Data Definitions

This section collects and defines all the data needed to build the instance models.

RefnameDD:ctrOfMass
LabelCenter of Mass
Symbol\({\boldsymbol{p}\text{(}t\text{)}_{\text{CM}}}\)
Units\({\text{m}}\)
Equation\[{\boldsymbol{p}\text{(}t\text{)}_{\text{CM}}}=\frac{\displaystyle\sum{{m_{j}}\,{\boldsymbol{p}\text{(}t\text{)}_{j}}}}{{m_{T}}}\]
Description
  • \({\boldsymbol{p}\text{(}t\text{)}_{\text{CM}}}\) is the Center of Mass (\({\text{m}}\))
  • \({m_{j}}\) is the mass of the j-th particle (\({\text{kg}}\))
  • \({\boldsymbol{p}\text{(}t\text{)}_{j}}\) is the position vector of the j-th particle (\({\text{m}}\))
  • \({m_{T}}\) is the total mass of the rigid body (\({\text{kg}}\))
Notes
  • All bodies are assumed to be rigid (from A:objectTy).
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RefByIM:transMot and IM:col2D
RefnameDD:linDisp
LabelLinear displacement
Symbol\(u\text{(}t\text{)}\)
Units\({\text{m}}\)
Equation\[u\text{(}t\text{)}=\frac{\,d\boldsymbol{p}\text{(}t\text{)}\left(t\right)}{\,dt}\]
Description
  • \(u\text{(}t\text{)}\) is the linear displacement (\({\text{m}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(\boldsymbol{p}\text{(}t\text{)}\) is the position (\({\text{m}}\))
Notes
  • All bodies are assumed to be rigid (from A:objectTy).
Source–
RefByIM:transMot
RefnameDD:linVel
LabelLinear velocity
Symbol\(v\text{(}t\text{)}\)
Units\(\frac{\text{m}}{\text{s}}\)
Equation\[v\text{(}t\text{)}=\frac{\,d\boldsymbol{u}\left(t\right)}{\,dt}\]
Description
  • \(v\text{(}t\text{)}\) is the linear velocity (\(\frac{\text{m}}{\text{s}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(\boldsymbol{u}\) is the displacement (\({\text{m}}\))
Notes
  • All bodies are assumed to be rigid (from A:objectTy).
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RefByIM:transMot
RefnameDD:linAcc
LabelLinear acceleration
Symbol\(a\text{(}t\text{)}\)
Units\(\frac{\text{m}}{\text{s}^{2}}\)
Equation\[a\text{(}t\text{)}=\frac{\,d\boldsymbol{v}\text{(}t\text{)}\left(t\right)}{\,dt}\]
Description
  • \(a\text{(}t\text{)}\) is the linear acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(\boldsymbol{v}\text{(}t\text{)}\) is the velocity (\(\frac{\text{m}}{\text{s}}\))
Notes
  • All bodies are assumed to be rigid (from A:objectTy).
Source–
RefByIM:transMot
RefnameDD:angDisp
LabelAngular displacement
Symbol\(θ\)
Units\({\text{rad}}\)
Equation\[θ=\frac{\,dϕ\left(t\right)}{\,dt}\]
Description
  • \(θ\) is the angular displacement (\({\text{rad}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(ϕ\) is the orientation (\({\text{rad}}\))
Notes
Source–
RefByIM:rotMot
RefnameDD:angVel
LabelAngular velocity
Symbol\(ω\)
Units\(\frac{\text{rad}}{\text{s}}\)
Equation\[ω=\frac{\,dθ\left(t\right)}{\,dt}\]
Description
  • \(ω\) is the angular velocity (\(\frac{\text{rad}}{\text{s}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(θ\) is the angular displacement (\({\text{rad}}\))
Notes
Source–
RefByIM:rotMot
RefnameDD:angAccel
LabelAngular acceleration
Symbol\(α\)
Units\(\frac{\text{rad}}{\text{s}^{2}}\)
Equation\[α=\frac{\,dω\left(t\right)}{\,dt}\]
Description
  • \(α\) is the angular acceleration (\(\frac{\text{rad}}{\text{s}^{2}}\))
  • \(t\) is the time (\({\text{s}}\))
  • \(ω\) is the angular velocity (\(\frac{\text{rad}}{\text{s}}\))
Notes
Source–
RefByIM:rotMot
RefnameDD:chaslesThm
LabelChasles’ theorem
Symbol\({\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}\)
Units\(\frac{\text{m}}{\text{s}}\)
Equation\[{\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}={\boldsymbol{v}\text{(}t\text{)}_{\text{O}}}+ω\times{\boldsymbol{u}_{\text{O}\text{B}}}\]
Description
  • \({\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}\) is the velocity at point B (\(\frac{\text{m}}{\text{s}}\))
  • \({\boldsymbol{v}\text{(}t\text{)}_{\text{O}}}\) is the velocity at point origin (\(\frac{\text{m}}{\text{s}}\))
  • \(ω\) is the angular velocity (\(\frac{\text{rad}}{\text{s}}\))
  • \({\boldsymbol{u}_{\text{O}\text{B}}}\) is the displacement vector between the origin and point B (\({\text{m}}\))
Notes
  • The linear velocity \({\boldsymbol{v}\text{(}t\text{)}_{\text{B}}}\) of any point B in a rigid body is the sum of the linear velocity \({\boldsymbol{v}\text{(}t\text{)}_{\text{O}}}\) of the rigid body at the origin (axis of rotation) and the resultant vector from the cross product of the rigid body’s angular velocity \(ω\) and the displacement vector between the origin and point B \({\boldsymbol{u}_{\text{O}\text{B}}}\).
  • All bodies are assumed to be rigid (from A:objectTy).
SourcechaslesWiki
RefBy
RefnameDD:torque
LabelTorque
Symbol\(\boldsymbol{τ}\)
Units\(\text{N}\text{m}\)
Equation\[\boldsymbol{τ}=\boldsymbol{r}\times\boldsymbol{F}\]
Description
  • \(\boldsymbol{τ}\) is the torque (\(\text{N}\text{m}\))
  • \(\boldsymbol{r}\) is the position vector (\({\text{m}}\))
  • \(\boldsymbol{F}\) is the force (\({\text{N}}\))
Notes
  • The torque on a body measures the tendency of a force to rotate the body around an axis or pivot.
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RefBy
RefnameDD:kEnergy
LabelKinetic energy
Symbol\(KE\)
Units\({\text{J}}\)
Equation\[KE=m\,\frac{|\boldsymbol{v}\text{(}t\text{)}|^{2}}{2}\]
Description
  • \(KE\) is the kinetic energy (\({\text{J}}\))
  • \(m\) is the mass (\({\text{kg}}\))
  • \(\boldsymbol{v}\text{(}t\text{)}\) is the velocity (\(\frac{\text{m}}{\text{s}}\))
Notes
  • Kinetic energy is the measure of the energy a body possesses due to its motion.
  • All bodies are assumed to be rigid (from A:objectTy) and two-dimensional (from A:objectDimension).
  • No damping occurs during the simulation (from A:dampingInvolvement).
Source–
RefBy
RefnameDD:coeffRestitution
LabelCoefficient of restitution
Symbol\({C_{\text{R}}}\)
UnitsUnitless
Equation\[{C_{\text{R}}}=-\left(\frac{{{\boldsymbol{v}\text{(}t\text{)}_{\text{f}}}^{\text{A}\text{B}}}\cdot{}\boldsymbol{n}}{{{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\boldsymbol{n}}\right)\]
Description
  • \({C_{\text{R}}}\) is the coefficient of restitution (Unitless)
  • \({{\boldsymbol{v}\text{(}t\text{)}_{\text{f}}}^{\text{A}\text{B}}}\) is the final relative velocity between rigid bodies of A and B (\(\frac{\text{m}}{\text{s}}\))
  • \(\boldsymbol{n}\) is the collision normal vector (\({\text{m}}\))
  • \({{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\) is the initial relative velocity between rigid bodies of A and B (\(\frac{\text{m}}{\text{s}}\))
Notes
  • The coefficient of restitution \({C_{\text{R}}}\) determines the elasticity of a collision between two rigid bodies. \({C_{\text{R}}}=1\) results in an elastic collision, \({C_{\text{R}}}\lt{}1\) results in an inelastic collision, and \({C_{\text{R}}}=0\) results in a totally inelastic collision.
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RefBy
RefnameDD:reVeInColl
LabelInitial Relative Velocity Between Rigid Bodies of A and B
Symbol\({{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\)
Units\(\frac{\text{m}}{\text{s}}\)
Equation\[{{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}={\boldsymbol{v}\text{(}t\text{)}^{\text{A}\text{P}}}-{\boldsymbol{v}\text{(}t\text{)}^{\text{B}\text{P}}}\]
Description
  • \({{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\) is the initial relative velocity between rigid bodies of A and B (\(\frac{\text{m}}{\text{s}}\))
  • \({\boldsymbol{v}\text{(}t\text{)}^{\text{A}\text{P}}}\) is the velocity of the point of collision P in body A (\(\frac{\text{m}}{\text{s}}\))
  • \({\boldsymbol{v}\text{(}t\text{)}^{\text{B}\text{P}}}\) is the velocity of the point of collision P in body B (\(\frac{\text{m}}{\text{s}}\))
Notes
  • In a collision, the velocity of a rigid body A colliding with another rigid body B relative to that body \({{\boldsymbol{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\) is the difference between the velocities of A and B at point P.
  • All bodies are assumed to be rigid (from A:objectTy).
Source–
RefBy
RefnameDD:impulseV
LabelImpulse (vector)
Symbol\(\boldsymbol{J}\)
Units\(\text{N}\text{s}\)
Equation\[\boldsymbol{J}=m\,Δ\boldsymbol{v}\]
Description
  • \(\boldsymbol{J}\) is the impulse (vector) (\(\text{N}\text{s}\))
  • \(m\) is the mass (\({\text{kg}}\))
  • \(Δ\boldsymbol{v}\) is the change in velocity (\(\frac{\text{m}}{\text{s}}\))
Notes
  • An impulse (vector) \(\boldsymbol{J}\) occurs when a force \(\boldsymbol{F}\) acts over a body over an interval of time.
  • All bodies are assumed to be rigid (from A:objectTy).
Source–
RefBy

Detailed derivation of impulse (vector):

Newton’s second law of motion states:

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

Rearranging:

\[\int_{{t_{1}}}^{{t_{2}}}{\boldsymbol{F}}\,dt=m\,\left(\int_{{\boldsymbol{v}\text{(}t\text{)}_{1}}}^{{\boldsymbol{v}\text{(}t\text{)}_{2}}}{1}\,d\boldsymbol{v}\text{(}t\text{)}\right)\]

Integrating the right hand side:

\[\int_{{t_{1}}}^{{t_{2}}}{\boldsymbol{F}}\,dt=m\,{\boldsymbol{v}\text{(}t\text{)}_{2}}-m\,{\boldsymbol{v}\text{(}t\text{)}_{1}}=m\,Δ\boldsymbol{v}\]

RefnameDD:potEnergy
LabelPotential energy
Symbol\(PE\)
Units\({\text{J}}\)
Equation\[PE=m\,\boldsymbol{g}\,h\]
Description
  • \(PE\) is the potential energy (\({\text{J}}\))
  • \(m\) is the mass (\({\text{kg}}\))
  • \(\boldsymbol{g}\) is the gravitational acceleration (\(\frac{\text{m}}{\text{s}^{2}}\))
  • \(h\) is the height (\({\text{m}}\))
Notes
  • The potential energy of an object is the energy held by an object because of its position to other objects.
  • All bodies are assumed to be rigid (from A:objectTy) and two-dimensional (from A:objectDimension).
  • No damping occurs during the simulation (from A:dampingInvolvement).
Source–
RefBy
RefnameDD:momentOfInertia
LabelMoment of inertia
Symbol\(\boldsymbol{I}\)
Units\(\text{kg}\text{m}^{2}\)
Equation\[\boldsymbol{I}=\displaystyle\sum{{m_{j}}\,{d_{j}}^{2}}\]
Description
  • \(\boldsymbol{I}\) is the moment of inertia (\(\text{kg}\text{m}^{2}\))
  • \({m_{j}}\) is the mass of the j-th particle (\({\text{kg}}\))
  • \({d_{j}}\) is the distance between the j-th particle and the axis of rotation (\({\text{m}}\))
Notes
  • The moment of inertia \(\boldsymbol{I}\) of a body measures how much torque is needed for the body to achieve angular acceleration about the axis of rotation.
  • All bodies are assumed to be rigid (from A:objectTy).
Source–
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