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For D'Alembert Forces, draw both a force (FBD) and acceleration diagram.
D'Alembert Forces always directly oppose acceleration. |
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Definition
[image]
Acceleration in y is zero. |
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| When dealing with Equations of Motion a useful substitution is:[image] |
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By inputting the initial and final conditions you can skip the need to find the value of '+ C'. For example:
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Numerical integration is based upon the Taylor series, we usually use first order only.
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| How do you check if a simulation is accurate? |
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1. Check a case/aspects of the simulation you already know the answer to.
2. Check convergence: make [image]t smaller until the answer stops changing. |
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| If a force model is conservative:[image] |
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| The total work done going from A to B is the same regardless of the route. |
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| Potential energy equations derived from models |
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Local Gravity:
V = mgh
Gravity via inverse square law:
V = -GMm/r
Springs:
V = 1/2 kx^2
Torsional Springs:
V = 1/2 k[image] |
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| General case for Potential energy situations that are not conservative |
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[image]
Which can be rearranged to:
[image] |
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Potential energy, equilibrium, and stability in a conservative system
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The stable points occur at a minima (when V''(x) > 0).
The unstable points occur when at a maxima (V''(x) < 0). |
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Energy Principle
Linear Momentum Principle |
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| Special Momentum Case when it is elastic (all energy is conserved) |
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[image]
Which results in:
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An explanation for the equation of moment q about point O:
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What is [image] and how do you find it?
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[image] is a vector quantity. [image] is the moment of F about an axis in the direction of n passing through the origin.
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| Angular momentum [image] of a particle about point O: |
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What is [image] and how do you find it?
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[image] is a vector quantity. [image] is the angular momentum of a particle about an axis in the direction of n passing through the origin.
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What does this mean and how is it derived?
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Rate of increase of angular momentum about n axis is equal to the moment of force about the same axis.
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| When is angular momentum conserved? |
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Definition
Angular momentum is conserved about a point/axis if the moment of force about the point/axis is zero or if the line of action of the net force passes through (or is parallel to - for axis) point O/the axis.
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