73 lines
2.1 KiB
Markdown
Executable File
73 lines
2.1 KiB
Markdown
Executable File
---
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author: Akbar Rahman
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date: \today
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title: MMME1028 // Dynamics
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tags: [ uni, mmme1028, dynamics ]
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uuid: e6d3a307-b2e6-40e3-83bb-ef73512d69ad
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---
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# Circular Motion
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$$a_c = r\omega^2$$
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$$a = r\alpha \hat{e}_\theta - r\omega^2\hat{e}_r$$
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## Moment of Inertia
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$$J = mr^2 = \frac{M}{\ddot\theta}$$
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The unit of $J$ is kgm$^2$.
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Consider a particle of mass $m$ attached to one end of a rigid rod of length $r$.
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The rod is pivoting at its other end about point $O$, and experiences a torque $M$.
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This torque will cause the mass and the rod to rotate about $O$ with angular velocity
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$\dot{\theta}$ an angular acceleration $\ddot{\theta}$.
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What is the expression for $M$?
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Well if break down the moment $M$ into a force, $F$, acting on the mass, we know that the
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moment $M = Fr$.
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We know $F = ma$, and this case $a = r\ddot{\theta}$ so $M = mr^2\ddot\theta$.
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The moment of inertia is $J = mr^2$ so $M = J\ddot\theta$.
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If multiple torques are applied to a body the *rotational equation* of the motion is
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$$\overrightarrow{M} = \sum_i \overrightarrow{M}_i = J \overrightarrow{\ddot\theta} = J \overrightarrow{\alpha}$$
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The moment of inertia of any object is found by considering the object to be made up of lots of
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small particles and adding the moments of inertia for each small particle.
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The moments of inertia for a body depends on the mass and its distribution about the axis in
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consideration.
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$$J = \sum_i m_ir^2_i \rightarrow \int\! r^2 \mathrm{d}m$$
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### Perpendicular Axis Rule
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The perpendicular axis rule states that, for lamina object:
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$$J_z = J_x + J_y$$
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where $J_x$, $J_y$, and $J_z$ are the moments of inertia along their respective axes.
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### Parallel Axes Rule (Huygens-Steiner Theorem)
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The parallel axes rule states that:
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$$J_A = J_G = md^2$$
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where $d$ is the perpendicular distance between the two axes.
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### Moment of a Compound Object
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The moment of inertia for any compound object can be calculated by adding and subtracting the
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moments of inertia for its 'standard' components.
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### Moment of Inertia of Standard Objects
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// TODO
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