College Physics Supplementary Handout 1: Review Vectors
and Coulomb Force
Adding Vectors:
Vectors, as oppose to such scalar quantities as time, mass, charge, etc. that do not have direction but have just magnitude, are quantities that have direction as well as magnitude (size, strength). When adding or subtracting and multiplying or dividing vectors, we use the scalar algebra for the components of the vector. Another words, adding two vectors is not like adding two scalars. Here is an example that contrasts adding two quantities .
Adding vectors
(Geometric method):
To add two vectors graphically, for example, _{}, bring B’s tail to A head without changing the direction of B. Draw an arrow from the tail of A to the tip (or head) of B. This is Head to Tail or toe method! You cam also use parallelogram method to obtaion the same result. Both methods are shown below.
Adding Vectors (algebraic Methods):
Add two distances: A movement of 3 mi north than 4 miles east. The resultant total distance (adding two distances) is 7 mi. Here distance is a scalar. Therefore, we used the scalar (one that you learned in High School) algebra to add them |
Add two vectors: A displacement of 3 mi north, then a second displacement of 4 mi east. The resultant displacement is not 7 mi! Instead, we must consider the directions as well the magnitudes. |
In the example above left, the resultant vector C is obtained by adding two vectors A and B. The arrows above each symbol indicate that these quantities have directions assigned to them. Note that the resultant is in the general direction of NE and 5 miles (_{}). When we need to decompose a vector into its components that is into two vectors mutually perpendicular to each other we need to undo what we did in the above example. To do this we need a little trigonometry, namely sine, cosine, and cotangent.
Here are the definitions of sine and cosine.
We can apply this to a vector. Suppose we have a force of 20 N in the direction of 30o north of east. Let’s decompose this vector into its Cartesian components (an x direction component and a y direction component).
_{}
Therefore, a 20 N force is equivalent to a 17.3 N force in the north direction (y-direction, and a 10 N force in the east direction (x-direction).
What if the force direction were opposite, that is SW? Let’s work out this case, keeping in mind that vector components in the south( negative y direction) and west (negative x –direction) should be fitted with a negative sign in front after using the process above.
Note that the calculations are the same,
except the signs,
_{}
In mathematical language this is written as
_{}
where the sing above x and y are hats,
_{}denote x and y directions.
When we add several vectors, these signs
will be important (see the upcoming example).
An application of Coulomb’s law:
Three charges 1 C, 2 C and -3 C are situated at the corners of a right triangle as shown. Find the net force on 1 C charge.
We first need to compute the magnitude of the forces F_{2}, and F_{-3}. Distance needed for F_{-3} can be obtained from definition of sine,
opposite side = 10xSin (30) =10 (0.5) = 5 m
_{}
The total force is not 1.8x10^{+8}+10.8x10^{+8} N!!!Because these forces are not in the same direction. We will find the total force in class! See your notes.