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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
Description
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.
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En el caso el stico (resorte) la fuerza es
| $$ |
con
| $ dW = \vec{F} \cdot d\vec{s} $ |
\\n\\nLa diferencia\\n\\n
$\Delta x = x_2 - x_1$
\\n\\ncorresponde al camino recorrido por lo que\\n\\n
$\Delta W=k,x,\Delta x=k(x_2-x_1)\displaystyle\frac{(x_1+x_2)}{2}=\displaystyle\frac{k}{2}(x_2^2-x_1^2)$
y con ello la energ a potencial el stica es
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
(ID 3246)
En el caso el stico (resorte) la fuerza es
| $$ |
con
| $ dW = \vec{F} \cdot d\vec{s} $ |
\\n\\nLa diferencia\\n\\n
$\Delta x = x_2 - x_1$
\\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
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
(ID 3246)
(ID 3687)
Since kinetic energy is equal to
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
and momentum is
| $ p = m_i v $ |
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
| $ K =\displaystyle\frac{ p ^2}{2 m_i }$ |
(ID 4425)
(ID 10283)
(ID 12338)
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
(ID 15848)
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
| $ \omega_0 ^2=\displaystyle\frac{ k }{ m_i }$ |
(ID 15563)
(ID 15851)
The total Energy ($E$) corresponds to the sum of the total Kinetic Energy ($K$) and the potential Energy ($V$):
| $ E = K + V $ |
(ID 3687)
The kinetic energy of a mass $m$
| $ K_t =\displaystyle\frac{1}{2} m_i v ^2$ |
can be expressed in terms of momentum as
| $ K =\displaystyle\frac{ p ^2}{2 m_i }$ |
(ID 4425)
En el caso el stico (resorte) la fuerza es
| $$ |
la energ a
| $ dW = \vec{F} \cdot d\vec{s} $ |
se puede mostrar que en este caso es
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
(ID 3246)
En el caso el stico (resorte) la fuerza es
| $$ |
la energ a
| $ dW = \vec{F} \cdot d\vec{s} $ |
se puede mostrar que en este caso es
| $ V =\displaystyle\frac{1}{2} k x ^2$ |
(ID 3246)
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:
| $ \omega_0 ^2=\displaystyle\frac{ k }{ m_i }$ |
(ID 1242)
The moment ($p$) is calculated from the inertial Mass ($m_i$) and the speed ($v$) using
| $ p = m_i v $ |
(ID 10283)
The period ($T$) is determined from the inertial Mass ($m_i$) and the hooke Constant ($k$) by means of:
| $ T =2 \pi \sqrt{\displaystyle\frac{ m_i }{ k }}$ |
(ID 7106)
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:
| $ \nu =\displaystyle\frac{1}{ T }$ |
Frequency is indicated in Hertz (Hz).
(ID 4427)
The angular frequency ($\omega$) is with the period ($T$) equal to
| $ \omega = \displaystyle\frac{2 \pi }{ T }$ |
(ID 12335)
The relationship between the angular frequency ($\omega$) and the sound frequency ($\nu$) is expressed as:
| $ \omega = 2 \pi \nu $ |
(ID 12338)
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
| $ x = x_0 \cos \omega_0 t $ |
(ID 14074)
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
| $ v = - x_0 \omega_0 \sin \omega_0 t $ |
(ID 14076)
ID:(1425, 0)