Pendulum movement: simple pendulum, simple harmonic movement

A pendulum is an object (ideally a point mass) hung by a thread (ideally without mass) of a fixed point and that oscillates thanks to the force of gravity, that mysterious invisible force that, among other things, keeps stuck to the universe.

The pendular movement is the one that occurs in an object from one side to another, hanging from a fiber, cable or thread. The forces that intervene in this movement are the combination of the force of gravity (vertical, towards the center of the Earth) and the tension of the thread (direction of the thread).

Pendulum oscillating, showing speed and acceleration (wikipedia.org)

It's what pendulum clocks do (hence its name) or the playground swings. In an ideal pendulum the oscillatory movement would continue perpetually. In a real pendulum, however, the movement ends up stopping over time due to friction with the air.

Thinking of a pendulum makes it inevitable to evoke the image of the pendular clock, the memory of that old and imposing clock of the grandparents' country house. Or maybe Edgar Allan Poe's horror story, The well and the pendulum whose narrative is inspired by one of the many methods of torture used by the Spanish Inquisition.

The truth is that the different types of pendulums have various applications beyond measuring time, such as, for example, determine the acceleration of gravity in a given place and even demonstrate the rotation of the Earth as did the French physicist Jean Bernard Léon Foucault

Foucault pendulum. Author: Veit Froer (wikipedia.org).

The simple pendulum and the simple harmonic vibratory movement

Simple pendulum

The simple pendulum, although it is an ideal system, allows to carry out a theoretical approach to the movement of a pendulum.

Although the equations of the movement of a simple pendulum can be somewhat complex, the truth is that when the amplitude ( TO ), or displacement from the equilibrium position, of the movement is small, this can be approximated with the equations of a simple harmonic movement that are not excessively complicated.

Simple harmonic movement

The simple harmonic movement is a periodic movement, that is, that repeats itself in time. Furthermore, it is an oscillatory movement whose oscillation occurs around a point of equilibrium, that is, a point at which the net result of the sum of the forces applied to the body is zero.

In this way, a fundamental characteristic of the movement of the pendulum is its period ( T ), which determines the time it takes to do a complete cycle (or complete oscillation). The period of a pendulum is determined by the following expression:

being, l = the length of the pendulum; Y, g = the value of the acceleration of gravity.

A magnitude related to the period is the frequency ( F ), which determines the number of cycles the pendulum travels in a second. In this way, the frequency can be determined from the period with the following expression:

Dynamics of the pendulum movement

The forces that intervene in the movement are the weight, or what is the same the force of gravity ( P ) and the thread tension ( T ). The combination of these two forces is what causes the movement.

While the tension is always directed in the direction of the thread or rope that joins the mass with the fixed point and, therefore, it is not necessary to decompose it; the weight is always directed in the vertical towards the center of mass of the Earth, and therefore, it is necessary to decompose it in its tangential and normal or radial components.

The tangential component of the weight P _t = mg sen θ , whereas the normal component of the weight is P _N = mg cos θ . This second one is compensated with the tension of the thread; The tangential component of the weight acting as a recuperating force is therefore ultimately responsible for the movement.

Displacement, speed and acceleration

The displacement of a simple harmonic movement, and therefore of the pendulum, is determined by the following equation:

x = A ω cos (ω t + θ )

where ω = is the angular speed of rotation; t = is time; Y, θ = is the initial phase.

In this way, this equation allows you to determine the pendulum position at any time. In this regard, it is interesting to highlight some relationships between some of the magnitudes of simple harmonic motion.

ω = 2 Π / T = 2 Π / f

On the other hand, the formula that governs the speed of the pendulum as a function of time is obtained by deriving the displacement as a function of time, thus:

v = dx / dt = -A ω sen ( ω t + θ )

Proceeding in the same way, we obtain the expression of the acceleration with respect to time:

a = dv / dt = - A ω ² cos ( ω t + θ )

Maximum speed and acceleration

Observing both the expression of velocity and that of acceleration, some interesting aspects of pendulum movement are appreciated.

The speed takes its maximum value in the equilibrium position, at which time the acceleration is zero, since, as already stated above, at that moment the net force is zero.

On the contrary, at the extremes of the displacement the opposite occurs, there the acceleration takes the maximum value, and the velocity takes a null value.

From the equations of speed and acceleration it is easy to deduce both the maximum speed module and the maximum acceleration module. Simply take the maximum possible value for both the sin (ω t + θ ) as for the cos (ω t + θ ), which in both cases is 1.

│ v _max │ = A ω

│ to _max │ = A ω ²

The moment in which the pendulum reaches the maximum speed is when it passes through the point of balance of forces since then sin (ω t + θ ) = 1 . On the contrary, the maximum acceleration reaches it at both ends of the movement, since then cos (ω t + θ ) = 1

conclusion

A pendulum is an object easy to design and apparently with a simple movement although the truth is that in the background it is much more complex than it seems.

However, when the initial amplitude is small, its movement can be explained with equations that are not excessively complicated, given that it can be approximated with the equations of simple harmonic vibratory motion.

The different types of pendulums that exist have different applications for both daily life and in the scientific field.

References

1. Van Baak, Tom (November 2013). "A New and Wonderful Pendulum Period Equation". Horological Science Newsletter. 2013 (5): 22-30.
2. Pendulum. (n.d.). In Wikipedia. Retrieved on March 7, 2018, from en.wikipedia.org.
3. Pendulum (mathematics) . (n.d.). In Wikipedia. Retrieved on March 7, 2018, from en.wikipedia.org.
4. Llorente, Juan Antonio (1826). The history of the Inquisition of Spain. Abridged and translated by George B. Whittaker . Oxford University. pp. XX, preface.
5. Poe, Edgar Allan (1842). The Pit and the Pendulum . Booklassic. ISBN 9635271905.