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Pendulum Swing
Storyboard

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.

>Model

ID:(1426, 0)


Calculation of the Potential Energy of the Pendulum
Definition

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)


Pendulum Swing
Description

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.

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$m_g$
m_g
Gravitational mass
kg
$h$
h
Height in Case Pendulum
m
$L$
L
Pendulum Length
m
$V$
V
Potential Energy Pendulum
J
$V$
V
Potential Energy Pendulum, for small Angles
J
$\theta$
theta
Swing angle
rad

Calculations

First, select the equation:   to ,  then, select the variable:   to 
Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used


Equations

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$

(ID 4514)

Examples

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)

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 )$

(ID 4523)

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 g z $



will be

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

where $g$ is the acceleration due to gravity.

(ID 4513)

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$



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

(ID 4514)

ID:(1426, 0)