Pendulum movement: simple pendulum, simple harmonic

A pendulum is an object (ideally a point mass) hanging by a string (ideally massless) from a fixed point and oscillating thanks to the force of gravity, that mysterious invisible force that, among other things, holds the universe together.

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

That’s what pendulum clocks do (hence the name) or swings on children’s playgrounds. In an ideal pendulum, the oscillating motion would perpetually continue. In a real pendulum, on the other hand, 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 pendulum clock, the memory of that ancient and imposing clock in the grandparents’ country house. Or perhaps Edgar Allan Poe’s horror story The Well and the Pendulum whose narration is inspired by one of the many methods of torture used by the Spanish Inquisition.

The truth is that different types of pendulums have different applications beyond measuring time, such as determining the acceleration of gravity at a given location and even demonstrating the Earth’s rotation, as did the French physicist Jean Bernard Léon Foucault

The simple pendulum and the simple harmonic vibratory movement

simple pendulum

The simple pendulum, while an ideal system, allows for a theoretical approach to the movement of a pendulum.

Although the equations of motion of a simple pendulum can be a little complex, the truth is that when the amplitude ( A ), or displacement from the equilibrium position, of motion is small, it can be approximated with the equations of harmonic motion. Simple they are not overly complicated.

simple harmonic motion

Simple harmonic movement is periodic movement, that is, repeated over time. Furthermore, it is an oscillatory movement whose oscillation occurs around an equilibrium point, 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 pendular movement is its period ( T ), which determines the time needed to complete a complete cycle (or complete oscillation). The period of a pendulum is determined by the following expression:

Pendulum movement: simple pendulum, simple harmonic 3

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

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

Pendulum movement: simple pendulum, simple harmonic 4

Dynamics of pendulum movement

The forces involved 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 towards the thread or rope that joins the mass to the fixed point and, therefore, it is not necessary to break it; the weight is always directed vertically towards the Earth’s center of mass and therefore it is necessary to break it down into its tangential and normal or radial components.

The tangential component of weight t = mg sin θ , while the normal component of weight is N = mg cos θ . This second is compensated with the line voltage; therefore, the tangential component of the weight that acts as a recovery force is ultimately responsible for the movement.

Displacement, velocity and acceleration

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

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x = A ω cos (ω t + θ )

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

Thus, this equation allows you to determine the position of the pendulum at any time. In this sense, 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 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 sin ( ω t + θ )

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

a = dv / dt = – A ω 2 cos ( ω t + θ 0 )

Velocity and Acceleration

Looking at both the expression of velocity and acceleration, some interesting aspects of pendulum motion are appreciated.

The velocity assumes its maximum value in the equilibrium position, when the acceleration is zero, because, as mentioned above, at that moment the net force is zero.

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

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

│ max │ = A ω

│ max │ = A ω 2

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

Conclusion

A pendulum is an object that is easy to project and apparent with a simple movement, although the truth is that, underneath, it is much more complex than it looks.

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

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

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