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Pendulum Swing
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In the case of a pendulum it is gravity that generates a torque opposite to the mass leaving the resting point. However, the torque is not proportional to the angle, there being a non-linear relationship which makes the movement more complex.

Since the torque is not proportional to the angle, the oscillation frequency depends on the amplitude, which makes it difficult to apply it to mark the passage on a clock. However, the effect is minimal if the angle is small which leads to the application of the pendulum on clocks is achieved with long bars.

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ID:(1426, 0)


Calculation of the Potential Energy of the Pendulum
Description

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When a pendulum of length $l$ is deflected at an angle $\theta$, the mass gains height, which is calculated as

$l - l \cos\theta = l (1 - \cos\theta)$

this is associated with the gain in gravitational potential energy.

ID:(1239, 0)


Height of the center of mass in a pendulum
Equation

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For a pendulum with length $L$ that is deflected at an angle $\theta$, the mass is raised



by a height equal to:

$ h = L (1-\cos \theta )$

$h$
Height in Case Pendulum
$m$
$L$
Pendulum Length
$m$
$\theta$
Swing angle
$rad$

ID:(4523, 0)


Potential energy of a mathematical pendulum
Equation

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For the case of a mass $m$ hanging from a string of length $L$ and being deflected at an angle $\theta$ from the vertical, the mass will gain a height of

$ h = L (1-\cos \theta )$



which means that the gravitational potential energy

$ V = m g z $



will be

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

$g$
Gravitational Acceleration
9.8
$m/s^2$
$m_g$
Gravitational mass
$kg$
$L$
Pendulum Length
$m$
$U$
Potential Energy Pendulum
$J$
$\theta$
Swing angle
$rad$

where $g$ is the acceleration due to gravity.

ID:(4513, 0)


Potential energy of a mathematical pendulum for small angles
Equation

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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$

$g$
Gravitational Acceleration
9.8
$m/s^2$
$m_g$
Gravitational mass
$kg$
$L$
Pendulum Length
$m$
$V$
Potential Energy Pendulum, for small Angles
$J$
$\theta$
Swing angle
$rad$

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$



It's important to note that the angle must be expressed in radians.

ID:(4514, 0)


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Video

Video: Oscillators of a pendulum