PENDULUM MOTION: SIMPLE PENDULUM, SIMPLE HARMONIC - PHYSICAL - 2024

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

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)

This is what pendulum clocks (hence its name) or playground swings do. In an ideal pendulum the oscillatory motion would continue perpetually. In a real pendulum, on the other hand, the movement ends up stopping after time due to friction with the air.

Thinking of a pendulum makes it inevitable to evoke the image of the pendulum clock, the memory of that old and imposing clock from the grandparents' country house. Or perhaps Edgar Allan Poe's horror tale, The Well and the Pendulum, whose narration is inspired by one of the many torture methods used by the Spanish Inquisition.

The truth is that the different types of pendulums have varied applications beyond measuring time, such as, for example, determining the acceleration of gravity in a certain place and even demonstrating the rotation of the Earth as the French physicist Jean Bernard Léon did. 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 motion of a simple pendulum can be somewhat complex, the truth is that when the amplitude (A), or displacement from the equilibrium position, of the motion is small, it can be approximated with the equations of a harmonic motion simple that are not overly complicated.

Simple harmonic motion

The simple harmonic movement is a periodic movement, that is, it is repeated 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.

Thus, a fundamental characteristic of the pendulum's movement is its period (T), which determines the time it takes to make a complete cycle (or complete oscillation). The period of a pendulum is determined by the following expression:

where, l = the length of the pendulum; and, g = the value of the acceleration due to gravity.

A quantity related to the period is the frequency (f), which determines the number of cycles the pendulum goes through in one second. In this way, the frequency can be determined from the period with the following expression:

Pendulum movement dynamics

The forces that intervene in the movement are the weight, or what is the same, the force of gravity (P) and the tension of the thread (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 vertically towards the center of mass of the Earth, and therefore, it is necessary to decompose it into its tangential and normal or radial components.

The tangential component of the weight P _t = mg sin θ, while the normal component of the weight is P _N = mg cos θ. This second is compensated with the tension of the thread; The tangential component of the weight, which acts as a restoring force, is therefore ultimately responsible for the movement.

Displacement, velocity, 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 ₀)

where ω = is the angular velocity of rotation; t = is the time; and, θ ₀ = is the initial phase.

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

ω = 2 ∏ / T = 2 ∏ / f

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

v = dx / dt = -A ω sin (ω t + θ ₀)

Proceeding in the same way, the expression of the acceleration with respect to time is obtained:

a = dv / dt = - A ω ² cos (ω t + θ ₀)

Maximum speed and acceleration

Observing both the expression of the velocity and the acceleration one can appreciate some interesting aspects of the pendulum's motion.

The velocity takes its maximum value in the equilibrium position, at which time the acceleration is zero, since, as previously stated, at that instant 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 velocity and acceleration it is easy to deduce both the modulus of maximum velocity and the modulus of maximum acceleration. It is enough to take the maximum possible value for both sin (ω t + θ ₀) and for cos (ω t + θ ₀), which in both cases is 1.

│ v _max │ = A ω

│ a _max │ = A ω ²

The moment at which the pendulum reaches its maximum speed is when it passes through the equilibrium point of forces since then sin (ω t + θ ₀) = 1. On the contrary, the maximum acceleration is reached at both ends of the motion since then cos (ω t + θ ₀) = 1

conclusion

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

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

The different types of pendulums that exist have different applications both for 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. (nd). In Wikipedia. Retrieved on March 7, 2018, from en.wikipedia.org.
3. Pendulum (mathematics). (nd). 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.