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2 Statics of Particles

Statics is a quantitative description of forces acting on a structure in equilibrium. Hence, we need a mathematical description of force – action of one body on another, a vector quantity.

Mass, length, volume are scalar quantities and have only magnitude, while force is vector quantity and has magnitude and direction.

Example: Ball hit by a bat.

Types of vectors reminder:

Free vector is not attached to any point or a line in a body.

Sliding vector has unique line of application.

Fixed vector has unique point of application (and, thereof, line of application).

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In statics forces are sliding vectors. Hence, they have unique magnitude, direction, and line of action. However, we can slide the force along its line of action.

Resultant of Forces

As vector quantities, forces are added as vectors and the sum is called resultant.

Note: Graphical method doesn’t provide a line of action for the resultant, unless the forces are applied at the same point. To find line of action of resultant, draw resultant applied at a point of intersection of lines of actions for F1 and F2 (3D?)

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Particle Equilibrium (2D)

For the particle to be in equilibrium, resultant of all forces acting on it should be equal to zero. In mathematical representation this condition is called an equilibrium equation.

Equilibrium equation (EE) in vector form:

∑ F=0

For 2D this equation can be written as two scalar equations to be satisfied simultaneously:

∑ F=(∑ F x ) i+(∑ F y) j=0

∑ F x=0 ,∑ F y=0

Force components can be found using vector algebra.

It is critical to account all forces. To do so a Free Body Diagram is drawn.

Free Body Diagram

Diagram of a particle, with all forces applied to it, is a Free Body Diagram (FBD). There are three steps in drawing FBD:

1. Imagine and draw the particle isolated from its surroundings.

2. Indicate all forces acting on the particle. Include active forces, that would otherwise set the particle in motion, and reactive forces, that are caused by constraints and

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supports, so the particle stays in equilibrium. Encircling the particle might help with accounting for all forces.

3. Mark known forces with their directions and magnitudes, use letters for unknown quantities. If line of action of a force is known, then a direction is assumed. However, if during solving magnitude is found to be negative, the direction of the force will be opposite to the one assumed.

Analysis of Equilibrium

1. Draw FBD to include all acting forces.2. Apply equations of equilibrium.In case of 2D there are two equations per particle. Hence, to solve them, there should be maximum 2 unknowns per particle.Possible unknowns:

Force vectors (two components, or magnitude and angle of direction)

Magnitudes of two forces with known directions Orientations of two forces with known magnitudes Other parameters (geometric, etc.) – 2 max.

Elimination Technique

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To eliminate an unknown force from equation(s) of equilibrium, one should consider equilibrium of forces in the direction perpendicular to the unknown force.

Repeat it for the second unknown force.

In 2D this technique breaks down a system of equations of equilibrium into a set of independent equations. This greatly simplifies the computational complexity and is less likely to produce a computational error.

Note: Sometimes it is beneficial to eliminate a known force.

Example 1:

Notes:

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Note 1: Often, it is convenient to eliminate even known force to reduce the amount of computations (see step ii).

Note 2: Axes in the force EE’s should not necessarily be perpendicular. Any two non-parallel axes will do.

Note 3: The advantage of ET would be especially spectacular if we were asked to evaluate just one of two unknown tensions. By straightforward method we’d still have to solve for the system of 2 simultaneous equations. By ET we’d get it from a single equation (i) or (ii).

Example 2:

Example 3:

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Particle in Equilibrium Example 1:

Example 2:

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Example 3:

Example 4:

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Example 5:

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Example 6: