Often times an object we are trying to analyze with FBDs sits atop a surface that is not parallel to the ground. This slight variation adds some noteworthy complexities that we will cover in this section.
Inclined planes can be used for mechanical advantage. It is considered a Simple Machine.
Read the BoxSand Introduction and watch the pre-lecture videos before doing the pre-lecture homework or attending class. If you have time, or would like more preparation, please read the OpenStax textbook and/or try the fundamental examples provided below.
Objects on a non-horizontal surface are considered to be on an inclined plane. If they slide up or down the incline their acceleration is not horizontal. Some forces acting on the object, with the normal force being an example, are off at an angle and not straight vertical (or horizontal). This presents a unique example of a system where rotating your coordinate system is beneficial to the analysis. Consider the block below and think about the direction of forces acting on it. Using a standard horizontal and vertical coordinate system (not pictured), you would have to break the tension and normal force into components. Furthermore, If the block was accelerating, without flying off the incline or falling through it, the block's acceleration would be along the incline, off at an angle from the horizontal. The acceleration would also have to be broken into components. To illustrate how cumbersome this choice of coordinate system makes the calculation, consider the case where the hanging mass is not large enough to hold the block and it is accelerating down the incline. Newton's 2nd law would look as follows with a standard positive x to the right and positive y upward. Note: the open resource diagram below doesn't use the letter F for all the forces like we encourage you do.
$\sum\overrightarrow{F}= m \overrightarrow{a} \rightarrow$ $\langle T\cos{\alpha}-N\cos{\alpha}, T\sin{\alpha}+N\sin{\alpha}-F_g\rangle = m \langle -a\cos{\alpha},-a\sin{\alpha} \rangle$
What if instead of using a horizontal and vertical coordinate system you were to define one where the positive x direction pointed parallel to the incline, as shown in the figure. Now the tension and the normal force, are 100% in the x direction. Movement along the incline will now be along the x direction and so will the any acceleration. The only thing then that needs to be broken into components is the force of gravity. Newton's 2nd law is reduced to the following.
$\sum \overrightarrow{F} = m \overrightarrow{a} \rightarrow$ $\langle F_g\sin{\alpha}-T, N-F_g\cos{\alpha}\rangle = m \langle a,0 \rangle$
The algebra and arithmetic will be much easier using a rotated coordinate system. The biggest issue it to figure out what the new x and y components are for a rotated system. Watch my lecture video for a technique to get the components correct in a rotated coordinate system. Once you've done it a few times, I believe you'll agree the work to learn how to rotate the coordinate system is worth the savings in algebra headaches of not rotating your coordinate system. If you're still not convinced, imagine having to then do kinematics problems with the acceleration you have just calculated doing a force analysis. With a standard coordinate system it is a 2D problem and you'll have a set of kinematic equations for both the x and y directions. With a rotated coordinate system, assuming the object doesn't leave the incline, the velocity, acceleration, and change in position are zero in the direction perpendicular to the plane, making it a 1D problem.
Here's a short video to visualize the components of gravitational force (they use the letter P), and the normal force (N).
inclined plane
Now, take a look at the pre-lecture reading and videos below.
