Chapter 2, Part 2 Vectors in Physics - St. Monica Academy · Copyright © 2010 Pearson Education,...

Copyright © 2010 Pearson Education, Inc.

Vectors in Physics Chapter 2, Part 2


Distance: A Scalar Quantity

A scalar quantity: Contains magnitude only and consists of a number and a unit. (20 m, 40 mi/h, 10 gal)

A

B

§  Distance is the length of the actual path taken by an object.

d = 20 m


Displacement—A Vector Quantity

A vector quantity: Contains magnitude AND direction, a number, unit & angle. (12 m, 200; 8 km/h, N)

A

B s = 12 m, 20o

•  Displacement is the straight-line separation of two points in a specified direction.

θ


3-1 Symbols for Vectors

Vectors are often written as a capital letter or a lowercase letter with an arrow over it: V or v

Magnitudes of vectors are written with single or double lines around the letter: V or v


3-3 Adding and Subtracting Vectors Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last.

This is known as the tip-to-tail method.


Vector Addition

To add vectors v and u:

1. Place the initial point of u at the terminal point of v.

2. Draw the vector with the same initial point as v and the same terminal point as u.

u v

v + u

Vector Addition v u


Sum of Two Vectors


Trigonometry Review •  Application of Trigonometry to Vectors

y

x

R

θ

sin yR

θ =

cos xR

θ =

tan yx

θ =

R2 = x2 + y2

Trigonometry


Finding Components of Vectors

A component is the effect of a vector along other directions. The x and y components of the vector (R,θ) are illustrated below.

x

y R

θ

x = R cos θ y = R sin θ

R = x + y


Example 2: A person walks 400 m in a direction of N 60.0o E. How far is the displacement east and how far north?

x

y R

θ

x = ?

y = ? 400 m

60ο

E

N

The y-component (N) is OPP:

The x-component (E) is ADJ: x = R cos θ

y = R sin θ

E

N

θ=30.0°

= 346 m, E

= 200 m, N


Resultant of Perpendicular Vectors Finding the resultant of two perpendicular vectors is like adding components and finding direction.

θ is measured from the positive x axis

tanθ = yx

x

y R

θ

R = x2 + y2


Example 3: A 30-lb southward force and a 40-lb eastward force act on a donkey at the

same time. What is the NET or resultant force on the donkey?

30 lb

40 lb

Draw a rough sketch.


Finding Resultant: (Cont.)

40 lb

30 lb

40 lb

30 lb

Finding (R,θ ) from given (x,y) = (40, -30)

R

φ θ

Ry

Rx

R = x2 + y2 R = (40)2 + (30)2 = 50 lb

tan φ = 30 40

φ = –37o


Example 7. Find the components of the 240-N force exerted by the boy on the girl if his arm

makes an angle of 28.00 with the ground.

28.00

F = 240 N F Fy

Fx

Fy

Fx = -|(240 N) cos 28.00| = -212 N

Fy = |(240 N) sin 28.00| = 113 N


Example 8. Find the components of a 300-N force acting along the handle of a lawn-

mower. The angle with the ground is 32.00.

32.00

F = 300 N

F Fy

Fx

Fy

Fx = -|(300 N) cos 32.00| = -254 N

Fy = -|(300 N) sin 32.00| = -159 N

32o


3-3 Adding and Subtracting Vectors

Adding Vectors Using Components:

1.  Find the components of each vector to be added.

2.  Add the x- and y-components separately.

3.  Find the resultant vector.


Adding Vectors Using Components A = 5.00m B = 4.00m

xxx BAC +=

yyy BAC +=


Ax = (5.00m)cos60.0o = 2.50m

Ay = (5.00m)sin60.0o = 4.33m

Bx = (4.00m)cos20.0o = 3.76m

By = (4.00m)sin20.0o =1.37m

2.50m

4.33m

3.76m

1.37m


mmmBAC xxx 26.676.350.2 =+=+=

mmmBAC yyy 70.537.133.4 =+=+=

θ = tan−1Cy

Cx

"

#$

%

&'

= tan−1 5.70m6.26m"

#$

%

&'

= 42.3o

mmmCCC yx 47.8)70.5()26.6( 2222 =+=+=

6.26m

5.70m 8.47m

= 42.3°


Positive and Negative Vectors


3-3 Adding and Subtracting Vectors

Subtracting Vectors: The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, .

D= A −B


Difference of Two Vectors


Homework

p. 46 Multiple Choice (11-19 odd) pp. 49-51 Probs. 43, 51, 65, 69, 71, 75, 87

Chapter 2, Part 2 Vectors in Physics - St. Monica Academy · Copyright © 2010 Pearson Education,...

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