Physics 12 notes VECTORS Page # 1 VECTORS

Physics 12 notes

VECTORS

Page # 1

VECTORS

Headings

Headings can be given several ways.

i.

3600 co-ordinates: Assuming that the x-axis on the right is 00, than you can give headings

according to that. Ex. 2200

00

ii. The more common method is use compass headings Ex. 400 N of W ) 400

Vectors

The above angle can also be read as 500 W of N, however, the convention is to use the smaller angle (both 400 N of W & 500 W of N are correct).

? Scalar quantities - have magnitude, but no direction ( mass, volume, distance, speed, time, energy, etc. . .)

? Vector has magnitude and direction (usually written as bolded, or with an arrow above)

o Notice the difference between distance(d), and displacement (d), or speed (v), and velocity (v).

? Resultant vector (VR) "sum" of several vectors;

VR

Concurrent vectors - act on the same point at the same time

Equilibrant vector produces equilibrium and is equal and opposite to resultant vector

Vector 1

VR

Vequilibrant

Vector 2

In other words, the two vectors on the left (solid lines) together add up to make a resultant vector on the left (the dashed line). In order to balance off the two vectors on the left, a vector that is equal but opposite to the VR is needed (the vector on the right).

Physics 12 notes

Properties of vectors

VECTORS

Page # 2

1. A = B if the magnitude and direction of each are the same

2. C = A + B , adding two vectors will give you a resultant vectors 3. A - B = A + (-B)

It is not possible to subtract vectors, however, you are allowed to add a negative (opposite direction) of a vector

Ex. A = B = - B = A + (-B) =

therefore, resultant is

SOLVING VECTOR PROBLEMS

There are several ways to solve vector problems. We will look at three different methods this year.

i.

Graphical Method (not covered this year)

ii.

Analytical Method (Triangles)

iii.

Component Method (best way to solve vectors, but overkill for High School Physics)

i. Graphical method of vector addition: (very briefly)

1. Vectors are represented graphically by using arrows and drawing to scale.

2. Vectors are added graphically by placing them "tips to tails."

3. The resultant vector is drawn graphically by placing its tail at the tail of the first vector and its tip at

the tip of the last vector. (From the very start to the very end)

Example 3: A small motorboat is aimed directly across a river (S) and travels at 5.0 m/s relative to the shore. The river water is flowing West with a velocity of 3.0 m/s relative to the shore. What will the resultant velocity of the boat be relative to the shore?

All vectors can be broken down into simpler x & y components. vx

vy

v

vy

v

vx

vx

vx

v

Physics 12 notes

VECTORS

Example 6: Break down 9 m 300 W of N into X and Y components

Page # 3

When you have a vector like 7 m/s, 20o N of E, it tells you that something is moving in two directions at once. It is moving in the x direction (to the East) and also moving in the y direction (North) at the same time.

ii. Vectors- The Analytical Method

The graphical method is limited to two dimensions, will never give very precise results, and takes a lot more time. Therefore, we must now learn the analytical method.

The key behind the Analytical method is that any vector can be expressed as the sum of two perpendicular vectors, called the component vectors.

Any vector can be broken down into a vertical and horizontal components, just like the horizontal and vertical components can be added together to give you the vector. Even though you're solving analytically, you must always start off with a quick diagram. This will help you see the problem and help you avoid mistakes. In order to solve for vector components you will need to know basic TRIGONOMETRY

and the Pythagorean Theorem One can obtain

Example 6: Analytically break down 9 m 300 W of N into X and Y components

Physics 12 notes

VECTORS

Page # 4

Example 7: A small motorboat is aimed directly across a river (S) and travels at 5.0 m/s relative to the

shore. The river water is flowing West with a velocity of 3.0 m/s relative to the shore. What will the resultant

velocity of the boat be relative to the shore?

o When solving vector problems, you'll usually have to assign which way is positive and which way is negative. The simplest way (and the way that you're most used to) is to assign everything to right (or East) as positive (meaning everything to the left (West) is negative) and assigning everything going up (North) as positive (meaning everything down(south) is negative).

Example 8: A 110 kg sign is suspended by two ropes that make a 900 angle with each other. What is the

tension of the rope on the right?

1. First, start by making a free body diagram

T1

T2

Fg

2. Make a vector diagram, remembering that the tension in the two ropes pulling upward have to add up a resultant force that is equal and opposite to the force of gravity.

T2 Fg

T1

3. Since T1 and T2 make an angle of 900 with respect to each other, than the angle on the vector diagram that is opposite

of Fg must be 900 . We also know that T1 and T2 are the same, than the angle at the top and the bottom of the triangle above must be 450

4. Since this is a right triangle, that we can use Soh Cah Toa. In order to Find T2, Sin = opp / Hyp =T2 / Fg T2 = (Sin 450) (110 kg)(9.8 m/s2) = 762.2 N = 800 N

SIN AND COS LAWS

The problem is that Soh Cah Toa only works when you have a right triangle. Unfortunately, you'll often get triangles that aren't right triangles. What to do? Cosine Law and Sine Law. I know, I know, scary stuff. Well, not really. Once you learn the laws and when to use them, they're actually quite easy. Its only a matter of plugging in numbers and getting an answer.

Cos Law: c2 = a2 + b2 ? 2 a b CosC

Sin Law: Sin A -------- =

a

Sin B -------

b

Sin C = -------

c

Physics 12 notes

VECTORS

Page # 5

In order to use these formulas, you must be able to label a triangle with the appropriate letters and angles.

1. Label all the angles (A, B, C) try to label clockwise. o If you are looking for a side of the triangle, then label the angle opposite the side your looking for as Angle C

Let's say you're looking for side X C

B

A

X

2. Label all the sides with small letters (a, b, c). The letter for the side must be the same letter as the angle that is opposite the side.

C

a

b

B

A

c X

USING THE COS LAW: c2 = a2 + b2 ? 2 a b CosC o if you are looking for side x, plug in the values and solve for c

USING THE SIN LAW If you are using the Sin Law, you must label all the sides and angles as above. The only difference is that it doesn't matter where you start labeling, or which side you decide to make side C. Just make sure that Side a is opposite angle A, side B is opposite Angle B, and side c is opposite Angle C.

Example 10: An bird flying North at 5.2 m/s encounters a wind heading 300 W of N at 3.2 m/s. What will the resultant velocity of the bird be? Solution: 1. Draw a quick vector diagram. (See diagram on the left)

300

3.2 m/s

VR 5.2 m/s

VR 5.2 m/s

2. Find what angle C () will be: (See diagram on the right) o If the angle between the 3.2 m/s vector and the North co-ordinate is 300, then the angle () between the 3.2 m/s vector and the 5.2 m/s N vector must be 1800 ? 300, meaning angle is 1500

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