ Kinetic energy= 12 mv2 = 12 m 2 (a2  – x2)

As km= 2 

∴ k = m 2

Kinetic energy= 12k ( a2  – x2). The equations can be used for calculating the kinetic energy of the particle.

Potential Energy(P.E.) of Particle Performing S.H.M.

A particle’s potential energy is the amount of energy it has while it is not moving. Let us learn how to calculate the potential energy of a particle doing S.H.M. Let us learn how to calculate the potential energy of a particle performing S.H.M. Think of a particle with mass m moving in a simple harmonic motion at a distance x from its mean location.  Knowing that the restoring force operating on the particle is F= -kx, where k is the force constant, you may calculate the particle’s restoring force. 

The particle is now subjected to an additional infinitesimal displacement dx in opposition to the restoring force F. Let dw represent the amount of labor required to dislodge the particle. Therefore, While displaced, the work done by dw is considered. 

dw = – fdx = – (- kx)dx = kxdx

Therefore, the total work done to displace the particle now from 0 to x is

∫dw=  ∫kxdx = k ∫x dx

Hence Total work done = 12K x2 = 12m 2×2

Potential energy is used to store the entire amount of work done in this location. 

Therefore Potential energy = 12kx2 = 12m 2×2

Equations IIa and IIb are equations of the potential energy of the particle. Thus, potential energy is directly proportional to the square of the displacement, that is P.E. α x2.

Total Energy in Simple Harmonic Motion (T.E.)

In simple harmonic motion, the total energy is equal to the sum of the potential energy and the kinetic energy of the motion. 

Thus, T.E. = K.E. + P.E.  = 12k ( a2–x2) + 12K x2 = 12k a2

Hence, T.E.= E = 12m 2a2

Equation III is the equation of total energy in a simple harmonic motion of a particle performing the simple harmonic motion. As 2 , a2 are constants, the total energy in the simple harmonic motion of a particle performing simple harmonic motion remains constant. Therefore, it is independent of displacement x.

As ω= 2πf , E= 12 m ( 2πf )2a2 

∴ E=  2m2f2a2

As 2 and 2 constants, we have T.E. m, T.E. f 2, and T.E. a2

Types of oscillations

Oscillations may be classified into four categories:

Free oscillations

Exercising free oscillations is the term used to describe when a body vibrates at its natural frequency. The frequency of oscillations is determined by the inertial component and spring factor, which are denoted by the equations.

Damped oscillations

The vast majority of oscillations in air, or in any medium, are dampened or attenuated. When an oscillation occurs, a damping force may be generated as a result of the friction or air resistance provided by the medium. As a result, some of the energy is used to overcome the resistive force. Therefore, the amplitude of oscillation diminishes with time until it eventually reaches a value equal to zero. 

Maintained oscillations

An oscillating system may be made to have a constant amplitude by supplying it with a constant amount of energy. If enough energy is supplied to the system to make up for the energy that has been lost, the amplitude will remain constant. 

Forced oscillations

Forced vibrations are created when a vibrating body is kept in a state of vibration by a periodic force with a frequency (n) that is different from the natural frequency of the body. Forced vibrations are created when a vibrating body is kept in a state of vibration by a periodic force with a frequency (n) that is different from the natural frequency of the body. The external force serves as the driver, while the body serves as the vehicle. 

An external periodic force is applied to the body, causing it to vibrate. The difference between the frequencies of the driver and the frequencies of the driven determines the magnitude of forced vibration. The amplitude of the forced oscillations will decrease as the frequency difference between the two frequencies increases. 

Resonance

The amplitude of forced vibration will be considered if the frequency difference is minimal in comparison to the natural frequency difference. At the end, when the two frequencies are the same, the amplitude is at its maximum. This is an example of forced vibration in its most extreme form. 

It is known as resonance when the frequency of an external periodic force is equal to the natural frequency of oscillation of the system, resulting in a significant increase in the amplitude of the system’s oscillations. 

Conclusion

Total Energy refers to the total amount of energy used by a branch or variable in its ultimate state. When compared to Final Energy Intensity, Total Energy is differentiated by the fact that energy data is entered directly, rather than as the result of a certain activity level and given energy intensity. 

It is possible to calculate the total energy possessed by a particle in S.H.M. by multiplying its mass (m) by the amplitude (a) with which it is executing S.H.M. and by the constant angular frequency. 

The energy E of a particle oscillating in SHM is dependent on the mass m of the particle, the frequency n of the oscillation, and the amplitude of the oscillation, among other things.