Now that we are getting into the statics portion of the guide, we will start off by looking at how we can leave particles in equilibrium in 2D and 3D scenarios. We will heavily use Newton's 2nd Law of Motion in order to analyze these systems.
Remember that
[ 1 ]
$$ \Sigma \vec{F} = m \vec{a} $$
which means that $ \Sigma \vec{F} $, or the sum of all forces on a body, will equal the mass of the body times its acceleration vector. Now for 2D spaces, we will have to make sure the forces in the x-direction and the y-direction are in equilibrium, so we will have to write
[ 2 ]
$$ \Sigma F_x = m a_x = 0 $$ $$ \Sigma F_y = m a_y = 0 $$
Notice how here, we are working with the magnitudes $ F_x $, $ a_x $ and $ F_y $, $ a_y $ because we are working with the magnitudes, or lengths, of those values in that specific dimension rather than working with the x and y magnitudes all at once.
Now say we are in a 3D space, then in order to have a static system, we would have to determine
[ 3 ]
$$ \Sigma F_x = m a_x = 0 $$ $$ \Sigma F_y = m a_y = 0 $$ $$ \Sigma F_z = m a_z = 0 $$
This then confirms that in all 3 dimensions, the body is balanced!