These notes are based on the requirements of the M1 A Level mathematics module.
This is quite a large chapter and possibly the most challenging the book has to offer, but if you heed the key principles I am going to list here, you will be fine. One basic overseeing principle is, ALWAYS use EXACT VALUES whenever possible. For instance use that button on your calculator that gives you exact fractions, examiners like these. Yet, if this is impossible don’t worry.
Much like in the vectors chapter, to resolve a force, you can join the given vector lines, or force lines: nose to tail. In this way the vector lines can be followed round using the arrows, and the resultant force will follow suit and can be drawn in as connecting the first and last vector lines. However, this does not come up in exams (its more a physics principle, drawing vector polygons and triangles).
In this chapter when you have forces acting on a particle, you simply draw yourself a small diagram and label all angles and forces you can, with respect to the horizontal and vertical lines that you just imagine are there. Using these angles and trigonometry (sin and cos), you can form equations.
When resolving the key idea is that you can calculate the resultant force along the horizontal and then the vertical, and you can then use Pythagorean theorem to calculate the magnitude of the resultant force. Using a small diagram you can then use trigonometry to calculate the angle between the resultant and either the horizontal, or vertical (for bearings).
Equilibrium of coplanar forces
In this case, you resolve horizontally and vertically, as was discussed in the last section. However, weight may be introduced in this case if the object is hanging. This becomes part of the vertical resolution. In the case of equilibrium the forces in the horizontal and vertical planes are equal exactly. Therefore you can form an equation for the horizontal and vertical equilibria. These two equations will allow you to calculate certain unknowns, such as angles o r magnitudes of certain forces.
When the forces lie on a plane, you should remember to check whether or not the plane is ROUGH. This will affect the question heavily, as you will need to consider friction and its direction based on which way the object is moving. When considering forces on a plane you must remember the following rhyme, despite its silliness:
Weight is SIN all the TIME, Friction is COS just beCOS.
This is so you know which ratio to use in conjunction with the angle on the inclined plane when resolving your forces. Weight always acts down the plane, but friction does not always act in the same direction, as is discussed now.
There are key things to remember about friction. As my maths teacher says:
- “Friction is LAZY and AWKWARD, like you teenagers.”
The reason the man says this is this: Friction IS LAZY because it only acts to oppose a force when something is moving, when something is still, it doesn’t act on it. Friction IS AWKWARD because it acts to oppose motion and NEVER works with the direction of motion.
You must remember how to calculate friction:
- F = Frictional force,
- f = the coefficient of friction (represented by ),
- R = Reactionary force on the object.
- m = mass of object,
- g = gravity and
- x = angle of inclination of the plane.
When an object is said to be in limiting equilibrium, the friction is just enough to prevent it from moving, which means you can resolve along the plane to find the value of the frictional coefficient (a common demand) and other unknowns.
Originally written by RobbieC on TSR forums.