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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.

>Model

ID:(1420, 0)


Mathematical Pendulum
Description

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.

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$\omega_0$
omega_0
Angular Frequency of Mathematical Pendulum
rad/s
$\omega$
omega
Angular Speed
rad/s
$m_g$
m_g
Gravitational mass
kg
$m_i$
m_i
Inertial Mass
kg
$\theta_0$
theta_0
Initial Angle
rad
$K$
K
Kinetic energy of point mass
J
$L$
L
Pendulum Length
m
$T$
T
Period
s
$V$
V
Potential Energy Pendulum, for small Angles
J
$\nu$
nu
Sound frequency
Hz
$\theta$
theta
Swing angle
rad
$t$
t
Time
s
$E$
E
Total Energy
J

Calculations

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Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

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Equations

The gravitational potential energy of a pendulum with mass m, suspended from a string of length L and deflected by an angle \theta is given by

$ U = m g L (1-\cos \theta )$



where g is the acceleration due to gravity.

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

$ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$

(ID 4514)

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:

$ K =\displaystyle\frac{1}{2} m_i L ^2 \omega ^2$



Similarly, the potential Energy Pendulum ($V$), as a function of the gravitational Acceleration ($g$) and the gravitational mass ($m_g$), is determined by:

$ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$



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:

$ \omega_0 ^2=\displaystyle\frac{ g }{ L }$

(ID 4516)

Using the complex number

$ z = x_0 \cos \omega_0 t + i x_0 \sin \omega_0 t $



introduced in

$ \dot{z} = i \omega_0 z $



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

$ v = - x_0 \omega_0 \sin \omega_0 t $

(ID 14076)

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:

(ID 15849)

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.

(ID 7098)

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.

(ID 1180)


(ID 15852)

ID:(1420, 0)