# Pendulum motion

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**What is the pendulum motion?**

The **pendulum movement** is a back and forth movement carried out by a more or less heavy object, called a pendulum, suspended by a rope or light rod, fixed at its other end.

The pendulum is given an initial impulse and is allowed to oscillate, in this way the object describes arcs back and forth. This is the principle of the operation of pendulum clocks, swings, rocking chairs and pendulum *metronomes* , used to mark the times in music.

It is said that around 1581, Galileo Galilei observed the oscillation of a lamp in the cathedral of Pisa, observing that, although the amplitude of the oscillation of the candelabrum was decreasing due to friction with the air, the duration of the cycle.

This caught the attention of Galileo, who decided to continue with the study and determined that the period of the pendulum does not depend on the mass, but on the square root of the length of the string, as will be seen later.

**Pendulum motion characteristics**

A pendulum is very easy to build, since a plumb line hung from a cotton thread and held by the other end with the fingers or by attaching it to a support like a nail is enough.

After the small initial impulse, the weight is in charge of keeping the pendulum oscillating, although the friction decreases the amplitude of the movement, until it finally ceases completely.

The main characteristic of the pendulum movement is to be repetitive, as it is a back and forth movement. Now, to facilitate its study, it is convenient to make some simplifications to focus on a simpler model, called the *simple pendulum* .

**The simple pendulum**

It is an ideal system consisting of a plumb line, considered as a point mass *m* , subject to a light and inextensible string of length *L* . The characteristics of this system are:

- Have a repetitive and periodic movement, consisting of going back and forth an arc of circumference of radius equal to L.
- It does not take friction into account.
- The range of motion is small (<5º).
- The period is independent of the mass
*m*, and depends only on the length*L*of the pendulum.

**Formulas and equations**

The following is a diagram of the simple pendulum, on which two forces act: weight **P** of magnitude mg, which is directed vertically downward, and tension **T** in the string. They are not considered friction.

The reference axis is the vertical axis and coincides with the position θ = 0, from there the angular displacement θ is measured, either in one direction or another. The + sign can be assigned to the right shift in the figure.

To study the motion of the pendulum, a coordinate system with the origin at the pendulum itself is chosen. This system has a tangential coordinate to the arc of circumference A’CA described by the pendulum, as well as a radial coordinate, directed towards the center of the trajectory.

At the instant shown in the figure, the pendulum is moving to the right, but the tangential component of gravity, called F _{t} , is responsible for making it return. It is noted from the figure that this component has the opposite direction to movement.

As for the tension in the rope, it is balanced by the weight component mgcosθ.

**Angular displacement **

We must express the equation in terms of a single variable, remembering that the angular displacement θ and the arc traveled are related by the equation:

s = L.θ

The mass cancels on both sides and if the amplitude is small, the angle θ also, so the following approximation is valid:

sen θ ≈ θ

With this, the following differential equation is obtained for the variable θ (t):

This equation is very easy to solve, since its solution is a function whose second derivative is the function itself. There are three alternatives: a cosine, a sine, or an exponential. The cosine function is chosen for the angular displacement θ (t), since it is a well-known and easy-to-handle function.

The reader can verify, by differentiating twice, that the following function satisfies the differential equation:

θ (t) = θ _{m} cos (ωt + φ)

Where θ _{m} is the maximum angle that the pendulum moves with respect to the vertical and the angular frequency ω is:

**Equation of the period**

The period T of the movement is the time it takes to execute a cycle and is defined as:

Substituting ω:

As stated above, the period does not depend on the mass of the pendulum, but only on its length.

**Examples of pendulum motion**

**Heart rate measurement**

Galileo had the idea of measuring the heart rate of people, adjusting the length of the pendulum to make the period coincide with the beats of a person’s heart.

**The pendulum clock**

This is undoubtedly one of the most familiar examples of pendulum motion. Pendulum clock making is as much about science as it is about art. Dutch physicist Christian Huygens (1629-1695) developed the first pendulum clock in 1656, based on a study done years ago by Galileo.

**Foucault’s pendulum**

It is a somewhat different pendulum from the one described above, since it is capable of rotating in any vertical plane. It was created by the French physicist Léon Foucault (1819-1868) and is used to visualize the rotation of the Earth.

**Exercise resolved**

A simple pendulum passes every 0.5 s through the equilibrium position. What is the length of the thread?

**Solution**

Since the period is the time it takes to carry out a complete cycle, in which it passes through the equilibrium position twice: one going and the other back, then:

T = 2 × 0.5 s = 1 s

From:

The length L of the thread is cleared:

The thread is 0.25 m or 25 cm long.