Vectors - Physics - Lecture Slides, Slides of Physics

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Vectors

• 2-D Force &

Motion Problems

• Trig Applications
• Relative Velocities
• Free Body Diagrams
• Vector Operations
• Components
• Inclined Planes
• Equilibrium

Vector Addition

• Tip to tail method
• Parallelogram method

8 N

4 N

3 N

Suppose 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 F net? The answer

is to add the vectors ... not their

magnitudes, but the vectors

themselves. There are two basic

ways to add vectors w/ pictures:

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).

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

Place the vectors tip to tail and draw a vector from the tail of the first to the tip of the second.

Commutative Property

As with scalars quantities and 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.

(We’ll learn how to find the resultant’s magnitude soon.)

Parallelogram Method

This time we’ll add red & blue by

placing the tails together and

drawing a parallelogram with

dotted lines. The resultant’s tail

is at the same point as the other

tails. It’s tip is at the intersection

of the dotted lines.

Note: Opposite sides of a parallelogram are congruent.

Comparison of Methods

Tip to tail method

Parallelogram method

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

Scalar Multiplication

x

-2 x

3 x

Scalar multiplication means multiplying a vector by a real number, such as 8.6. The result is a parallel vector of a different length. If the scalar is positive, the direction doesn’t change. If it’s negative, the direction is exactly opposite.

Blue is 3 times longer than red in the same direction. Black is half as long as red. Green is twice as long as ½ x red in the opposite direction.

Vector Subtraction

red - blue

blue - red

Put vector tails together and complete the triangle, pointing to the vector that ―comes first in the subtraction.‖

Why it works: In the first diagram, blue and black are tip to tail, so blue + black = red red – blue = black.

Note that red - blue is the opposite of blue - red.

Comparison of Vectors

15 N

43 m

0.056 km

27 m/s

Which vector is bigger?

The question of size here doesn’t make sense. It’s like

asking, ―What’s bigger, an hour or a gallon?‖ You can

only compare vectors if they are of the same quantity.

Here, red’s magnitude is greater than blue’s, since

0.056 km = 56 m > 43 m, so red must be drawn longer

than blue, but these are the only two we can compare.

Vector Components

150 N

Horizontal component

Verticalcomponent

A 150 N force is exerted up and to the right. This force can be thought of as two separate forces working together, one to the right, and the other up. These components are perpendicular to each other. Note that the vector sum of the components is the original vector (green + red = black). The components can also be drawn like this:

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

14.369 m/s^25

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 25 30.814 m/s. This is how fast the copter is traveling to the west. The magnitude of the vertical component is 34 sin 25 14.369 m/s. This is how fast it’s moving to the south.

Pythagorean Theorem

34 m/s

30.814 m/s

14.369 m/s^25

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.

Component Form

Instead of a magnitude and an angle, vectors are often specified by listing their horizontal and vertical components. For example, consider this acceleration vector:

3 m/s^2

m/s

2

a = 10 m/s^2 at 53.13 N of W

In component form:

a = -3, 4 m/s^2 Some books use parentheses rather than angle brackets. The vector F = 2, -1, 3 N indicates a force that is a combination of 2 N to the east, 1 N south, and 3 N up. Its magnitude is found w/ the Pythag. theorem:

F = [2^2 + (-1)^2 + 3^2 ]1/2^ = 3.742 NDocsity.com

Finding the direction of a vector

x = 5, -2 meters is clearly a position to the southeast of a given reference point. If the reference pt. is the origin, then x is in the 4th^ quadrant. The tangent of the angle relative to the east is given by:

5 m

2 m

tan = 2 m / 5 m = tan -1(0.4) = 21.

The magnitude of x is (25 + 4)1/2^ = 5.385 m. Thus, 5, -2 meters is equivalent to 5.385 m at 21.801 S of E.