ENSC2004 - Lecture 21

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ENSC2004 - Lecture 20
Impulse and Momentum (Rigid Bodies) - Slides

Linear and Angular Momentum

• The linear momentum of a rigid body is mass times speed:
  $\mathbf{L}=\text{m}{\mathbf{v}}_{\text{G}}$
  This equation states that the linear momentum vector $\mathbf{L}$ has a magnitude equal to ($\text{m}{\mathbf{v}}_G$) and a direction defined by ${\mathbf{v}}_G$
• The angular momentum of a rigid body is inertia times angular velocity:
  ${\mathbf{H}}_{\text{G}}={\text{I}}_{\text{G}}\omega$
  When body undergoes purely rectilinear or curvilinear translation, $\omega =0$ thus, angular momentum is zero (${\mathbf{H}}_{\text{G}}=0$)

When a rigid body is rotating about a fixed axis passing through $O$, then:
${\mathbf{H}}_{\text{O}}=({\mathbf{r}}_{\text{G}}\times \text{m}{\mathbf{v}}_{\text{G}})+{\text{I}}_{\text{G}}\omega ={\text{I}}_{\text{O}}\omega$
The equations above still apply

When a rigid body is subjected to general plane motion, both the linear momentum and the angular momentum computed about G are required

The angular momentum about point A is: ${\text{H}}_{\text{A}}={\text{I}}_{\text{G}}\omega +{\text{mv}}_{\text{G}}(d)$

Principle of Impulse and Momentum

• Force → creates impulse → creates change in momentum
• Use these equations to solve problems with force, velocity, and time

Linear impulse-linear momentum equation:
${\mathbf{L}}_1+\sum {\int }_{t_1}^{t_2}\mathbf{F}dt={\mathbf{L}}_2$
$\mathbf{L}=\text{m}{\mathbf{v}}_{\text{G}}$

Angular impulse-angular momentum equation:
$({\mathbf{H}}_{\text{G}}{)}_1+\sum {\int }_{t_1}^{t_2}{\mathbf{M}}_{\text{G}}dt=({\mathbf{H}}_{\text{G}}{)}_2$
${\mathbf{H}}_{\text{G}}={\text{I}}_{\text{G}}\omega$

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Conservation of Momentum 2 - Slides

Conservation of Momentum 2

• Recap: Conservation of Linear & Angular Momentum
• ! When linear impulses are negligible or non-impulse, we can use the conservation of linear momentum to solve problems
  • You can identify negligible forces if they act over a very short/instantaneous period of time
• See example in slides

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End of Unit!