General Definitions

This section collects the laws and equations that will be used to build the instance models.

The \(x\)-component of velocity of the pendulum

Refname	GD:velocityIX
Label	The \(x\)-component of velocity of the pendulum
Units	\(\frac{\text{m}}{\text{s}}\)
Equation	\[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Description	\({v_{\text{x}}}\) is the \(x\)-component of velocity (\(\frac{\text{m}}{\text{s}}\)) \(ω\) is the angular velocity (\(\frac{\text{rad}}{\text{s}}\)) \({L_{\text{rod}}}\) is the length of the rod (\({\text{m}}\)) \({θ_{p}}\) is the displacement angle of the pendulum (\({\text{rad}}\))
Source	–
RefBy

Detailed derivation of the \(x\)-component of velocity:

At a given point in time, velocity may be defined as

\[\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}\]

We also know the horizontal position

\[{p_{\text{x}}}={L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

Applying this,

\[{v_{\text{x}}}=\frac{\,d{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt}\]

\({L_{\text{rod}}}\) is constant with respect to time, so

\[{v_{\text{x}}}={L_{\text{rod}}}\,\frac{\,d\sin\left({θ_{p}}\right)}{\,dt}\]

Therefore, using the chain rule,

\[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

The \(y\)-component of velocity of the pendulum

Refname	GD:velocityIY
Label	The \(y\)-component of velocity of the pendulum
Units	\(\frac{\text{m}}{\text{s}}\)
Equation	\[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Description	\({v_{\text{y}}}\) is the \(y\)-component of velocity (\(\frac{\text{m}}{\text{s}}\)) \(ω\) is the angular velocity (\(\frac{\text{rad}}{\text{s}}\)) \({L_{\text{rod}}}\) is the length of the rod (\({\text{m}}\)) \({θ_{p}}\) is the displacement angle of the pendulum (\({\text{rad}}\))
Source	–
RefBy

Detailed derivation of the \(y\)-component of velocity:

At a given point in time, velocity may be defined as

\[\boldsymbol{v}\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}}{\,dt}\]

We also know the vertical position

\[{p_{\text{y}}}=-{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

Applying this,

\[{v_{\text{y}}}=-\left(\frac{\,d{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt}\right)\]

\({L_{\text{rod}}}\) is constant with respect to time, so

\[{v_{\text{y}}}=-{L_{\text{rod}}}\,\frac{\,d\cos\left({θ_{p}}\right)}{\,dt}\]

Therefore, using the chain rule,

\[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

The \(x\)-component of acceleration of the pendulum

Refname	GD:accelerationIX
Label	The \(x\)-component of acceleration of the pendulum
Units	\(\frac{\text{m}}{\text{s}^{2}}\)
Equation	\[{a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Description	\({a_{\text{x}}}\) is the \(x\)-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \(ω\) is the angular velocity (\(\frac{\text{rad}}{\text{s}}\)) \({L_{\text{rod}}}\) is the length of the rod (\({\text{m}}\)) \({θ_{p}}\) is the displacement angle of the pendulum (\({\text{rad}}\)) \(α\) is the angular acceleration (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source	–
RefBy

Detailed derivation of the \(x\)-component of acceleration:

Our acceleration is:

\[\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}\]

Earlier, we found the horizontal velocity to be

\[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{x}}}=\frac{\,dω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{x}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)-ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt}\]

Simplifying,

\[{a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

The \(y\)-component of acceleration of the pendulum

Refname	GD:accelerationIY
Label	The \(y\)-component of acceleration of the pendulum
Units	\(\frac{\text{m}}{\text{s}^{2}}\)
Equation	\[{a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Description	\({a_{\text{y}}}\) is the \(y\)-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \(ω\) is the angular velocity (\(\frac{\text{rad}}{\text{s}}\)) \({L_{\text{rod}}}\) is the length of the rod (\({\text{m}}\)) \({θ_{p}}\) is the displacement angle of the pendulum (\({\text{rad}}\)) \(α\) is the angular acceleration (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source	–
RefBy

Detailed derivation of the \(y\)-component of acceleration:

Our acceleration is:

\[\boldsymbol{a}\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}}{\,dt}\]

Earlier, we found the vertical velocity to be

\[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{y}}}=\frac{\,dω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{y}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt}\]

Simplifying,

\[{a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

Horizontal force on the pendulum

Refname	GD:hForceOnPendulum
Label	Horizontal force on the pendulum
Units	\({\text{N}}\)
Equation	\[\boldsymbol{F}=m\,{a_{\text{x}}}=-\boldsymbol{T}\,\sin\left({θ_{p}}\right)\]
Description	\(\boldsymbol{F}\) is the force (\({\text{N}}\)) \(m\) is the mass (\({\text{kg}}\)) \({a_{\text{x}}}\) is the \(x\)-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \(\boldsymbol{T}\) is the tension (\({\text{N}}\)) \({θ_{p}}\) is the displacement angle of the pendulum (\({\text{rad}}\))
Source	–
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Detailed derivation of force on the pendulum:

\[\boldsymbol{F}=m\,{a_{\text{x}}}=-\boldsymbol{T}\,\sin\left({θ_{p}}\right)\]

Vertical force on the pendulum

Refname	GD:vForceOnPendulum
Label	Vertical force on the pendulum
Units	\({\text{N}}\)
Equation	\[\boldsymbol{F}=m\,{a_{\text{y}}}=\boldsymbol{T}\,\cos\left({θ_{p}}\right)-m\,\boldsymbol{g}\]
Description	\(\boldsymbol{F}\) is the force (\({\text{N}}\)) \(m\) is the mass (\({\text{kg}}\)) \({a_{\text{y}}}\) is the \(y\)-component of acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \(\boldsymbol{T}\) is the tension (\({\text{N}}\)) \({θ_{p}}\) is the displacement angle of the pendulum (\({\text{rad}}\)) \(\boldsymbol{g}\) is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
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Detailed derivation of force on the pendulum:

\[\boldsymbol{F}=m\,{a_{\text{y}}}=\boldsymbol{T}\,\cos\left({θ_{p}}\right)-m\,\boldsymbol{g}\]

The angular frequency of the pendulum

Refname	GD:angFrequencyGD
Label	The angular frequency of the pendulum
Units	\({\text{s}}\)
Equation	\[Ω=\sqrt{\frac{\boldsymbol{g}}{{L_{\text{rod}}}}}\]
Description	\(Ω\) is the angular frequency (\({\text{s}}\)) \(\boldsymbol{g}\) is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \({L_{\text{rod}}}\) is the length of the rod (\({\text{m}}\))
Notes	The torque is defined in TM:NewtonSecLawRotMot and frequency is \(f\) is defined in DD:frequencyDD.
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RefBy	GD:periodPend and IM:calOfAngularDisplacement

Detailed derivation of the angular frequency of the pendulum:

Consider the torque on a pendulum defined in TM:NewtonSecLawRotMot. The force providing the restoring torque is the component of weight of the pendulum bob that acts along the arc length. The torque is the length of the string \({L_{\text{rod}}}\) multiplied by the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:

\[\boldsymbol{τ}=-{L_{\text{rod}}}\,m\,\boldsymbol{g}\,\sin\left({θ_{p}}\right)\]

So then

\[\boldsymbol{I}\,α=-{L_{\text{rod}}}\,m\,\boldsymbol{g}\,\sin\left({θ_{p}}\right)\]

Therefore,

\[\boldsymbol{I}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\boldsymbol{g}\,\sin\left({θ_{p}}\right)\]

Substituting for \(\boldsymbol{I}\)

\[m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\boldsymbol{g}\,\sin\left({θ_{p}}\right)\]

Crossing out \(m\) and \({L_{\text{rod}}}\) we have

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\boldsymbol{g}}{{L_{\text{rod}}}}\right)\,\sin\left({θ_{p}}\right)\]

For small angles, we approximate sin \({θ_{p}}\) to \({θ_{p}}\)

\[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\boldsymbol{g}}{{L_{\text{rod}}}}\right)\,{θ_{p}}\]

Because this equation, has the same form as the equation for simple harmonic motion the solution is easy to find. The angular frequency

\[Ω=\sqrt{\frac{\boldsymbol{g}}{{L_{\text{rod}}}}}\]

The period of the pendulum

Refname	GD:periodPend
Label	The period of the pendulum
Units	\({\text{s}}\)
Equation	\[T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\boldsymbol{g}}}\]
Description	\(T\) is the period (\({\text{s}}\)) \(π\) is the ratio of circumference to diameter for any circle (Unitless) \({L_{\text{rod}}}\) is the length of the rod (\({\text{m}}\)) \(\boldsymbol{g}\) is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Notes	The frequency and period are defined in the data definitions for frequency and period respectively
Source	–
RefBy

Detailed derivation of the period of the pendulum:

The period of the pendulum can be defined from the general definition for the equation of angular frequency

\[Ω=\sqrt{\frac{\boldsymbol{g}}{{L_{\text{rod}}}}}\]

Therefore from the data definition of the equation for angular frequency, we have

\[T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\boldsymbol{g}}}\]

Software Requirements Specification for Single Pendulum