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 first object

Refname	GD:velocityX1
Label	The \(x\)-component of velocity of the first object
Units	\(\frac{\text{m}}{\text{s}}\)
Equation	\[{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]
Description	\({v_{\text{x}1}}\) is the horizontal velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) \({w_{1}}\) is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{1}}\) is the length of the first rod (\({\text{m}}\)) \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\))
Source	–
RefBy

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

At a given point in time, velocity is defined in DD:positionGDD

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

We also know the horizontal position that is defined in DD:positionXDD1

\[{p_{\text{x}1}}={L_{1}}\,\sin\left({θ_{1}}\right)\]

Applying this,

\[{v_{\text{x}1}}=\frac{\,d{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt}\]

\({L_{1}}\) is constant with respect to time, so

\[{v_{\text{x}1}}={L_{1}}\,\frac{\,d\sin\left({θ_{1}}\right)}{\,dt}\]

Therefore, using the chain rule,

\[{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]

The \(y\)-component of velocity of the first object

Refname	GD:velocityY1
Label	The \(y\)-component of velocity of the first object
Units	\(\frac{\text{m}}{\text{s}}\)
Equation	\[{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]
Description	\({v_{\text{y}1}}\) is the vertical velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) \({w_{1}}\) is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{1}}\) is the length of the first rod (\({\text{m}}\)) \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\))
Source	–
RefBy

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

At a given point in time, velocity is defined in DD:positionGDD

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

We also know the vertical position that is defined in DD:positionYDD1

\[{p_{\text{y}1}}=-{L_{1}}\,\cos\left({θ_{1}}\right)\]

Applying this,

\[{v_{\text{y}1}}=-\left(\frac{\,d{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt}\right)\]

\({L_{1}}\) is constant with respect to time, so

\[{v_{\text{y}1}}=-{L_{1}}\,\frac{\,d\cos\left({θ_{1}}\right)}{\,dt}\]

Therefore, using the chain rule,

\[{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]

The \(x\)-component of velocity of the second object

Refname	GD:velocityX2
Label	The \(x\)-component of velocity of the second object
Units	\(\frac{\text{m}}{\text{s}}\)
Equation	\[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]
Description	\({v_{\text{x}2}}\) is the horizontal velocity of the second object (\(\frac{\text{m}}{\text{s}}\)) \({v_{\text{x}1}}\) is the horizontal velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) \({w_{2}}\) is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{2}}\) is the length of the second rod (\({\text{m}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\))
Source	–
RefBy

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

At a given point in time, velocity is defined in DD:positionGDD

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

We also know the horizontal position that is defined in DD:positionXDD2

\[{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)\]

Applying this,

\[{v_{\text{x}2}}=\frac{\,d{p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt}\]

\({L_{1}}\) is constant with respect to time, so

\[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]

The \(y\)-component of velocity of the second object

Refname	GD:velocityY2
Label	The \(y\)-component of velocity of the second object
Units	\(\frac{\text{m}}{\text{s}}\)
Equation	\[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]
Description	\({v_{\text{y}2}}\) is the vertical velocity of the second object (\(\frac{\text{m}}{\text{s}}\)) \({v_{\text{y}1}}\) is the vertical velocity of the first object (\(\frac{\text{m}}{\text{s}}\)) \({w_{2}}\) is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{2}}\) is the length of the second rod (\({\text{m}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\))
Source	–
RefBy

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

At a given point in time, velocity is defined in DD:positionGDD

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

We also know the vertical position that is defined in DD:positionYDD2

\[{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)\]

Applying this,

\[{v_{\text{y}2}}=-\left(\frac{\,d{p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt}\right)\]

Therefore, using the chain rule,

\[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]

The \(x\)-component of acceleration of the first object

Refname	GD:accelerationX1
Label	The \(x\)-component of acceleration of the first object
Units	\(\frac{\text{m}}{\text{s}^{2}}\)
Equation	\[{a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]
Description	\({a_{\text{x}1}}\) is the horizontal acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) \({w_{1}}\) is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{1}}\) is the length of the first rod (\({\text{m}}\)) \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\)) \({α_{1}}\) is the angular acceleration of the first object (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source	–
RefBy	IM:calOfAngle2

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

\[{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{x}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{x}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\cos\left({θ_{1}}\right)-{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt}\]

Simplifying,

\[{a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]

The \(y\)-component of acceleration of the first object

Refname	GD:accelerationY1
Label	The \(y\)-component of acceleration of the first object
Units	\(\frac{\text{m}}{\text{s}^{2}}\)
Equation	\[{a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]
Description	\({a_{\text{y}1}}\) is the vertical acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) \({w_{1}}\) is the angular velocity of the first object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{1}}\) is the length of the first rod (\({\text{m}}\)) \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\)) \({α_{1}}\) is the angular acceleration of the first object (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source	–
RefBy	IM:calOfAngle2

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

\[{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{y}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{y}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt}\]

Simplifying,

\[{a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]

The \(x\)-component of acceleration of the second object

Refname	GD:accelerationX2
Label	The \(x\)-component of acceleration of the second object
Units	\(\frac{\text{m}}{\text{s}^{2}}\)
Equation	\[{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]
Description	\({a_{\text{x}2}}\) is the horizontal acceleration of the second object (\(\frac{\text{m}}{\text{s}^{2}}\)) \({a_{\text{x}1}}\) is the horizontal acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) \({w_{2}}\) is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{2}}\) is the length of the second rod (\({\text{m}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\)) \({α_{2}}\) is the angular acceleration of the second object (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source	–
RefBy	IM:calOfAngle2

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

\[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{x}2}}=\frac{\,d{v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]

The \(y\)-component of acceleration of the second object

Refname	GD:accelerationY2
Label	The \(y\)-component of acceleration of the second object
Units	\(\frac{\text{m}}{\text{s}^{2}}\)
Equation	\[{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]
Description	\({a_{\text{y}2}}\) is the vertical acceleration of the second object (\(\frac{\text{m}}{\text{s}^{2}}\)) \({a_{\text{y}1}}\) is the vertical acceleration of the first object (\(\frac{\text{m}}{\text{s}^{2}}\)) \({w_{2}}\) is the angular velocity of the second object (\(\frac{\text{rad}}{\text{s}}\)) \({L_{2}}\) is the length of the second rod (\({\text{m}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\)) \({α_{2}}\) is the angular acceleration of the second object (\(\frac{\text{rad}}{\text{s}^{2}}\))
Source	–
RefBy	IM:calOfAngle2

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 horizontal velocity to be

\[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]

Applying this to our equation for acceleration

\[{a_{\text{y}2}}=\frac{\,d{v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt}\]

By the product and chain rules, we find

\[{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]

Horizontal force on the first object

Refname	GD:xForce1
Label	Horizontal force on the first object
Units	\({\text{N}}\)
Equation	\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)+{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)\]
Description	\(\boldsymbol{F}\) is the force (\({\text{N}}\)) \(m\) is the mass (\({\text{kg}}\)) \(\boldsymbol{a}\text{(}t\text{)}\) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \({\boldsymbol{T}_{1}}\) is the tension of the first object (\({\text{N}}\)) \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\)) \({\boldsymbol{T}_{2}}\) is the tension of the second object (\({\text{N}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\))
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the first object:

\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{1}}\,\sin\left({θ_{1}}\right)+{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)\]

Vertical force on the first object

Refname	GD:yForce1
Label	Vertical force on the first object
Units	\({\text{N}}\)
Equation	\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{1}}\,\cos\left({θ_{1}}\right)-{\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\boldsymbol{g}\]
Description	\(\boldsymbol{F}\) is the force (\({\text{N}}\)) \(m\) is the mass (\({\text{kg}}\)) \(\boldsymbol{a}\text{(}t\text{)}\) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \({\boldsymbol{T}_{1}}\) is the tension of the first object (\({\text{N}}\)) \({θ_{1}}\) is the angle of the first rod (\({\text{rad}}\)) \({\boldsymbol{T}_{2}}\) is the tension of the second object (\({\text{N}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\)) \({m_{1}}\) is the mass of the first object (\({\text{kg}}\)) \(\boldsymbol{g}\) is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the first object:

\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{1}}\,\cos\left({θ_{1}}\right)-{\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\boldsymbol{g}\]

Horizontal force on the second object

Refname	GD:xForce2
Label	Horizontal force on the second object
Units	\({\text{N}}\)
Equation	\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)\]
Description	\(\boldsymbol{F}\) is the force (\({\text{N}}\)) \(m\) is the mass (\({\text{kg}}\)) \(\boldsymbol{a}\text{(}t\text{)}\) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \({\boldsymbol{T}_{2}}\) is the tension of the second object (\({\text{N}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\))
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the second object:

\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}_{2}}\,\sin\left({θ_{2}}\right)\]

Vertical force on the second object

Refname	GD:yForce2
Label	Vertical force on the second object
Units	\({\text{N}}\)
Equation	\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\boldsymbol{g}\]
Description	\(\boldsymbol{F}\) is the force (\({\text{N}}\)) \(m\) is the mass (\({\text{kg}}\)) \(\boldsymbol{a}\text{(}t\text{)}\) is the acceleration (\(\frac{\text{m}}{\text{s}^{2}}\)) \({\boldsymbol{T}_{2}}\) is the tension of the second object (\({\text{N}}\)) \({θ_{2}}\) is the angle of the second rod (\({\text{rad}}\)) \({m_{2}}\) is the mass of the second object (\({\text{kg}}\)) \(\boldsymbol{g}\) is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
Source	–
RefBy	IM:calOfAngle2

Detailed derivation of force on the second object:

\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\boldsymbol{g}\]

Software Requirements Specification for Double Pendulum