Chapter One: Units, Vectors, and Trigonometry

Part 1: Units

To learn the units used in physics, let us develop a chart.  If you participate in its development, you will learn it.  First draw a chart like the one shown below on a piece of paper.  You should actually draw the chart and write its content if you really want to learn, but do not write the numbers 1 through 23 as are shown.  As we proceed, you will be asked what to write down for each number.   Do not copy and paste.  Also, do not look at the next chart yet.

                  Unit

System

   Fundamental Units

 Area

 A=L2

 Volume

 V=L3   

 Velocity

9

Acceleration 

14

Force

19

Mass Length Time

Metric Consistent (SI)

 

   1 kg

  1m

 1s

1

5

10

15

20

Metric Gravitational

 

    1 kg

  1m

 1s

2

6

11

16

21

American Consistent

1 slug

= 32 lbm

  1 ft

 1s

3

7

12

17

22

 

American     Gravitational

 

   1 lbm= (1/32)slug

  1ft

1s

4

8

13

18

23

Continue, if you have used paper and pencil, and have already drawn the above chart except for the numbers.

Answer the following questions:

a)  What is the unit of mass in Metric Gravitational system?   To check your answer, click here.

b) What is the unit of mass in the American Consistent system?

c) What is the unit of mass in the American Gravitational system?   To check your answer, click here.

d) What is the unit of mass in the Metric Consistent system? 

e) What is the unit of length in the Metric systems?   To check your answer, click here.

f) What is the unit of time in all systems? 

Area

g) If area is (Length times Width) or in fact ( L times L = L2),  what is the unit of area in Metric Consistent System?  Note that width itself has the dimension of length (L).  To check your answer, click here.  Write down the correct answer in the chart where it belongs.

h) What are the units of area for positions 2,3, and 4 in the chart?  To check your answer, click here.

Volume (V)

i ) If volume (V) is the product (Length x Width x Height) or in fact (L· L· L = L3), what is the correct unit for volume in Metric Cons. System?  To check your answer, click here.

j ) What are the units of volume for positions 6,7, and 8?  To check your answer, click here.

Speed (v)

In Physics, speed (v) and velocity are two different quantities.  When we use the term velocity, the direction of motion must be mentioned in addition to how fast (speed) the object is moving.  The difference will become clear to you soon.   How do you define speed?  Think for a while before you continue.  If car A travels a certain distance in 1 hr, and car B travels the same distance in 1/2 hr, which one has a higher speed? Ans. = ..........?  If you were to come up with a formula for speed, what variables would you use to write a mathematical expression for it?  If your answer is "distance and time", it is correct.  In what way should distance (x) be mathematically combined with time (t) to make a formula for velocity?  Think!  You know that the shorter the travel time of car B is, the higher its speed.  Let's repeat, "the shorter the travel time is, the higher the speed."  What is this telling us?  In a mathematical formula, where should the variable (t) be placed so that when it is smaller, it makes the speed (v) greater?  In the numerator of a fraction or its denominator?  Ans.:........ ?  If your answer is "the denominator," it is correct.

(k) What formula for speed do you suggest?  Fill item 9 in your chart.  To check your answer, click here. If your answer is "distance over time," it is correct. If distance is (x) and time is (t), speed becomes (v=x/t).

Now, if length is expressed in meters (m) and time in seconds (s) as they are in Metric systems, answer the following:

l ) What are the units of speed in both Metric systems?   For answer, click here.  Fill 10 and 11 in the chart.

m) What are the units of speed in both American systems?  For answer, click here.  Fill 12 and 13 in the chart.

In math and physics, change is denoted by Δ, pronounced as "Delta".  For example, Δx means a change in distance and Δt means a change in time.  Based on the above discussion on the definition of speed, we may write:

v = x / t.    More precisely,    v = Δx / Δt.    Def.:   Speed is the change in distance per unit of time.

Example 1: Car A travels a distance of 450 miles in 7.5hrs.  Car B travels the same distance in 9.0hrs.  Find the average speed of each car.  Make sure you solve it completely on your note book. When writing it down, make sure you use horizontal fraction bars and not slashes. 

Solution: vA = 450mi / 7.5hr = 60 mi/hr.        vB = 450 mi / 9.0hr = 50. mi/hr.

Acceleration (a)

Let's think for a while and come up with a definition for acceleration.  What does change when you accelerate your car?  In other words, what variables does acceleration depend on?  Ans. :.........  If your answer is "speed (v) and time (t)," it is correct.  Again, if the change in speed occurs in a shorter period of time, wouldn't it result in a greater acceleration?  If you go once from 50mi/h speed to 70mi/h speed, in 10 seconds, and once in 5 seconds, which one does give you a greater acceleration?  Ans. :.........  If your answer is " in 5 seconds," it is correct.

Now you know that acceleration depends on speed (v) and time (t).  How would you combine these two variables mathematically to reflect the fact that a shorter time of speed-change results in a greater acceleration?  Ans. :.......  To check your answer, (mn) click here.  This can be mathematically written as

a = v / t.   More precisely,    a = Δv / Δt.    Def.:   Acceleration is the change in speed per unit of time.

Note: Δ =  (a final value) - ( its initial value ),  for example,  Δv = vf - vi  and  Δt = tf - ti .

Acceleration equation takes the form:  a = (vf - vi) / (tf - ti).  Rewrite this but with a horizontal fraction bar.

Example 2: A car changes its speed from 15m/s to 24m/s in 6.0 seconds.  Find its acceleration during this period.  Make sure when solving (writing) it in your notebook to use horizontal fraction bars (not slashes).

Solution: a = (vf - vi) / (tf - ti) = (24m/s - 15m/s) / 6.0s  = 1.5 m/s2.

Note that the unit of acceleration is m/s2 that means meter per second per second.  Dimensionally, speed is L / T and when L / T is further divided by another T, it results in L / T2.  In fact, it is better to write L / T in box #9 rather than x / t.  Answer the following questions now:

n) What is the unit of acceleration in the Metric Consistent system (item 15)?   For answer, click here.

o) What is the unit of acceleration in the American Consistent system (item 17)?   For answer, click here.

To figure out One Unit of Acceleration in the Metric Gravitational system, one may drop a tiny and solid object ant let it fall freely under the influence of gravity.  In doing this, if a vacuum chamber is used and different objects are dropped to fall freely in vacuum, it will be observed that they all fall together and arrive at the ground at the same time.  In other words, gravity gives the same acceleration to all objects in the vicinity of the Earth regardless of their shapes and masses.  Many have measured the acceleration of a freely falling object and they all have obtained the same value of 9.8m/s2.  This acceleration has been assigned the symbol "g"; therefore, g = 9.8 m/s2 is called the "acceleration of gravity."

p) Is it correct to write (1g = 9.8 m/s2) for item 16?  Ans.:........  If your answer is "yes," it is correct.

q) Knowing that (1m = 3.28 ft), calculate the value of "g" in ft /s2, and write the result in (18). To check your answer, click here.

 

Force

What is force?  How do you define force?  Stop reading for a few minutes and think for a while.   You shouldn't be reading this now, you are supposed to think about the definition of force.

Force may be defined as the cause of motion and deformation.  If the force (a push or a pull) applied to an object is great enough, it puts the object into motion; otherwise, it causes some deformation, even if we cannot detect it with naked eyes.  If you put a concrete block on a horizontal and solid table, the Earth's gravity pulls the block downward.  This causes the block to exert a force onto the table straight down.  If the table is strong enough to hold the block stationary, it must then be pushing the block upward with an equal force.  These two forces form a pair of action and reaction.  Later on we will learn that: there is a reaction for every action, equal in magnitude but opposite in direction.  These two forces do not cause motion, but they cause deformation that in most cases a naked eye cannot detect.  This paragraph was used to support the definition of force.

To define a unit for force, let's first figure out what the quantity force depends on or is related to?  It is easy to see that when a force is applied to a less massive object, it accelerates that object better.   Also, it takes a greater force to give a certain mass a greater acceleration.  It all boils down to the fact that force depends on mass and acceleration.

On one hand, experiments have shown that in order to give masses: M, 2M, 3M, ... the same acceleration (a), forces of magnitudes: F, 2F, 3F, ... are needed.  This shows the proportionality of force (F) and mass (M).  Using the proportionality symbol, ~ , we may write:  F ~ M.  (1)

On the other hand, other series of experiments have shown that in order to generate accelerations of magnitudes: a, 2a, 3a, ... in the same mass (M), forces of magnitudes: F, 2F, 3F,  .... are needed.  This shows the proportionality of force (F) and acceleration (a).  We may write:  F ~ a.  (2)

Comparing (1) and (2) above, one may conclude that F ~ Ma.  (3)

Is it really correct to conclude (3) from (1) and (2)?  Ans.:........  If your answer is "Yes", you are right.  As an example, consider the area of a rectangular room.  If you keep the length the same while doubling, tripling, or quadrupling the width, the area (A) also doubles, triples, or quadruples.  This shows the proportionality of area (A) and width (w).  Now if you keep the width the same while doubling, tripling, or quadrupling the length, the area (A) also doubles, triples, or quadruples.  This shows the proportionality of area (A) and length ( L ).  As we see, area (A) is a quantity that is proportional to two other quantities: w and L, and as we know A ~ wL.  With proper choice of units, we even write A = wL.

The same is true for F ~ Ma.   With proper choice of units, it is correct to write **F = Ma **.  This equation is called the "Newton's 2nd Law" of motion and expresses that force is equal to mass times acceleration.

In Metric Consistent System, M is in (kg) and a is in (m/s2).  The product Ma is therefore in (kgm/s2) called Newton in the honor of Isaac Newton.

Example 3:  In the figure shown, a 4.0-kg block is being pushed to the right on a horizontal frictionless surface at an acceleration of 3.5 m/s2.  Calculate the magnitude of the horizontal force applied to it.

   

Solution:  F = Ma;     F = (4.0kg)(3.5m/s2) = 14 kgm/s2  = 14 N.

 

Example 4:  In the figure shown, calculate and show all forces acting on the 5.0-kg stationary block placed on a horizontal surface.

      

Solution: No force acts on the block in the horizontal direction.  Two forces act on the block in the y-direction. Gravity pulls the block down with force w =Mg.  The acceleration of gravity (g = 9.8 m/s2) is imposed on all objects near the earth.  It causes a downward force that is called the "weight" of the object w =Mg.   The horizontal surface reacts to w and pushes the block up with an equal and opposite force called the "Normal" force, N.    In Mathematics and physics, "normal" means "perpendicular."

  w = Mg   ;   w = (5.0kg)(9.8 m/s2) = 49N    ; therefore, the magnitude of N is also 49N.

 

Example 5:  In the figure shown, the 10.0-kg block is being pushed to the left at an acceleration of 4.0m/s2.  Calculate and show all forces acting on the block.  Friction is negligible.

Solution:  In the vertical direction, gravity pulls the block down by the weight force, w =Mg = 98N, and the surface pushes it up with an equal and opposite force of  N  = 98N as well.

In the horizontal direction, the applied force  F = Ma  acts to the left.     F = (10.kg)( 4.0m/s2) = 40.N  (to the left)

Note: this answer is good only if there is no friction, like pushing a 10-kg chunk of ice on a flat and horizontal sheet of glass.  F = 40N is not enough to do the same if friction is present.

Now, answer the following questions:

r) What formula should be written for item 19?  To check your answer, click here.

s) What is the unit of force in Metric Consistent system?  To check your answer, click here.

In Metric Gravitational system, the unit of mass is (kg) and one unit of acceleration is g =9.8 m/s2.  How much is one unit of force in this system?  Ans.:........  If your answer is 9.8kgm/s2, it is correct.  Do you know that this used to be called " One kilogram force?"  The abbreviation is 1kgf.

t) What is the unit of force in Metric Gravitational system?  To check your answer, click here.

In American Consistent system, the unit of mass is 1slug and that of acceleration is 1ft/s2What is the unit of force in this system?  Ans.:.........  If your answer is 1slug ft/s2, you are correct.  Do you know that it is called  " pound force?"   It is abbreviated as lbf.  Write this in your chart.

In American Gravitational system, the unit of mass is lbm (pound mass).   lbm = (1/32)slug.   1 unit of acceleration is 1g = 32 ft/s2.  What should therefore the unit of force be in this system?   Ans.:........  If your answer is slug ft/s2, you are correct.

u) What is interesting about the units of force in the American systems?  Ans.:......  If your answer is " The unit of force is the same ( lbf ) in both systems, you are right.  Fill items 19 through 23.

Your completed chart should look like the following where the area and volume columns are not shown:

       Units

    System

      Mass

M

Length

    L

Time

 T

Velocity

v = L / T

Acceleration

a = v / t

 =  L / T

Force

  F = M a

=  M L / T2

Metric

Consistent

(SI)

1 kg

1 m

1 s

1 m/s

1 m/s2

1kgm/s2 = 1N

1 Newton

Metric

Gravitational

1 kg

1 m

1 s

1 m/s

9.8 m/s2=1g

9.8kgm/s2=

1 kgf = 9.8N

American Consistent

1slug= 32 lbm

1 ft

1 s

1 ft/s

1 ft/s2

1 slug ft/s2=

1 lbf

American

Gravitational

1lbm=

(1/32)slug

1 ft

1 s

1 ft/s

32 ft/s2=1g

(1/32slug)*32ft/s2

=1 lbf

Repeat the process from the beginning until you feel confident that you know the chart.

Homework: Do the assigned problems of Chapter 1 under units.

Test Yourself #1:

1) A car travels 480km in 8.0 hours.  Its average speed is (a) 40 km/h  (b) 60 km/h   (c) 80 km/h.    click here.

2) What distance does a car travel in 30 seconds if its speed is 20 m/s?  (a) 600 m   (b) 600km   (c) 0.67m.

3) One meter (1m) is (a) 3ft    (b) 4ft   (c) 3.28 ft.    click here.

4) How long does it take for a train moving at 65 mi/h to travel 13 miles?   (a) 5.0h   (b) 0.20h  (c) 14min.

5) When a car changes its speed from 5m/s to 12m/s in 3.5s, its acceleration is (a) 2.0m/s  (b) 20m/s2  (c) 2.0m/s2.

6) A train moving at 12m/s comes to stop in 96s.  Its acceleration is (a) 0.125 m/s2  (b) -0.125m/s2  (c) -8.0m/s2.

7) Average speed is (a) the change in distance times time (b) the change in distance over elapsed time.    click here.

8) Average acceleration is the change in (a) speed per unit of time (b) speed times time  (c) distance over speed.

9) Force is (a) the same thing as mass times acceleration  (b) the cause of motion (c) both a and b.

10) The net force acting on a 1200kg car accelerating it at 3.0m/s2 is (a) 400N (b) 300N (c) 3600N.

11) A net force of 150N can accelerate a 30.0-kg object at (a) 4500m/s2  (b) 5.0m/s2   (c) 180m/s2.    click here.

12) Gravity pulls a 10-kg object down toward the Earth's center by a force of (a) 98N  (b) 9.8N  (c) 10N.

13) The weight of an object is (a) its mass times g  (b) Mg  (c) the force of gravity on it (d) a, b, and c.

14) The mass of an object is (a) the amount of matter it is made of (b) the same as its weight (d) changes from planet to planet.

15) The weight of an object (a) changes from planet to planet (b) is the same everywhere in the Universe (c) is mass over gravity.

16) A 5.0kg object weighs (a) 98N  (b) 49N  (c)15N.    click here.

17) The word "normal" in mathematics and physics means (a) not abnormal  (b) perpendicular  (c) parallel.

18) A 10-kg object is placed on a horizontal table.  The normal force from the table on the object is (a)10kg  (b) 98N  (c) 0.

19) A 10-kg object is on an incline.  The normal force from the incline on the object is (a) 98N  (b) more then 98N  (c) less than 98N.

20) The normal force on an object that is under the gravitational pull only, is maximum when the object is placed (a) on a horizontal surface  (b) on an incline  (c) is hanging but slightly in touch with a vertical wall.    click here.

Part 2: Vectors

A scalar quantity requires a magnitude only.  Examples are: mass, volume, and temperature.

A vector quantity requires a magnitude as well as a direction.  Examples are: velocity, acceleration, and force.

Arrows are used to show vectors.  The length of the arrow represents the magnitude and the angle it makes with a reference axis counts as its direction.  The reference axis is usually chosen to be the positive x-axis.

 

Example 6:  Draw the following vectors: A = (5.0N, 30.0),   B = (10.0N, 70.0),  C = (8.0N, 230.0), and  D = (8.0N, -130.0).

Solution: Note that counterclockwise (CCW) is positive for angles and clockwise (CW) is negative. On a set of x-y coordinates system, select the 30.0, 70.0, 230.0, and -130.0 angles as shown and using an appropriate scale (1cm = 1N, for example) mark appropriate lengths on the corresponding lines.  Another point to consider is that bold letters are used to show vectors and regular letters to show magnitudes.  Finally, in this example, vectors C and D have the same direction and magnitude and are therefore equal.  +230.0 and -130.0 specify the same direction.

 

Make sure you practice this example by using a ruler and a protractor.  Without actually doing it completely on paper, your study is not effective.

Equal Vectors

Equal vectors have the same magnitude and the same direction.  Equal vectors are necessarily parallel because of having the same direction.

 

 Example 7:  From Point P(3,2) draw vector B equal to A = (5.0m/s, 120) that is drawn from the origin.

 Solution: First draw a tiny x-axis from P.  Then select a 120 angle from P by a line segment as shown, and then select a length that represents 5.0m/s based on the scale used.

 

The Negative of a Vector

The negative of a vector is a vector that has the same magnitude but opposite direction.

 Example 8: Draw A = (3.0 m/s2, 27) from the origin and then -A from point Q(2, 6).

 Solution: First draw a line at 27 that passes through the origin and select 3.0in. or 3.0cm on it depending on your choice of scale.  Next, locate point Q and draw another line through it parallel to A. Finally select 3.0in. or 3.0cm on it but in the opposite direction of A.  Complete the arrowhead and label it -A.

Note that the direction of -A is (+/-) 180 different from that of A.

 

Addition of Vectors:

The addition of vectors is very easy.  There are two cases.

Case I ) Parallel vectors:

Vectors that are parallel, they act in the same direction.  If they are anti-parallel, they act exactly in opposite directions.  Both types have parallel lines of action. Simply add or subtract their magnitudes algebraically depending on whether they all act in one direction or some act in the opposite direction.  The following 3 figures with their corresponding solutions clarify the concept:

Examples 9, 10, and 11: In each case, find the vector sum (resultant) R of the forces acting on the box.  Assume that to the right is positive.

Solution:   

In the left figure, both forces are parallel and act to the right and therefore the total force to the right is +120N.  The (+) sign shows the direction of the resultant.

In the middle figure, the stronger force acts to the right and the weaker force acts to the left; therefore, a net force of 30N acts to the right.  The (+) sign shows the direction of the resultant.

In the right figure, overall 120N acts to the right and overall 120N acts to the left; therefore the net force acting on the block is 0.

It is also possible to have a number of parallel upward and downward forces.  If so, since they all act in vertical direction, they are parallel and anti-parallel as well.  Again, to find the sum (resultant), simply add their magnitudes algebraically with the assumption that upward is positive and downward is negative.

Examples 12, 13, and 14:

Solution:  

Note that bold letters are used to show vector quantities.  On paper you should show vector R as R with a tiny arrow (→) on top of it.

Before discussing the addition of non-parallel vectors a brief review on trigonometry is vital.

Trigonometry Review:

In every triangle, the sum of internal angles is 180.  Do you know why?  Are you interested in knowing why?  If yes, all you need to know is the fact that the acute angles in a "Z" are equal.  First, when two parallel lines are intersected with a third line, 8 angles develop.   The four acute angles are equal.  The four obtuse angles are also equal.  Look at the following figure and its related explanation for the proof.

Second, look at the following triangle.  You can easily follow the reason why the sum of angles in any triangle is 180.  The trick is to draw (from A) a line like d parallel to base BC as shown.

Right Triangle Trigonometry

In every right triangle, the following are true:  If you do not draw the figures and completely write the relations, your study will not be effective.  You may have to repeat it a few times before feeling confident that you have learned it. "tan" means "tangent", and "cot" means "cotangent."

Answer the following questions:

1) Which angle is the greatest angle in a right triangle?  For answer, click here.

2) Is it correct that in a triangle, the longer a side, the grater its opposite angle is?  For answer, click here.

3) There are two sides to angle B, which side is considered as the adjacent side, side "c", or the hypotenuse?  For answer, click here.

4) Are "sine" and "cosine" hypotenuse related?  For answer, click here.

5) Are "tan" and "cot" hypotenuse related?  For answer, click here.

6) Is the "sine of an angle" defined as "opposite" over "hypotenuse?"  For answer, click here.

7) Is the "cosine of an angle" defined as "Adjacent" over "hypotenuse?"  For answer, click here.

8) Are "tangent" and "cotangent" hypotenuse related?  For answer, click here.

9) Is the tangent of an angle defined as "opposite" over "adjacent?"  For answer, click here.

10) Is the cotangent of an angle defined as " adjacent" over the "Opposite?"  For answer, click here.

11) Based on the above sin(B) = ?  Ans.: .......  For answer, click here.

12) Based on the above cos(B) = ?  Ans.: .......  For answer, click here.

13) Based on the above tan(B) = ?  Ans.: .......  For answer, click here.

14) Based on the above cot(B) = ?  Ans.: .......  For answer, click here.

15) Based on the above sin(C) = ?  Ans.: .......  For answer, click here.

16) Based on the above cos(C) = ?  Ans.: .......  For answer, click here.

17) Based on the above tan(C) = ?  Ans.: .......  For answer, click here.

18) Based on the above cot(C) = ?  Ans.: .......  For answer, click here.

19) Write the expression for sin(B) and cos(B) and then divide the two expressions side by side to get the ratio of sin(B) over cos(B).  What is the result of your division? For answer, click here.

20) Is the answer to (19) the ratio (b/c)?  If "Yes", what is this (b/c) equal to?  For answer, click here.

21) Did you conclude from 19 and 20 that   tan(B) = (sinB) / (cosB) ?  Rewrite this using a horizontal fraction bar.

Example 15: In the following figure, find sides b and c in the right triangle shown:

Since the angles are in degrees, you calculator must be in "degrees mode."

Solution:  sin33 = b/5.  Make sure to write this on paper but definitely with a horizontal fraction bar and not with a "slash".  Then give sin33 a denominator of 1.  It becomes (sin33)/1 = b/5.   Again make sure you use a horiz. fraction bar.  You end up with a proportion.   Now, what is a proportion?

Math Review:

Back to Example 15:

We had:  (sin33)/1 = b/5.     Write this with horizontal fraction bars and then cross multiply.  You should get            b = 5sin33        Do you expect the result to be more than 5 or less? Ans.:.......  Why?  What maximum value can sine of an angle have?   Ans.:......  If you answered "1", you are correct.  Therefore b cannot be more that 5.     b = 5sin33 =  2.7 units.   To find c (the side adjacent to the 33 angle), cosine must be used.   c = 5cos33  =  4.2 units.

Example 16: In the following figure sides b and c of the right triangle are given.  Find the hypotenuse (a) and angle θ.

Now, review the past two examples and ignore the part on proportions.  You notice that in the first example, a and θ were given and we wanted to find b and c.  In the second example, b and c were given, and we wanted to find a and θ.

On this basis, we are going to treat the hypotenuse of a right triangle as a typical vector and the angle θ shown as its direction (relative to the +x-axis).  If, for example, the vector acts Northeast, it then has a Northward effect and an Eastward effect.  This can be viewed as a (+ x) effect and a (+ y) effect. The sum of the East and North effects (Ax and Ay) is equivalent to vector A itself acting through angle θ.

As can be seen, vector A can be replaced by Ax and Ay .    A does the job of Ax and Ay .   The reverse is also true, Ax and Ay  can be replaced by A.  This means that any vector can be replaced by two vectors, one that acts along the x-axis and one that acts along the y-axis.  This suggests a method for adding vectors.

You may have already figured out the method.  Stop and see if you have.  .........Before answering, if you need to go back and review, do not hesitate;  it will be helpful.

Case II ) Non-Parallel Vectors:

If you came up with the idea of resolving each vector into its x- and y- components and then adding the x-components separately and also adding the y-components separately, you were right.  Once all x-components are added, it may be named Rx .   Once all y-components are added, it may be named Ry .  It is then Rx  and Ry that must be added together to result in R, the sum of all participant components The above formulas take care of the entire process.  When the above method of  calculation is used, it is called the "Analytical Method" of vector addition.

From now on, we will always solve each vector problem by two methods: 1) Graphical, and 2) Analytical.  Analytical means by calculation and use of trigonometry.

1) Graphical Method:

a) The Polygon Method:  By graphical, we mean using a ruler and a protractor.  Even when you hand-draw and rely on your good judgment of estimating the size of each vector and its angle, it is still graphical because you do not use trigonometric calculations to find the resultant.  In adding any number of vectors A, B, C, . . .  ,  graphically, you may choose an arbitrary point, P, anywhere on the page.  Next, draw vector A from P.  Next draw vector B from the tip of A.  Next, draw C from the tip of B, and so on . . . .  In other words, place the tail of each vector at the tip of the previous vector (in any order) until all vectors are used as shown below.  Finally, the vector that connects the tail of the 1st vector to the tip of the last vector is the resultant.  This method is called the "Polygon Method."

Compare each vector in (1) with its equal in (2).  Also, compare each vector in (1) with its equal in (3).  As can be seen, the order in which the vectors are connected in a tail-to-tip fashion is not important.  It is important to pay attention to the relations in (4).  Although, vector R equals the sum of vectors A, B, C, and D, the magnitude R does not equal the sum of the magnitudes A, B, C, and D.

b) The Triangle Method:  The polygon method becomes the "Triangle Method" if only two vectors are to be added.  This is simply because the polygon of adding two vectors is a triangle as shown below:

c) Parallelogram Method:  Parallelogram method is convenient for adding two vectors.  Although it can be applied to more than two vectors; however, for more than two vectors, the polygon method is preferred.

In parallelogram method, both vectors are drawn from the same  point, P.  Next, a parallelogram is constructed on them.  The diagonal that is drawn from P is the resultant as shown.  As you know, each parallelogram has two diagonals.  The one that lies in between the two vectors is the resultant, not the other one!  The following figure shows the parallelogram method in part (4).  You can see that if (2) and (3) are put together, (4) will be resulted; in other words, two triangle methods are embedded in one parallelogram application. 

As can be seen in (4) both A and B are drawn from the same point, P.  The diagonal in between (the vectors) is the resultant not the other one.

One can see the triangle method in (2) on the right half of the parallelogram in (4), and the the triangle in (3) on the left half of (4).  This shows how the 3 graphical methods: triangle, polygon and parallelogram are embedded in each other.

 

2) Analytical Method:

It is a good idea to repeat what we have covered regarding the calculation method as follows:

Three Useful Links:  http://www.walter-fendt.de/ph14e/resultant.htm  ,    http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=68  , and  http://lectureonline.cl.msu.edu/~mmp/kap4/cd082.htm .

Example 17:  Given A = (4N, -30 ) and B = (8N, 70) , find R = A + B by calculation.   Use the parallelogram method for its graphical part.

Solution:  First let's draw a sketch of the problem.

 

Example 18: Given A = (5N, -20 ),  B = (10N, 60) ,  and C = (15N, 180),  find R = A + B + C by calculation.  Use the polygon method for its graphical part.

Test Yourself 2:  For answer, click here.

1) To be defined, a vector quantity requires (a) a magnitude only  (b) a direction only  (c) both a magnitude and a direction.

2) To be defined, a scalar quantity requires (a) a magnitude only  (b) a direction only  (c) both a magnitude and a direction.

3) Equal vectors are equal in magnitude and (a) are parallel (b) have the same direction  (c) make the same angle with respect to a reference axis  (d) all of the above.  For answer, click here.

4) The negative of vector A = (14N, 23) is  (a) 14N, 203   (b) -14N, 23   (c) -14N, -23.

5) The negative of vector B = (29N, 0) is  (a) 29N, -0   (b) -29N, 0   (c) 29N, 180.   click here.

6) The x and y components of C =(20.0m/s, 30.0) are: (a) 10.0m/s and 8.3m/s   (b) 17.3m/s and 10.0m/s   (c) 5m/s, 12m/s.

7) The x and y components of D =(10.0N, 135.0) are: (a) 7.07m/s and -7.07m/s  (b) -7.07m/s and 7.07m/s  (c) -7.07m/s and -7.07m/s.   click here.

8) Vector D =(10.0N, 135.0) is in the (a) 1st quadrant  (b) 2nd quadrant (c) 4th quadrant (d) none of the above.

9) The way we may determine the direction (or angle) of a vector is (a) to draw a horizontal line segment from its tail (b) to draw a horizontal line segment from its tip  (c) to rotate the vector and line it up with the y-axis.  click here.

10) Adding vectors A = (10.0N, 30.0) and B = ( 20.0N, 80.0) results in (a) R = (30.0N, 110)    (b) R = (27.5N, 63.8)    (c) R = (27.5N, 26.2).

11) Adding vectors A = (15.0m/s, 30.0) and B = ( 20.0m/s, -60.0) results in (a) R = (25.0m/s, -23.1)   (b) R = (25.5m/s, 23.1)   (c) R = (20.8m/s, -23.1).

12) The resultant of C = (15.0m, 0.0) and D = (30.0m, 120.0) is (a) R = (45.0m, 120)    (b) R = (50.0m, 45.0)    (c) R = (26.0m, 90.0).  click here.

13) The x and y components of vector A are -4N and +4N, respectively.  If A starts from the origin, in which quadrant is this vector?  What is its direction angle?  Ans. : ........,  ........   click here.

14) Vector B has Bx = -5 m/s and By = -12m/s.   (a) Draw a diagram and show the angle θ1 this vector makes with the negative x-axis.  (b) Calculate its direction (the angle with the positive x-axis).  (c) Calculate the magnitude of B?    click here.

Exercises:

In each of the following right triangles, find the unknown sides and angles.  Solve these problems in a row by row sequence.