Physics 101ah- Lesson 2

8/3/2019 Physics 101ah- Lesson 2

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PHYSICS 101AH: LESSON 2

SCALARS AND VECTORS

2.1. General Categories of Physical Quantities

Scalar quantities are those quantities which are completely specified by their magnitude,

expressed in some convenient units.

easy to handle since they can be manipulated by ordinary laws of algebra

(Usually can be added, subtracted, multiplied, divided directly)

Examples: length, mass , area, volume, time, density

Vector quantities are those which require for complete specification, both magnitude and

direction.

Direction is just as important as the magnitude when specifying quantities

It always represented by an arrow.

Examples: displacement, force, acceleration, velocity, momentum

The sum of the vectors is called the RESULTANT vector.

2.2. Addition of Vectors

Methods in Adding Vectors

A. Graphical Method also known as Geometrical Method and require no computation.

a. Parallelogram Method (Tail to Tail Method) used to add only two vectors

b. Triangle Method (Head to Tail Method) use to add only two vectors

c. Polygon Method (Head to Tail Method) use to add more than 2 vectors

An illustration will help for Head to Tail method: Note that we can

move vectors around as we wish on the graph paper, because only the

magnitude and direction matter. The location does not. Also note that

this is the way we would add together displacements if I say "goalong vectorA and then along B", then in the end we would end up in

the same place as if we had just gone along vectorR, the resultant. In

that sense,R is thesum ofA andB. We take this as a general definitionof adding any two vectors together, whether they are displacements,

velocities, or some other quantities.

To do this accurately, we need to use a ruler and protractor. The general technique foradding two vectors on graph paper is as follows:

1. Start at the origin and draw the first vector (based on the numbers given you)

2. At the end of the first vector, make a new origin

3. Draw the second vector starting at the new origin4. Connect the tail of the first vector to the head of the second

When you are finished, the last line you have drawn is the resultant. You can then

measure the length and angle of this vector using a protractor and ruler.


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B. Analytical Method (Component Method/Trigonometric Method)

Trigonometryis also important in physics. When you have a right-angled triangle, the

following relationships are true:

This method is based on the fact that we can specify a vector by specifying its magnitudein two perpendicular directions. We take these to be the x andy directions. We call the length of

the vector in the x direction as thex-componentof the vector, and similarly fory. The nice thing

about this method is that once we have the x and y components of the vectors we want to add,

adding them is simple. Let us say that we have two vectors, A and B, with the components

labeled asAx,Ay,Bx, andBy. It should be obvious which component each symbol stands for. If our

resultant, or sum vector, is called R, with componentsRx andRy, then we have

Rx =Ax +Bx Ry =Ay +By

But how do we get the components if we are given the angle and direction? We use

trigonometry. Consider the diagram:

We know from trigonometry that

Ax =A cos

Ay =A sin

Similar relations hold for the components ofB, or any other

vector for that matter. HereA is the magnitude ofA, and is

the angle.

We now need to be able to go back from the

components to the magnitude and direction. We have, again from trigonometry,

R2 =Rx2 +Ry2 = tan-1 (Ry /Rx)

The process for adding two vectors, A and B,is thus:

1. Find the components ofA andB using (2a) and (2b)2. Add the components together using (1a) and (1b)

3. Find the magnitude and direction ofR using (3a) and (3b)


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Note: Calculators are funny things. You should always make a sketch of your addition to

see if your values agree with what your calculator gives. The magnitudes should be correct, but

the angles will often come out wrong. To adjust the angles, note that

cos = cos (360 - )

sin = sin (180 - )

tan = tan (180+ )

The summarized steps for component method are as follows:

1. Resolve all the given vectors into their x and y components.

2. Find the algebraic sum of all the x-components (X), and the algebraic sum of all the

y-components (Y).

3. Find the magnitude of the resultant using the Pythagorean theorem.

R = sqrt ((X)2 + (Y)2)

4. determine the angle of the resultant and the specific direction using the tangent

function.

= arctan (Y / X)

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Physics 101ah- Lesson 2