Chapter 7

Question 1

What is a system?

Answer 1

A system is any portion of the universe that can be separated from the rest by a boundary

Question 2

What is work?

Answer 2

Work is a measure of the influence of forces on a system

Question 3

What is work influenced by?

Answer 3

Magnitude and direction of forces

Question 4

What are the formulas of work?

Answer 4

W = F∥ d (product of magnitude of the displacement x component of force parallel to displacement)

W = Fd cos θ

Question 5

What is the SI unit of work?

Answer 5

Joules ( J )

Question 6

What is θ in the work formula?

Answer 6

θ is the angle between force (f) and displacement (d)

Question 7

Is the force and displacement positive or negative and why?

Answer 7

They are always positive because they are magnitudes

Question 8

A train car is stalled on the tracks. Two horses are used to pull the train 20 meters using these forces :
F1 = (500N, 30°)
F2 = (500N, 45°)
Calculate the work done by each horse

Answer 8

W = Fd cos θ

   W1 = F1 * d * cos θ W1 = 500 * 20 * cos (30°) = 8660.3 J

   W2 = F2 * d * cos θ W2 = 500 * 20 * cos (45°) = 7071.1 J

Question 9

Work is a scalar, true or false?

Answer 9

True

Question 10

What are the possible ways to multiply vectors?

Answer 10

1) multiplication of a vector by a scalar (unit 3).
2) multiplication of one vector by a second vector to produce a scalar, called scalar product or dot product.
3) multiplication of one vector by a second vector to produce another vector called vector product (unit 11).

Question 11

What is a scalar product / dot product?

Answer 11

It is a measure of how closely two vectors align in terms of the directions they point.

Question 12

Dot product is a scalar, true or false?

Answer 12

True

Question 13

Example1

Answer 13

If we have vector a = (ax, ay, az) and vector b = (bx, by, bz) then its dot product is
a·b = (axbx) + (ayby) + (azbz)

Question 14

Vector a= ( 2, 4 )
Vector b= ( 1, -3 )
Find the dot product

Answer 14

a·b = (( 2 · 1) + (1 · -3 ))
a·b = 2 + (-12)
a·b = -10

Question 15

Properties of the dot product (Commutative)

Answer 15

a·b = b·a

Question 16

Properties of the dot product (distributive)

Answer 16

a · (b+c) = (a·b) + (a·c)

Question 17

If we have three vectors:
a = 5 i - 4 j
b = 7 i + 8 j
c = 3 i - 2 j
What is the value of a · (b+c) ?

Answer 17

First add b+c :
b+c= 7i + 8j + 3i - 2j
b+c= 10i + 6j

a · (b+c) = 5i (10i) + (-4j) 6j
a · (b+c) = 50 - 24
a · (b+c) = 26

Question 18

If we have two vectors:
a = 5 i - 4 j
b = 7 i + 8 j
What is the value of (3a) · b

Answer 18

First find 3a:
3a = 3 (5i - 4j)
3a= 15i - 12j

(3a) · b = 15 (7) - 12(8)
(3a) · b = 105 - 96 = 9

Question 19

If we have two vectors:
a = 5 i - 4 j
b = 7 i + 8 j
What is the value of 4(a · b)

Answer 19

4(a · b) = 4 ( 5 · (7) + (-4) · 8 )
4(a · b) = 4 (35 + -32)
4(a · b) = 4 (3)
4(a · b) = 12

Question 20

(ca) · b

Answer 20

c (a·b) = a · (cb)

Question 21

a·a

Answer 21

lal²

Question 22

0 · a

Answer 22

0

Question 23

What is the theorem of the dot product?

Answer 23

the dot product of A and B equals the length of A times the length of B times the cosine of the angle between them: A · B = |A||B| cos(θ).

Question 24

How to calculate an angle between 2 vectors?

Answer 24

cos θ = a·b ⁄ |a||b|

Question 25

Find the an angle between these vectors:
a = (2, 2, -1) and b = (5, -3, 2)

Answer 25

|a|= √(2)² + (2)² + (-1)² = 3
|b|= √(5)² + (-3)² + (2)² = √38

a·b = (2 · 5) + (2 · (-3) ) + ( (-1) · 2)
a·b = 10 + (-6) + (-2)
a·b = 2

cos θ = a·b ⁄ |a||b| = 2 / (3) ( √38 )
cos θ = 0.108
θ = cos-1 (0.108)
θ = 84°

Question 26

Find the magnitude of each vector:
a = 3i + 4j

Answer 26

|a|= √ax² + ay²
|a|= √3² + 4²
|a|= √9 + 16
|a|= √25
|a|= 5

Question 27

Find the magnitude of each vector:
b = -7i + 42j

Answer 27

|b|= √ax² + ay²
|b|= √ (-7)² + 24²
|b|= √ 49 + 576
|a|= √625
|a|= 25

Question 28

Find the magnitude of each vector:
a = 2i - 4j + 6k

Answer 28

|a|= √ax² + bx² + cx²
|a|= √2² + (-4)² + 6²
|a|= √4 + 16 + 36
|a|= √56
|a|= 2√14