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Statics (General Principles (Moment Occurs when the line of action of a…
Statics
General Principles
Newton's laws
- Particle stays at rest or constant velocity...
- \(F = ma\)
The acceleration is proportional to the force, and in the same direction
- Action/reaction are opposite, collinear and act on opposite bodies.
- \(F = \frac{GMm}{r^2}\), which is where \(W=mg\) is derived from.
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Vectors
Parallelogram law
- Join 2 vectors tail to tail
- Can be done in reverse to find the component of a force along two arbitrary axes
- Angles in a quadrilateral sum to 360.
Multiplication
Vector
Dot product
- also knows as scalar product. The result is a scalar.
- gives the perpendicular projection of one vector onto the other
- \(\vec A.\vec B= \lvert \vec A \rvert\lvert \vec B \rvert\cos\theta\)
where \(\vec A.\vec B = A_xB_x + A_yB_y + A_zB_z \)
Cross product
- The cross product of two vectors \( \vec A \times\vec B\) is a vector \( \vec C\)
- \( \vec C\) has magnitude \( \lvert \vec A \rvert\lvert \vec B \rvert\sin\theta\)
- \( \vec C\) = determinant of A and B.
- The direction of \( \vec C\) is perpendicular to the plane \( \vec A\) and \( \vec B\) is in.
Scalar
- Multiplying a vector \( \vec a \) with a scalar \(s \) gives a vector in the same direction, with magnitude \(\lvert\vec a\rvert |s|\)
Moment
- Occurs when the line of action of a force does not pass through the centroid of the body
- (A moment is due to a force not having an equal and opposite force directly along it's line of action.) ???
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About a specified axis
- It is sometimes useful to be able to calculate the moment a force exerts about a certain set axis that is relevant to the problem.
An example would be a force on the door with hinges. If we took the moment about a point (say one of the hinges on the door) we may find that the moment vector does not line up with the axis of this hinge.
In that case, the component of the moment vector that lines up with the axis of the hinge will cause a rotation, while the component of the moment vector that does not line up with the axis of the hinge will cause reaction moments in the hinge. If we are only interested in the rotation of the door, we will want to find the moment that the force exerts specifically about the axis of the hinges.
- To find the moment of a force a about a specific axis, we find the moment that the force exerts about some point on that axis and then we find the component of the moment vector that lines up with the axis we are interested in.
To do this mathematically, we use the cross product to calculate the moment of the force about any point along the axis, and then we take the dot product of a unit vector along the axis and the moment vector we just calculated.
- \(M=u.(r\times F)\)
Principle of transmissibility
- states that the point of application of a force can be moved anywhere along its line of action without changing the external reaction forces on a rigid body.
Any force that has the same magnitude and direction, and which has a point of application somewhere along the same line of action will cause the same acceleration and will result in the same moment. Therefore, the points of application of forces may be moved along the line of action to simplify the analysis of rigid bodies.
- When analyzing the internal forces (stress) in a rigid body, the exact point of application does matter. This difference in stresses may also result in changes in geometry which will may in turn affect reaction forces. For this reason, the principle of transmissibility should only be used when examining external forces on bodies that are assumed to be rigid.
Equilibrium of a particle
\(\sum F = 0\), is a necessary and sufficient condition
Coplanar force systems (2D)
- Resolve into components, then:
\(\sum F_x = 0\) and \(\sum F_y=0\)
- Can solve at most 2 unknowns, which are usually angles and magnitude
3D force systems
- Resolve into components. If the geometry appears difficult, express the forces in the FBD in their cartesian form and equate the respective components of \(\hat \imath, \hat \jmath, \hat k\) to zero.