Water pendulum

A water pendulum , also known as an oscillating water column, is generally a U-tube in which a previously deflected water column oscillates. In physics, the course of changes in the height of the water column can be idealized as a harmonic oscillation .

U-tube with deflected water column

The pushing force of a water pendulum is its weight

${\displaystyle F(t)=-m\cdot g}$

the protruding water column. The following applies:

${\displaystyle y}$	Deflection , i.e. half the height of the protruding water column
${\displaystyle m=2yA\rho }$	Mass of the protruding water column
${\displaystyle m_{\mathrm {ges} }=A\rho l}$	Mass of the entire water column with length ${\displaystyle l}$
${\displaystyle A=\pi r^{2}}$	Cross-sectional area of the water column
${\displaystyle \rho }$	Density of the liquid
${\displaystyle g}$	Gravity acceleration

Course of the deflection normalized to the amplitude y ₀ , initial phase φ = 0 and x = ωt

As the water column vibrates, the height of the protruding water column changes as a function of time. This also changes their mass and weight. At the time : ${\displaystyle t}$

${\displaystyle F(t)=-2y(t)A\cdot \rho \cdot g}$.

With the second Newton's law , one has the differential equation ${\displaystyle F(t)=m_{\mathrm {ges} }\cdot {\ddot {y}}(t)}$

${\displaystyle -2y(t)A\rho g=m_{\mathrm {ges} }\cdot {\ddot {y}}(t)}$

or.

${\displaystyle {\ddot {y}}(t)+{\frac {2A\rho g}{m_{\mathrm {ges} }}}\cdot y(t)=0}$.

The solution of the differential equation is a harmonic oscillation which z. B. by

${\displaystyle y(t)=y_{0}\sin(\omega t+\varphi )}$

is resolved. The amplitude , the circular frequency and the phase correspond to the oscillation. For the sake of simplicity, a phase shift of is calculated, since the starting point in time of the oscillation can be freely selected. ${\displaystyle y_{0}}$ ${\displaystyle \omega =2\pi f}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi =0}$

By deriving it twice and inserting it into the above equation, we get:

${\displaystyle -\omega ^{2}y_{0}\sin(\omega t)+{\frac {2A\rho g}{m_{\mathrm {ges} }}}\cdot y_{0}\sin(\omega t)=0}$ from which it follows that

${\displaystyle \omega ^{2}={\frac {2A\rho g}{m_{\mathrm {ges} }}}}$.

This result can be further simplified because . It follows: ${\displaystyle m_{\mathrm {ges} }=A\rho l}$

${\displaystyle \omega ^{2}={\frac {2g}{l}}}$and thus the angular frequency results in . ${\displaystyle \omega ={\sqrt {\frac {2g}{l}}}}$

The following applies to the period of the oscillation:

${\displaystyle T={\frac {2\pi }{\omega }}={\frac {2\pi }{\sqrt {\frac {2g}{l}}}}=2\pi {\sqrt {\frac {l}{2g}}}}$,

the period of oscillation only depends on the length of the water column and the gravitation.

For a more detailed treatment, consider the friction ( flow resistance ) with the wall of the pipe and the internal friction of the liquid, both of which dampen the vibration .

See also

• harmonic oscillator
• Pendulum
• Rolling tank

Web links

• Patrick Nordmann: Animation of the water pendulum