User:
Physical Pendulum
Storyboard 
In the case of a compound pendulum with a real mass, the potential energy is generated by raising the center of mass against the gravitational field as the pendulum deviates by a given angle.
ID:(1421, 0)
Oscillations with a physical pendulum
Description
Unlike the mathematical pendulum, the physical pendulum deals with a real, non-point mass. While the length $l$ is defined as the distance between the pivot point and the center of mass of the body, the potential energy of both pendulums is the same. However, the kinetic energy can no longer be approximated using expressions that depend solely on $l$ and $m$; instead, you need to know the actual moment of inertia of the body.
ID:(7097, 0)
Physical pendulum
Description
Unlike the mathematical pendulum, the physical pendulum deals with a real mass, not a point mass. As we define the length $l$ as the distance between the pivot and the center of mass of the body, the potential energy of both pendulums coincides. However, the kinetic energy can no longer be approximated by an expression that depends solely on $l$ and $m$; instead, it must incorporate the actual moment of inertia of the body.
ID:(1188, 0)
Model
Top
Parameters
Variables
Calculations
to 
, then, select the variable: 
to 
Calculations
Calculations
Variable 
Given Calculate 
Target : Equation To be used 
Equations
$ E = K_r + V $
E = K + V
$ K_r =\displaystyle\frac{1}{2} I \omega ^2$
K_r = I * omega ^2/2
$ \nu =\displaystyle\frac{1}{ T }$
nu =1/ T
$ \omega_0 = 2 \pi \nu $
omega = 2* pi * nu
$ \omega_0 = \displaystyle\frac{2 \pi }{ T }$
omega = 2* pi / T
$ \omega_0 ^2=\displaystyle\frac{ m g L }{ I }$
omega_0 ^2 = m * g * L / I
$ \omega = - \theta_0 \omega_0 \sin \omega_0 t $
v = - x_0 * omega_0 *sin( omega_0 * t )
$ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$
V = m_g * g * L * theta ^2/2
$ E =\displaystyle\frac{1}{2} m_g g L \theta_0 ^2$
V = m_g * g * L * theta ^2/2
$ \theta = \theta_0 \cos \omega_0 t $
x = x_0 *cos( omega_0 * t )
ID:(15853, 0)
Total Energy
Equation
The total energy corresponds to the sum of the total kinetic energy and the potential energy:
| $ E = K_r + V $ |
| $ E = K + V $ |
ID:(3687, 0)
Kinetic Energy of Rotation
Equation
In the case being studied of translational motion, the definition of energy
| $ \Delta W = T \Delta\theta $ |
is applied to Newton's second law
| $ T = I \alpha $ |
resulting in the expression
| $ K_r =\displaystyle\frac{1}{2} I \omega ^2$ |
The energy required for an object to change its angular velocity from $\omega_1$ to $\omega_2$ can be calculated using the definition
| $ \Delta W = T \Delta\theta $ |
Applying Newton's second law, this expression can be rewritten as
$\Delta W=I \alpha \Delta\theta=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta$
Using the definition of angular velocity
| $ \bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$ |
we get
$\Delta W=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta=I \omega \Delta\omega$
The difference in angular velocities is
$\Delta\omega=\omega_2-\omega_1$
On the other hand, angular velocity itself can be approximated with the average angular velocity
$\omega=\displaystyle\frac{\omega_1+\omega_2}{2}$
Using both expressions, we obtain the equation
$\Delta W=I \omega \Delta \omega=I(\omega_2-\omega_1)\displaystyle\frac{(\omega_1+\omega_2)}{2}=\displaystyle\frac{I}{2}(\omega_2^2-\omega_1^2)$
Thus, the change in energy is given by
$\Delta W=\displaystyle\frac{I}{2}\omega_2^2-\displaystyle\frac{I}{2}\omega_1^2$
This allows us to define kinetic energy as
| $ K_r =\displaystyle\frac{1}{2} I \omega ^2$ |
ID:(3255, 0)
Potential energy of a mathematical pendulum for small angles (1)
Equation
The gravitational potential energy of a pendulum is
| $ U = m g L (1-\cos \theta )$ |
which for small angles can be approximated as:
| $ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$ |
The gravitational potential energy of a pendulum with mass
| $ U = m g L (1-\cos \theta )$ |
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
| $ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$ |
It's important to note that the angle must be expressed in radians.
ID:(4514, 1)
Angular frequency for a physical pendulum
Equation
Regarding the physical pendulum:
The energy is given by:
$E=\displaystyle\frac{1}{2}I\omega^2+\displaystyle\frac{1}{2}mgl\theta^2$
As a result, the angular frequency is:
| $ \omega_0 ^2=\displaystyle\frac{ m g L }{ I }$ |
Given that the kinetic energy of the physical pendulum with moment of inertia $I$ and angular velocity $\omega$ is represented by
| $ K_r =\displaystyle\frac{1}{2} I \omega ^2$ |
and the gravitational potential energy is given by
| $ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$ |
where $m$ is mass, $l$ is string length, $\theta$ is the angle, and $g$ is angular acceleration, the energy equation can be expressed as
$E=\displaystyle\frac{1}{2}I\omega^2+\displaystyle\frac{1}{2}mgl\theta^2$
As the period is defined as
$T=2\pi\sqrt{\displaystyle\frac{I}{mgl}}$
we can determine the angular frequency as
| $ \omega_0 ^2=\displaystyle\frac{ m g L }{ I }$ |
ID:(4517, 0)
Potential energy of a mathematical pendulum for small angles (2)
Equation
The gravitational potential energy of a pendulum is
| $ U = m g L (1-\cos \theta )$ |
which for small angles can be approximated as:
| $ E =\displaystyle\frac{1}{2} m_g g L \theta_0 ^2$ |
| $ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$ |
The gravitational potential energy of a pendulum with mass
| $ U = m g L (1-\cos \theta )$ |
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
| $ V =\displaystyle\frac{1}{2} m_g g L \theta ^2$ |
It's important to note that the angle must be expressed in radians.
ID:(4514, 2)
Relación frecuencia angular - frecuencia
Equation
Como la frecuencia angular es con angular frequency $rad/s$, period $s$ and pi $rad$ igual a
| $ \omega_0 = \displaystyle\frac{2 \pi }{ T }$ |
y la frecuencia con frequency $Hz$ and period $s$ igual a
| $ \nu =\displaystyle\frac{1}{ T }$ |
se tiene que con frequency $Hz$ and period $s$ igual a
| $ \omega_0 = 2 \pi \nu $ |
| $ \omega = 2 \pi \nu $ |
ID:(12338, 0)
Angular frequency
Equation
The angular frequency ($\omega$) is with the period ($T$) equal to
| $ \omega_0 = \displaystyle\frac{2 \pi }{ T }$ |
| $ \omega = \displaystyle\frac{2 \pi }{ T }$ |
ID:(12335, 0)
Frequency
Equation
The 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:
| $ \nu =\displaystyle\frac{1}{ T }$ |
Frequency is indicated in Hertz (Hz).
ID:(4427, 0)
Oscillation amplitude
Equation
With the description of the oscillation using
| $ z = x_0 \cos \omega_0 t + i x_0 \sin \omega_0 t $ |
the real part corresponds to the temporal evolution of the amplitude
| $ \theta = \theta_0 \cos \omega_0 t $ |
| $ x = x_0 \cos \omega_0 t $ |
ID:(14074, 0)
Swing speed
Equation
When we extract the real part of the derivative of the complex number representing the oscillation
| $ \dot{z} = i \omega_0 z $ |
whose real part corresponds to the velocity
| $ \omega = - \theta_0 \omega_0 \sin \omega_0 t $ |
| $ v = - x_0 \omega_0 \sin \omega_0 t $ |
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, 0)