Explain the forces acting, including friction, when an object is stationary or moving on an inclined plane. What coordinate system do we use?

2 Answers
May 17, 2017

enter image source here

Three forces act:

F_"//" = F_w sin \theta = mg sin \theta

F_"|--" = F_w cos \theta = mg cos \theta

F_"frict" = \mu F_"|--"

Explanation:

The forces acting are a normal force at right angles to the plane, an accelerating force parallel to the plane and 'downhill', and a frictional force parallel to the plane and 'uphill'. The fundamental cause of all these forces is the weight force on the object, acting directly downward toward the centre of the earth.

The gravitational force is the key to the whole situation. It is given by F_w=mg where g=9.8 or 10 Nkg^-1 (the unit used is often ms^-2, which is equivalent to Nkg^-1, but I find the latter unit more direct and explanatory).

Using the image above, note that the slope is at an angle \theta to the ground, and that, by similar triangles, the angle between the weight force F_w and the normal force, F_"|--", is also \theta.

The constructed triangle is a right-angled triangle, so we can use trigonometry. F_w is the hypotenuse, F_"|--" is the adjacent side (to the angle \theta and F_"//" is the opposite side.

Using the definitions sin \theta = (opp)/(hyp) and cos \theta = (adj)/(hyp), and rearranging, gives:

F_"//" = F_w sin \theta = mg sin \theta

F_"|--" = F_w cos \theta = mg cos \theta

Finally, the frictional force will act in the opposite direction to F_"//" and will have a magnitude given by F_"frict" = \mu F_"|--" =\mumgcos\theta, where mu is the friction coefficient.

By rotating the axes to be aligned with the plane.

Explanation:

You take the axes and align them with the plane, so that the coordinates are 'parallel to the plane and perpendicular to the plane', rather than 'horizontal and vertical':

https://www.ilephysique.net/sujet-mouvement-sur-un-plan-incline-246708.htmlhttps://www.ilephysique.net/sujet-mouvement-sur-un-plan-incline-246708.html
You draw the forces involved:
https://fr.wikipedia.org/wiki/Plan_inclinéhttps://fr.wikipedia.org/wiki/Plan_incliné
If your object is not moving use Newton's first law:
sumF=0
If your object is moving use Newton's second law:
F=ma