Simple harmonic motion

The simple harmonic motion , also called vibrational motion simple harmonic is a rectilinear movement variable acceleration produced by the forces which arise when a body is separated from its equilibrium position, such as the pendulum of a clock or a mass suspended on a spring.

A body oscillates when it periodically moves about its equilibrium position. Simple harmonic motion is the most important of oscillatory movements, as it is a good approximation to many of the oscillations that occur in nature and is very easy to describe mathematically. It is called harmonic because the equation that defines it is a function of the sine or the cosine .

 

Summary

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  • 1 Simple harmonic motion in one direction
  • 2 Characteristics of a MAS
  • 3 Fundamental parameters
  • 4 Equation of motion
    • 1 Elongation
    • 2 Speed
    • 3 Acceleration
    • 4 Amplitude and initial phase
  • 5 Energy considerations in MAS
  • 6 Sources

Simple harmonic motion in one direction

In the case that the trajectory is rectilinear, the particle that performs a plus moves along the X axis, its position x being given as a function of time t by the equations.

where

  • A is the maximum width or elongation.
  • angular the angular frequency.
  • ωt + φ the phase.
  • φ the initial phase.

The particle oscillates away from and closer to a point, located in the center of its trajectory or equilibrium point, in such a way that its position as a function of time with respect to that point is a sinusoid. In this movement, the force acting on the particle is proportional to its movement with respect to said point and directed towards it, this force at all times directs the particle towards its equilibrium position and is called the restoring force. In MAS , position, speed , acceleration, and force vary with position as a function of time.

In simple harmonic motion, frequency and period are independent of amplitude , and acceleration is proportional to displacement, but in the opposite direction:

 

Characteristics of a MAS

  • Since the maximum and minimum values ​​of the sine or cos function are +1 and -1, the movement is performed in a region of the X axis between -A and + A.
  • The sine function is periodic and is repeated every 2p, therefore, the movement is repeated when the argument of the sine function or cos increases by 2p, that is, when a time P passes such that.

 

Fundamental parameters

The period T is the time required to make a complete oscillation, that is, for each successive repetition of the round-trip movement, its SI units are seconds.The frequency f of the movement is the number of oscillations per unit of time, its SI units is Hz or 1 / s , therefore the frequency is the reciprocal of the period T. T = 1 / f . Balance position, is the position for which no force acts on the particle, it is generally where the coordinate system is located to measure the distances. Elongation (linear or angular) is called the distance (linear or angular) of the particle that oscillates to its equilibrium position at any instant, its SI units are m. The amplitude of movement A is the maximum elongation.

 

Equation of motion

Elongation

A spring when we separate it from its equilibrium position, stretching it or compressing it, acquires a simple harmonic vibratory movement, since the recovery force of that spring is the one that generates an acceleration, which gives it that reciprocating movement. The position that the block occupies at each moment with respect to the central point is known as ELONGATION , x. To define the movement its equation is calculated, the relationship between the magnitudes that intervene and influence it. To find an equation that relates position (x) to time x (t) ,. For this, two well-known laws in Physics are taken as a starting point:

– Hooke’s Law : which determines that the spring recovery force is proportional to the position and has an opposite sign. The expression of the law is:

F = – Kx

– Newton’s 2nd law : that relates force, mass and acceleration, whose expression is:

F = ma

It is obvious that the spring recovery force is the one that causes the acceleration of the movement, which supposes that both forces, expressed above, are equal. Then:

where acceleration is expressed as the second derivative of position with respect to time. The solution of this equation for the position value as a function of time is:

where x (t) is the elongation, A the amplitude or maximum elongation, ω the angular frequency and φ the offset, which indicates the discrepancy between the origin of spaces (point where we begin to measure the space) and the origin of times.

The angular frequency value is related to the recovery constant by the equation below:

 

Speed

From the equation of position or elongation) and, deriving with respect to time, we obtain the velocity equation in the MAS:

Slightly modifying this equation we find an expression of the speed as a function of x , the elongation:

 

Acceleration

Deriving the velocity equation with respect to time, we obtain the acceleration equation in the MAS:

from which we can also obtain an equation that relates it to the position:

 

Amplitude and initial phase

Knowing the initial position x 0 and the initial velocity v 0 at time t = 0.

the amplitude A and the initial phase are determined φ

 

Energy considerations in the MAS

The forces involved in simple harmonic motion are central and therefore conservative. Consequently, a scalar field called potential energy ( Ep ) associated with force can be defined . To find the expression of potential energy, it is enough to integrate the expression of force (this is extensible to all conservative forces) and change its sign, obtaining for the specific case of a mass-spring system the following expression for energy potential:

Ep = ½ K x 2

The potential energy reaches its maximum at the ends of the path and has a null value (zero) at the point x = 0, that is, the equilibrium point. The kinetic energy will change throughout the oscillations as the speed does:

Ec = ½ mv 2

The kinetic energy is null at -A or + A (v = 0) and the maximum value is reached at the equilibrium point (maximum speed Aω). Substituting the velocity equation for the MAS we can obtain the expression for the maximum kinetic energy as follows.

Ec max = ½ ω 2 A 2

Since only conservative forces act, mechanical energy (sum of potential and kinetic energy) remains constant.

Ec + Ep = Em

Finally, being constant mechanical energy, it can be easily calculated by considering the cases where the particle velocity is zero and therefore the potential energy is high, i.e., at points x = – A and x = A . It is obtained then that,

Em = Ep max + 0 = ½ KA2

Or also when the velocity of the particle is maximum and the potential energy is zero, at the equilibrium point x = 0

Em = 0 + Ec max = ½ ω 2 A 2

 

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