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| What is the logic behind Relative motion of points within a rigid body? |
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Definition
[image]
The distance between the two points (R B/A) must be constant, therefore the velocities of both points must be equal in the direction of the line between them. |
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Term
| What are the equations derived from this logic? |
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Definition
[image], since r is fixed and therefore r dot is 0.
(There are fuller equations but I don't think they are useful, just differentiating RB = RA + RB/A in full without subbing in r dot = 0) |
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Term
| 2D Planar motion within a rigid body |
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Definition
| [image], basically the previous equations but written slightly different. |
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Term
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Definition
For 2D planar motion there is a point in the plane of a lamina that has zero velocity(instantaneously at rest). Therefore, the lamina behaves as if it were rotating about said point with an angular velocity.
[image]
Pay attention to the direction the angular velocity is, because that determines whether V is positive or negative.
[image]
If it was anticlockwise V would be negative. |
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Term
| Centre of mass integral derivation |
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Definition
[image]
M is the total mass of the body.
RG is the position vector of the centre of gravity. |
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Term
How can you use symmetry and area to dodge using the integral?
Plus extra info. |
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Definition
[image]
Also, if the object has a hole, the whole can be treated as negative mass. |
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Term
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Definition
The total external force is equal to the total mass times the acceleration of the centre of mass.
[image] |
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Term
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Definition
Any body can be describes by treating it as a particle with all the mass concentrated at the CoM.
The general expression relating torque to the angular acceleration of a 3D rigid body is:
[image]
When we consider this equation it is for the case of planar motion, where all the particles move in a plane parallel to a single 'plane of motion'. This happens when:
1. Rotation about a fixed axis.
2. General motion of a lamina within its own plane.
3. Some other cases beyond the scope of the course. |
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Term
| Rotation about a fixed axis |
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Definition
[image]
This is the rotational dynamics about a FIXED AXIS. Jo is the mass moment of inertia about the O axis |
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Term
| Mass moment of inertia ABOUT AN AXIS. |
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Definition
This is for each axis:
[image]
Where rn is the shortest distance to the n axis. |
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Term
| Mass moment of inertia about each axis when the system is a lamina |
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Definition
There is zero depth in the system so the z^2 terms can be neglected:
[image]
Giving the 'perpendicular axis theorem':
[image] |
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Term
| Mass moment of inertia about an axis that is not through the centre of mass |
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Definition
Io is the mass moment of inertia about this new axis.
IG is the mass moment of inertia of the axis parallel to the new axis and through the centre of gravity.
[image] |
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Term
| Moment of inertia of Composite bodies |
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Definition
1. Find IG for each component.
2. Use parallel axis theorem to find Io for each component.
3. Sum the contributions for each component.
[image] |
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