The pendulum in physics is a physical system made up of several elements: a **rigid, incompressible and inextensible thread** of zero mass and a **point mass** , which instead has its own mass. This point mass is subject to **gravitational attraction** , therefore it is drawn towards the surface by the force of gravity, although there is no friction in the pendulum.

This type of pendulum is defined as simple, while if the thread were replaced by a rigid body, we would have a **compound pendulum** . The pendulum maintains a **constant oscillation** , and for this reason we can observe that it describes a harmonic motion.

In this article we will find out more **about what a pendulum is, what its characteristics are and all the formulas** connected to it

## Characteristics of the pendulum

The pendulum (a ball hanging on a string) is made up of a **mass$m$which oscillates** attached to an **inextensible thread of negligible mass** .

A pendulum is in **equilibrium** when the mass$m$it is still and the wire is perfectly vertical.

If the mass is moved from the equilibrium position (from the vertical) it will begin to **oscillate** along a vertical plane due to the pull of its weight which attempts to bring it back to the equilibrium position. If there were no friction the pendulum would oscillate infinitely (the oscillation would continue the same).

The pendulum is an example of **a harmonic oscillator** as it is subject to a **restoring force directly proportional to the displacement with respect to the equilibrium position** .

The **trajectory** described by the motion is a **circumferential arc** which has its center at the anchoring point of the wire and a radius equal to the length of the wire itself.

## The motion of a pendulum

To analyze the motion of a pendulum it is necessary to create a Cartesian reference system in which:

- the
**axis$y$**has the**same direction**as the wire (of length L) - the
**axis$x$**is**tangential to the trajectory**of the motion.

This Cartesian reference **is not fixed** but rotates following the motion of the oscillating body.The **in component$x$**it has a negative sign because it is a **recall force** that tends to bring the body back to the equilibrium position and, in fact, it is opposite to the direction of motion.

## The period of the pendulum

**How is the period of the pendulum calculated?**

We derive from the pendulum force formula$F_{px}=–(Lm⋅g )⋅x$that the constant$k$is equal to$Lm⋅g $

We substitute the constant into the formula for the period of the harmonic motion of the spring$T=2πkm $

We obtain that the **period of the pendulum** is equal to$T=2πgL $(in which$L$is the length of the pendulum).

## Foucault’s pendulum

In **1851 Foucault** performed an experiment with a pendulum of mass equal to$30Kg__$and hanging from a long rope$68m$.

The experiment was carried out in the **Phanteon in Paris** and it was noted that, unlike an ideal pendulum which always oscillates along the same vertical plane, Foucalt’s pendulum, as time passes, describes a circular sector.

The experiment aimed to demonstrate the existence of the **Earth’s rotation** .

It is not the pendulum that has an altered oscillation compared to the ideal model but it is the Earth below which, by rotating, is as if it were causing the vertical plane on which it moves to rotate.

If the Earth were stationary, the pendulum would trace a single line on the floor. During the experiment, letting the pendulum swing was seen to draw lines under it. Since the plane of free swing of a pendulum does not change, the lines indicated that the ground beneath it was moving.

Foucault demonstrated that **the angle** that grouped these lines was related to the latitude of the place; in particular:

**at the equator,****the angle is zero**(there is no rotation because the pendulum plane is perpendicular to the Earth’s rotation axis)**at the North Pole it is 360°**(the Earth rotates beneath it, making a complete revolution in 24 hours, giving the impression that it is instead the pendulum that rotates).

Attention!

If the Earth were an inertial system, the pendulum would trace lines along the same direction.

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