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[image]
This means the magnitude of the moment is independent of a.
Thus the moment of a couple is the same about any point. |
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Any number of forces can be combined into one resultant force if they all act in plane and pass through the same point.
[image] |
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| Generalisation of force, couple systems |
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It can be shown that any system of forces and couples can be replaced by a single force F.
[image] |
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| Constant Fluid pressure equation and first use. |
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[image]
Where F is the force per unit width. |
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| Second Constant Fluid pressure use |
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If you want to take the pressure in a specific direction, you can utilise the projected area.
[image]
For instance, here the pressure in the x direction is p*Y [image].
Therefore, if an object has uniform pressure around it's entire surface, the resultant force would be zero.
[image]
Example:
[image], the resultant force ends up being a point force. |
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| Hydrostatic pressure at a depth h below the surface |
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| Converting distributive forces to point forces. |
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Here the pressure is varied due to height, but the surface normal is constant, so the equation for the force needs to be altered.
[image] Since, P(h) has been shown already to be [image]gh.
This means we know the magnitude of the force but not its location as a point force. To find the location we alter the original force equation to be taking moments instead. The moment represented by the integral must be equal to F h*.
[image] |
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| Trick that you can do with these types of pressure questions |
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[image]
This can be simplified by replacing the distribution of the weights by a single force for the whole body’s weight (Mg) acting at the centre of mass. |
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[image]
For the two general lamina, because the density and thickness are uniform, area dA has mass dm = [image]t dA. The earlier equation for centre of mass can therefore be restated as:
[image]
The centroid of area is a point which defines the geometric
centre of the area of a body.
For the simpler case of fig b:
[image]
The distributed forces due to gravity acting on a body can therefore be replaced by a single force acting at the centre of mass of the body, which for uniform laminae coincides with the centroid of area. |
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| Distributive forces acting on a surface |
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| Friction and the angle of friction |
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[image]
The reaction force between two surfaces is the sum of the normal force and friction.
[image]
If the angle between the normal and reaction force is less than the limit [image], then the object will not move. Therefore:
[image]
[image]
[image] |
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Different joints have different degrees of freedom.
[image] |
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