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Mathematical Pendulum
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In the case of a pendulum composed of a point mass the potential energy is given by the effect of raising the mass against the gravitational field as the pendulum deviates by a given angle.
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Mathematical Pendulum
Storyboard
In the case of a pendulum with a point mass, the potential energy is generated by raising the mass against the gravitational field as the pendulum deviates by a given angle.
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The gravitational potential energy of a pendulum with mass
where
For small angles, the cosine function can be approximated using a Taylor series expansion up to the second term
$\cos\theta\sim 1-\displaystyle\frac{1}{2}\theta^2$
This approximation leads to the simplification of the potential energy to
The kinetic energy of point mass ($K$), in relation to the inertial Mass ($m_i$), the pendulum Length ($L$), and the angular Speed ($\omega$), is expressed as:
Similarly, the potential Energy Pendulum ($V$), as a function of the gravitational Acceleration ($g$) and the gravitational mass ($m_g$), is determined by:
Considering the swing angle ($\theta$), the total energy equation is expressed as:
$E = \displaystyle\frac{1}{2}m r^2 \omega^2 + \displaystyle\frac{1}{2}m g r \theta^2$
Given that the period ($T$) is equal to:
$T = 2\pi\displaystyle\sqrt{\displaystyle\frac{m r^2}{m g r}} = 2\pi\displaystyle\sqrt{\displaystyle\frac{r}{g}}$
It is possible to establish the relation for the angular Frequency of Mathematical Pendulum ($\omega_0$) as:
Using the complex number
introduced in
we obtain
$\dot{z} = i\omega_0 z = i \omega_0 x_0 \cos \omega_0 t - \omega_0 x_0 \sin \omega_0 t$
thus, the velocity is obtained as the real part
Examples
An effective way to study the oscillation of a mathematical pendulum is by representing its motion in phase space, which describes the system in terms of momentum and position. In this case, the momentum corresponds to the angular momentum, while the position is described by the angular displacement:
A pendulum is described as a the gravitational mass ($m_g$) suspended from a string attached to the axis of rotation, at a distance the pendulum Length ($L$). It is called a mathematical pendulum because it represents an idealization of the physical pendulum, in which the mass is considered as a point mass, meaning it is concentrated at a single point.
A pendulum consists of the gravitational mass ($m_g$), suspended from a string attached to the axis of rotation of the pendulum Length ($L$). This model is known as a mathematical pendulum, as it represents an idealization of a physical pendulum in which all the mass is concentrated at a single point.
The total Energy ($E$) corresponds to the sum of the total Kinetic Energy ($K$) and the potential Energy ($V$):
The kinetic energy of a rotating body is given by
where $I$ is the moment of inertia and $\omega$ is the angular velocity. For a point mass $m$ rotating at a distance $L$ from an axis, the moment of inertia is
hence,
The gravitational potential energy of a pendulum is
which for small angles can be approximated as:
It's important to note that the angle must be expressed in radians.
The masses that Newton used in his principles are related to the inertia of bodies, which leads to the concept of the inertial Mass ($m_i$).
Newton's law, which is linked to the force between bodies due to their masses, is related to gravity, hence known as the gravitational mass ($m_g$).
Empirically, it has been concluded that both masses are equivalent, and therefore, we define
Einstein was the one who questioned this equality and, from that doubt, understood why both 'appear' equal in his theory of gravity. In his argument, Einstein explained that masses deform space, and this deformation of space causes a change in the behavior of bodies. Thus, masses turn out to be equivalent. The revolutionary concept of space curvature implies that even light, which lacks mass, is affected by celestial bodies, contradicting Newton's theory of gravitation. This was experimentally demonstrated by studying the behavior of light during a solar eclipse. In this situation, light beams are deflected due to the presence of the sun, allowing stars behind it to be observed.
The angular Frequency of Mathematical Pendulum ($\omega_0$) is determined as a function of the gravitational Acceleration ($g$) and the pendulum Length ($L$) through:
The angular frequency ($\omega$) is with the period ($T$) equal to
The sound frequency ($\nu$) corresponds to the number of times an oscillation occurs within one second. The period ($T$) represents the time it takes for one oscillation to occur. Therefore, the number of oscillations per second is:
Frequency is indicated in Hertz (Hz).
The relationship between the angular frequency ($\omega$) and the sound frequency ($\nu$) is expressed as:
With the description of the oscillation using
the real part corresponds to the temporal evolution of the amplitude
When we extract the real part of the derivative of the complex number representing the oscillation
whose real part corresponds to the velocity
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