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Forces & Trigonometry for Physicists
SCALAR: quantities that have magnitude only e.g. Energy, Temperature, Speed
VECTORS : quantities that have a magnitude and direction e.g. Forces, Velocity.
This section involves forces – but the rules below work with any vector.
RULE 1:
Formal definition: Two vectors which are the same magnitude, same direction are considered as the same.
Translation:
We can slide vectors around to make a “chain” or consider them from a point (Whichever
makes our lives easier!!).
e.g. An object having a mass of 1 kg rests on a very slippery horizontal surface. Two cords are attached to it
and these pass over pulleys and have weights at their ends such that one cord pulls on the object with a 10N
force and the other with a 5.0N force. The angle between the cords is 40o and the object starts off an equal
distance from the two pulleys.
=
=
Note: The length of the lines drawn is proportional to the magnitude of the forces. The direction of the lines
drawn is the same as the direction of the forces.
Why do we want to slide around the forces? Wouldn’t it be nice if we could combine all these forces to make
just one!
Method 1: Putting in a chain.
By putting all the forces in a chain (each force drawn to scale), we can find the “resultant” (what one force is
the equivalent of all the forces acting) by drawing a line starting at the beginning of the first force and
finishing at the end of the last force. The forces in the chain can be in any order – it will not matter!
e.g. Below is an object with a number of forces acting on it.
=
i.e. All the forces on the object can be brought together to be the equivalent of one force in one direction!!
is the same as
WorkFGDT 
If the chain of forces ends up in a ring, then there is no resultant, no overall force and the object is in
equilibrium.
=
=
However, this relies on accurate drawing – and the resultant may be a decimal!
Method 2: Resolving Forces
If we split each force into 2 forces at right angles to each other – for instance, horizontal and vertical
components, we can just add up all the horizontal bits and all the vertical bits. Our chain will boil down to
two forces at right angle with each other. Those two with the resultant will make a right-angled triangle.
Using Trigonometry and Pythagoras, we can work out the angle that is made by the resultant from, say, the
horizontal and its magnitude (length of the vector!).
Part 1 – Splitting up the forces
Here’s the initial problem – what’s the resultant force and the angle which it is acting?
Let’s look at one force at a time ……
The 12N can be split up by trigonometry. (Use the notes from Trig – intro to
help)
H1 = 12 cos 30o = 10.39N
V1 = 12 sin 30o = 6N
The 15N can also be split up by trigonometry
H2 = 15 cos 20o = 14.10N
V2 = 15 sin 20o = 5.13N
Part 2 – Putting together the “horizontal” bits and the “vertical” bits.
WorkFGDT 
Now H1 and H2 are in the same direction – so a chain would be just adding them …. H1 + H2 = 24.49N
V1 and V2 are going in the opposite direction to each other – so one will partially cancel out the other. If we
count upwards as positive (remember – direction matters with vectors!!)
V1 + V2 = 6 + (- 5.13) = 0.87N
Part 3 – Making the resultant
Using Pythagoras
Resultant =
24.49 2  0.87 2 = 24.51N
Using Trigonometry
 0.87 
 = 2.03o above the
 24.49 
  tan 1 
horizontal
The same principle applies regardless of how many forces or whether it is done horizontally and vertically or
parallel and perpendicular to a slope! If you change the directions, just think of tilting everything a bit.
WorkFGDT 