UNDERSTANDING SIMPLE HARMONIC MOTION

DEFINITION

A particle in a simple harmonic motion moves to the maximum and gains energy potential. The object’s mobility rises sharply when it returns to the equilibrium site. As a result, potential energy is transformed to kinetic energy and in the opposite way, too, in this scenario. The energy is preserved in an optimal simple harmonic motion.

Although it takes on different forms, this same ‘overall energy’ stays unchanged. So, at extreme values, the velocity is zero, but at the medium positions, it is maximal. The place in which the velocity is most significant is also where the element’s angular momentum is greatest.

Below, you can find more significant insights into the topic along with section-wise notes and derivations on every step to aid your understanding.

PRINCIPLES AND DERIVATIONS

For several sorts of motion, energy is the basic unit. Conformational changes can also be caused by energy based on its intensity. The premise of oscillations is related. Oscillations are cyclic fluctuations that repeat themselves. They are assessed around a center value, often called a ‘point of equilibrium’ or between two or more levels.

Oscillation is considered repeated changes of any measure or quantity around its equilibrium level in a given duration. Such fluctuations can be found in complex systems or, more precisely, in every scientific research discipline.

a) Sinusoidal Trajectory in Simple Harmonic Motion

• Take an object of velocity “v” and the mass “m”. Its kinetic energy would be derived by:

 KE= ½ mv2

• The entity’s velocity maintains a sinusoidal trajectory, which indicates that it peaks and drops in velocity.
• The magnitude of the velocity ( v = Aωcos(ωt + Φ)) could be inserted into the expression because the particle is in simple harmonic motion,

 KE = ½ m (Aωcos(ωt))2

⇒ K.E. =½ m A2ω2cos2(ωt)

⇒ KE =½ kA2cos2 (ωt)

• It’s worth noting that the K.E. is also written as a cyclical/periodic equation.
• At the equilibrium position, the maximum value of K.E. is ½ m A2ω2cos2(ωt), and
• At the absolute extreme, the minimum value of K.E. is 0.
• The displacement is commensurate/directly proportional to the force applied when it comes to conservative energies. Its potential energy is calculated as follows:

U = ½ kx2

b) Simple Harmonic Motion is Periodic

1. Simple harmonic motion is a type of oscillating resonance. The displacement occurs in a direct and straight line between both extreme points.
2. A restorative action is directed towards the central axis or the equilibrium point in this motion.
3. This mean position is a ‘durable equilibrium for the simple harmonic motion.’
4. Finally, a simple harmonic motion is not present in every sinusoidal wave (not every oscillatory movement is a simple harmonic motion).

c) Interchange of Potential and Kinetic Energy: Case by Case Basis

• In a simple harmonic oscillation, there is a constant interplay of potential and kinetic energy.
• The rhythmic harmonic oscillator is a device that conducts simple harmonic motion.

Let’s take a look at this, one at a time.

• This is our first case whenever the prospective potential energy is nil, and the angular momentum is at its greatest at the equilibrium point, where displacement occurs (as well as shown in part (a)).
• The maximal displacement point from the equilibrium state is visible in the following case, wherein the restoring force is maximal and the kinetic energy is zero.
• In the last case, the oscillatory body’s movement has different potential and kinetic energy results at different intervals.
• Displacement, velocity and Acceleration at any time (t) are expressed as:

Displacement x = A sin (ωt + Φ)

Velocity v = Aωcos(ωt + Φ)= A2-x2

Acceleration a =-ω2 Asin(ωt + Φ)= -ω2x

• The restoring force acting on the object becomes:

F = -kx,

Here, k = mω2

d) Simple Harmonic Energy: Potential Energy

• While displacing the substance from the equilibrium position (x = 0) to x, the restorative force performs the work that equates to x.
• Whenever the substance is shifted from location x to x + dx, the progress made by the restorative force is derived as:

 dw = F dx = -kx dx, wherein,

 w = ∫ dw= -∫ kxdx= -kx2/ 2

w =-mω2×2/2 [k= mω2]

w =-mω2/2. Asin(ωt + Φ)

As a result,

Potential Energy = (work done by restoring force), or simply

 Potential Energy U = mω2×2/2 =( mω2A2/2) sin2(ωt + Φ)

e) Total Mechanical Energy

 Have you ever wondered how much total mechanical energy a molecule with simple harmonic motion has?

The overall energy output of a block and a shock absorber combination is equivalent to the magnitude square. It is equivalent to the total of the potential stored energy in the elastic and the kinetic energy of the frame.

In this manner:

E =  ½ mω2 (A2-x2) + ½ mω2 x2

E = ½ mω2 A2

• As a result, the object’s energy output in simple harmonic motion is uniform and unaffected by its momentary displacement.
• At t = 0, and x = ±A, the correlation between total energy, time, and kinetic in simple harmonic motion is shown (refer to the formula in part (a)).

f) Examples of Simple Harmonic Motion

• We are surrounded by numerous examples of simple harmonic motion in our daily lives.
• Simple harmonic motion is demonstrated by hammocks in the park.
• The simple harmonic motion would be the swing’s to and fro, repeating movements against the force applied.
• A pendulum oscillating back and forth from its central axis exhibits simple harmonic motion.
• In essence, simple harmonic motion can be seen in the bouncing of a cradle, the cyclical movement of the arms of a clock, swaying of a swing, the leaves of a tree going to and forth owing to the wind, and so on.

CONCLUSION

Any movement in which a resistive force is provided proportionate to the dislocation and is in the reverse direction of the displacement is referred to as Simple Harmonic Motion. A metronome, for example, is an example, but only if it falls at low inclinations.

Simple harmonic motion is studied to determine the velocity, amplitude, acceleration, frequency, and position of an object performing the activity. As previously stated, the restorative force is applied when the pressure acting by the object at hand is directed towards the position of equilibrium. The resulting force would be F= – kx if the force is F and the displacement of the cord from the equilibrium position is x. The minus symbol signifies that the force is acting in the reverse direction. Also known as the force statistic, k is a constant. In the S.I. system, its metric is N/m.

 The periodic motion might or might not be oscillatory, while the simple harmonic motion always oscillates. To summarize, simple harmonic motion differs from a periodic motion in one substantial way: it involves to and fro frequency shifts.