Glencoe Physics Ch 4

Remember….When drawing vectors…

length = magnitude (with scale)

angle = direction of the vector quantity.

When drawing and moving vectors, these two characteristics must be maintained

Vector Addition= finding the sum or Resultant of two or more vector quantities

Resultant Vector

A Resultant vector is the vector that results from the addition of two or more vectors

VA VB

VR

Graphical Vector Addition

2. Parallelogram method

8 N

4 N

3 N

3 forces act on an object at the same time. Fnet is not 15 N because these forces aren’t working together. But they’re not completely opposing each either. So how do find Fnet ? The answer is to add the vectors ... not their magnitudes, but the vectors themselves. There are two basic ways to add vectors: 1. Tip to tail method

Tip to Tail Methodin-line examples

Place the tail of one vector at the tip of the other. The vector sum (also called the resultant) is shown in red. It starts where the black vector began and goes to the tip of the blue one. In these cases, the vector sum represents the net force. You can only add or subtract magnitudes when the vectors are in-line!

16 N

20 N

4 N

20 N16 N

12 N9 N

9 N

12 N

21 N

Tip to Tail – 2 Vectors

5 m

2 m

To add the red and blue displacement vectors first note: • Vectors can only be added if they are of the same quantity—in this case, displacement.

• The magnitude of the resultant must be less than 7 m (5 + 2 = 7) and greater than 3 m (5 - 2 = 3).

5 m

2 m

blue +

black

Interpretation: Walking 5 m in the direction of the blue vector and then 2 m in the direction of the black one is equivalent to walking in the direction of the red vector. The distance walked this way is the red vector’s magnitude.

Commutative Property

blue +

black

black +

blue

As with scalars (ordinary numbers), the order of addition is irrelevant with vectors. Note that the resultant (black vector) is the same magnitude and direction in each case.

If you’ve drawn everything to scale, and drawn the angles correctly, then you can simply measure the resultant vector and (using your scale) determine its magnitude.

Vector Properties Addition (Rule A+B = B+A)

A + B = C

Negative vector A-Ajust opposite direction

Subtractionadd negative vector

Tip to Tail – 3 Vectors

We can add 3 or more vectors by placing them tip to tail in any order, so long as they are of the same type (force, velocity, displacement, etc.).

blue + green + black

Parallelogram Method1. Create parallelogram using

“copies” of the two vectors

2. Draw Resultant vector from tail of first vector to tip of last vector.

3. You cannot add more than 2 vectors at a time with this method, so…how do you add 3 or 4 vectors with this method???

Note: Opposite sides of a parallelogram are congruent.

Comparison of Methodsblue +

black

Tip to tail method

Parallelogram method

The resultant has the same magnitude and direction regardless of the method used.

Opposite of a Vector

v

- v

If v is 17 m/s up and to the right, then -v is 17 m/s down and to the left. The directions are opposite; the magnitudes are the same.

Algebraic Solution

Pythagorean Theorem

34 m/s

30.814 m/s

2514.369 m/s

Since components always form a right triangle, the Pythagorean theorem holds: (14.369)2 + (30.814)2 = (34)2.

Note that a component can be as long, but no longer, than the vector itself. This is because the sides of a right triangle can’t be longer than the hypotenuse.

Law of Cosines

a b

c

R

A B

Law of Cosines: R2 = A2 + B2 - 2 AB cos Θ

This side is always opposite this angle.

These two sides are repeated.

It matters not which side is called A, B, and R, so long as the two rules above are followed. This law is like the Pythagorean theorem with a built in correction term of -2 AB cos Θ . This term allows us to work with non-right triangles. Note if Θ = 90, this term drops out (cos 90 = 0), and we have the normal Pythagorean theorem.

Θ

Example Vector Problem

A motorboat heads due east at 16 m/s across a river that flows due north at 9.0 m/s. What is the resultant velocity of the

boat? If the river is 136 m wide, how long

does it take the motorboat to reach the other side?

Graphical Solution

Draw vectors, tip to tail Using your scale, measure length of R

16 m/s

9 m/sR

Solution : Algebraic Method• What is the resultant velocity of the boat?

A2 + B2 = C2

• (9 m/s)2 + (16 m/s)2 = R2

Solution: Calculation of time• If the river is 136 m wide, how

long does it take the motorboat to reach the other side?

• V = Δd/Δt • 16 m/s = 136 m/t• 136 m /16 m/s =Δt • t = 8.5 s

Example Vector Problem

An airplane is flying 200

mph at 50°. Wind velocity

is 50 mph at 270°. What

is the velocity of the plane?

0o

90o

180o

270o

0o

90o

180o

270o

0o

90o

180o

270o

0o

90o

180o

270o

Practice Problems

P.67 1-4 Your turn!!

4.2 Components of Vectors

Any vector directed in two dimensions can be thought of as having two parts

Each part of a two-dimensional vector is known as a component.

The single two-dimensional vector could be replaced by the two components.

Vector Components

150 N

Horizontal component

Ver

tica

l co

mpo

nent

Ay

Ax

A

Θ

Ax=horizontal component of vector A

Ay=vertical component of vector A

Component Vectors

Ax = A cos Θ cos Θ = Adjacent sidehypotenuse

Ay = A sin Θ sin Θ = Opposite sidehypotenuse

tan Θ = Opposite side Adjacent side

Remember…SOH CAH TOA

Finding Components with Trig

v

v cos

v si

n

Multiply the magnitude of the original vector by the sine & cosine of the angle made with the horizontal. The units of the components are the same as the units for the original vector.

Here’s the correspondence:

cosine adjacent side

sine opposite side

Note that 30.814 + 14.369 > 34. Adding up vector components gives the original vector (green + red = black), but adding up the magnitudes of the components is meaningless.

Component Example

34 m/s

-30.814 m/s

205-14.369 m/s

A helicopter is flying at 34 m/s at 25 S of W (south of west). The magnitude of the horizontal component is 34 cos 205 -30.814 m/s. This is how fast the copter is traveling to the west. The magnitude of the vertical component is 34 sin 205 -14.369 m/s. This is how fast it’s moving to the south.

vWA = vel. of Wonder Woman w/ resp. to the air

vAG = vel. of the air w/ resp. to the ground (and Aqua Man)

vWG = vel. of Wonder Woman w/ resp. to the ground (and Aqua Man)

Wonder Woman Jet ProblemSuppose Wonder Woman is flying her invisible jet. Her onboard controls display a velocity of 304 mph 10 E of N. A wind blows at 195 mph in the direction of 32 N of E. What is her velocity with respect to Aqua Man, who is resting poolside down on the ground?

We know the first two vectors; we need to find the third. First we’ll find it using the laws of sines & cosines, then we’ll check the result using components. Either way, we need to make a vector diagram.

continued on next slide

The 80 angle at the lower right is the complement of the 10 angle. The two 80 angles are alternate interior. The 100 angle is the supplement of the 80 angle. Now we know the angle between red and blue is 132.

Wonder Woman Jet Problem (cont.)

continued on next slide

10

32 v

WA

vAG

vWG

vWA + vAG = vWG 80

195 mph

304

mph

vWG

8032

100

Wonder Woman Jet Problem (cont.)

195 mph

304

mph

v

132

The law of cosines says: v2 = (304)2 + (195)2 - 2 (304) (195) cos 132 So, v = 458 mph. Note that the last term above appears negative, but it’s really positive, since cos 132 < 0. The law of sines says:

sin 132 sin v 195

=

So, sin = 195 sin 132 / 458, and 18.45

80

This means the angle between green and the horizontal is 80 - 18.45 61.6

Therefore, from Aqua Man’s perspective, Wonder Woman is flying at 458 mph at 61.6 N of E.

Wonder Woman Problem: Component Method

32

vW

A =

304

mph

vAG = 195 mph

10

This time we’ll add vectors via components as we’ve done before. Note that because of the angles given here, we use cosine for the vertical comp. of red but sine vertical comp. of blue. All units are mph.

304

195103.3343

165.3694

52.789

299.3816

continued on next slide

Wonder Woman: Component Method (cont.)

304

195

103.

3343

165.3694

52.789

299.

3816

103.3343

52.789 165.3694

299.

3816

402.

7159

mph

218.1584 mph

458.

0100

mph

Combine vertical & horiz. comps. separately and use Pythag. theorem. = tan-1 (218.1584 / 402.7159) = 28.4452. is measured from the vertical, which is why it’s 10 more than was.

Comparison of Methods

We ended up with same result for Wonder Woman doing it in two different ways. Each way requires some work. You can only use the laws of sines & cosines if:

• you’re dealing with exactly 3 vectors. (If you’re adding three vectors, the resultant makes 4, and this method won’t work

• the vectors form a triangle.

Regardless of the method, draw a vector diagram! To determine which two vectors add to the third, use the subscript trick.

END of Ch 4