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Oscillators of a Spring
Storyboard 
In the case of the spring the force is proportional to the elongation of the spring so that the equations of motion are linear and the frequency of the oscillation is independent of the amplitude. This is the key to generate an oscillation that does not depend on the fact that the friction decreases over time. This is why old clocks used (circular) springs to generate stable oscillations to measure elapsed time.
ID:(1425, 0)
Oscillators of a Spring
Storyboard
In the case of the spring the force is proportional to the elongation of the spring so that the equations of motion are linear and the frequency of the oscillation is independent of the amplitude. This is the key to generate an oscillation that does not depend on the fact that the friction decreases over time. This is why old clocks used (circular) springs to generate stable oscillations to measure elapsed time.
Variables
Calculations
to 
,
then, select the variable: 
to 
Calculations
Variable
Given
Calculate
Target :
Equation
To be used
Equations
En el caso el stico (resorte) la fuerza es
con
$\Delta x = x_2 - x_1$
\\n\\ncorresponde al camino recorrido por lo que\\n\\n
$\Delta W=k,x,\Delta x=k(x_2-x_1)\displaystyle\frac{(x_1+x_2)}{2}=\displaystyle\frac{k}{2}(x_2^2-x_1^2)$
y con ello la energ a potencial el stica es
En el caso el stico (resorte) la fuerza es
con
$\Delta x = x_2 - x_1$
\\n\\ncorresponde al camino recorrido por lo que\\n\\n
$\Delta W=k,x,\Delta x=k(x_2-x_1)\displaystyle\frac{(x_1+x_2)}{2}=\displaystyle\frac{k}{2}(x_2^2-x_1^2)$
y con ello la energ a potencial el stica es
Since kinetic energy is equal to
and momentum is
we can express it as
$K_t=\displaystyle\frac{1}{2} m_i v^2=\displaystyle\frac{1}{2} m_i \left(\displaystyle\frac{p}{m_i}\right)^2=\displaystyle\frac{p^2}{2m_i}$
or
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
One of the systems it depicts is that of a spring. This is associated with the elastic deformation of the material from which the spring is made. By "elastic," we mean a deformation that, upon removing the applied stress, allows the system to fully regain its original shape. It's understood that it doesn't undergo plastic deformation.
Since the energy of the spring is given by
$E=\displaystyle\frac{1}{2}m_i v^2+\displaystyle\frac{1}{2}k x^2$
the period will be equal to
$T=2\pi\sqrt{\displaystyle\frac{m_i}{k}}$
and thus, the angular frequency is
The total Energy ($E$) corresponds to the sum of the total Kinetic Energy ($K$) and the potential Energy ($V$):
The kinetic energy of a mass $m$
can be expressed in terms of momentum as
En el caso el stico (resorte) la fuerza es
la energ a
se puede mostrar que en este caso es
En el caso el stico (resorte) la fuerza es
la energ a
se puede mostrar que en este caso es
The product of the hooke Constant ($k$) and the inertial Mass ($m_i$) is called the frecuencia angular del resorte ($\omega$) and is defined as:
The moment ($p$) is calculated from the inertial Mass ($m_i$) and the speed ($v$) using
The period ($T$) is determined from the inertial Mass ($m_i$) and the hooke Constant ($k$) by means of:
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 angular frequency ($\omega$) is with the period ($T$) equal to
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
ID:(1425, 0)