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