Working with Newton’s Second Law isn’t difficult, but you do need to be careful with directions and their signs.

Let’s look at a situation in which two forces are acting on an object. We will choose the positive direction to follow the usual mathematical convention of pointing to the right. The first force \(\color{black}{\vec{F}_1}\) points left (in the negative direction). The second force \(\color{black}{\vec{F}_2}\) is a little smaller and points to the right (the positive direction).

Here’s a quick video showing how to solve this for the acceleration.  A written description follows.

Two forces applied to an objectDraw the Free Body Diagram

First, we should draw a free body diagram of these two forces, which will look something like this…

Write and Modify the Vector Equation

Once this is done, we write down Newton’s Second Law in vector form as our starting equation.

\(\color{black}{m \vec{a} = \Sigma \vec{F}}\)


Notice that I wrote it down with the \(\color{black}{m \vec{a}}\) first. This helps keep my equal signs lined up on the page, since we won’t do much with this half of the equation.

Now we write down the forces that make up the left hand side of the equation.

\(\color{black}{m \vec{a} = \vec{F}_1 + \vec{F}_2}\)

Notice that I have left the equation as an addition. We still have a vector equation here.

Convert to an Algebra Equation by Including Signs

Next, we’ll put the positives and negatives, creating an algebra equation.

\(\color{black}{m a = {}^-F_1 + {}^+F_2}\)

I wrote the direction signs a little higher than normal just so we don’t lose them.

Solve for the Unknown Variable

Now we are in a position to solve for the unknown variable.  In basic examples, you would have been given four of the variables in this equation.  In more advanced problems, you might need to do something like use the kinematics equations to find the acceleration.