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Resolving Forces On A Particle


Introduction


A particle is an object of zero mass and size. To simplify some problems we can use the assumption that an object being acted upon by forces is a particle, this way we don't have to worry about things such as moments.

Resolving forces on a particle is quite simple and will help you learn some basic principles of static mechanics especially the equations of static equilibrium.

Firstly, it is important to understand this:
If the particle is not undergoing any acceleration, It is said to be in static equilibrium and all the forces applied to it must be equal to 0 when added together.

Particle
Particle


So if a particle in static equilibrium has a 10N force pulling it up, it can be counteracted with a 10N force pulling it down as shown in the image to the left (if F = 10N), simple really. It does get more complicated when there is more than one force acting on the particle and they are acting in all three dimensions, however it is always a case of basic vector addition and subtraction to work out what force is required for the particle to be in static equilibrium.




We can just use logic to determine that the force required to balance a single force is an equal and opposite force, and this can be used again and again and again in more complex problems but it isn't the best way. For example if you had 5 forces on one particle that are unbalanced and need the particle to be in static equilibrium, then you could add 5 different balancing forces, one for each force that is initially applied to the particle, or you could add one force. Generally it's much easier to look at a diagram with 6 arrows than one with 10.

How is The Resultant Calculated


The general situation for these types of questions is:
We have a particle with forces applied to it in three dimensions, the forces are all represented by vectors, we generally know all the forces except for one, we do not know it's direction or magnitude and this is what we must calculate. We only require one equation:
[math]\Sigma F = 0[/math] (Equation of Static Equilibrium)

so you take [math]F_1 + F_2 + ... + F_n + F_U = 0[/math] where [math]F_U[/math] is the unknown force. Each force in this equation is a two dimensional Vector, so you need to understand vector addition before you can complete this. Either that or you can convert all the vectors into their X and Y components and use the equations:
[math]\Sigma F_x = 0[/math] and [math]\Sigma F_y = 0[/math]
Then determine the X and Y components of the unknown force vector.

Here is an example: Forces on A Particle