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Physics 203 College Physics I Fall 2012. S. A. Yost. Chapter 3. Motion in 2 Dimensions – Part 1. Today’s Topics. Vectors We will introduce the concept of vectors, which have many applications throughout physics, and are the most important new mathematical concept used in the course.

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Physics 203 College Physics I Fall 2012

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Physics 203 college physics i fall 2012

Physics 203College Physics IFall 2012

S. A. Yost

Chapter 3

Motion in 2 Dimensions – Part 1


Today s topics

Today’s Topics

  • Vectors

  • We will introduce the concept of vectors, which have many applications throughout physics, and are the most important new mathematical concept used in the course.


Thursday s assignment

Thursday’s Assignment

  • Read Ch. 3, except section 8.

  • A problem set on HW3 on Ch. 3 will be due next Tuesday.

  • The first exam is now scheduled for Thursday, Sept. 20. The calendar in the syllabus posted on CitLearn has been updated.

  • You do not need to memorize equations: the essential ones will be provided for the exam.


Quiz question 2

Quiz: Question 2

  • Which of the equations gives the correct relation between the vectors in the figure?

  • A.A + B + C = 0

  • B. A = B + C

  • C. B = A + C

  • D. C = A + B

  • E. None of these

→ → →

→ → →

A

B

→ → →

→ → →

C


Quiz question 1

Quiz: Question 1

  • Which of the following is a vector?

  • A. Mass

  • B. Temperature

  • C. Distance

  • D. Displacement

  • E. Speed


Quiz question 3

Quiz: Question 3

→ → →

  • Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B?

  • A. Always

  • B. When A and B point in opposite directions.

  • C. Never

  • D. When A and B are parallel.

  • E. When A and B are perpendicular.

→ →

→ →

→ →

→ →


Quiz question 4

Quiz: Question 4

  • Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A?

  • A. – 20, 60o

  • B. 20, 240o

  • C. 20, – 30o

  • D. – 20, 240o

  • E. – 20, – 30o

A

y

θ = 60o

x


Vectors and scalars

Vectors and Scalars

  • Scalars are quantities described entirely by a number, with no need to specify a direction – the temperature, for example.

  • Vectorsrequire both a magnitude and direction to be fully specified.

  • Describing motion in 2 or more dimensions requires vectors.

  • Also forces, which must act in some direction, are described by vectors.


Quiz question 11

Quiz: Question 1

  • Which of the following is a vector?

  • A. Mass

  • B. Temperature

  • C. Distance

  • D. Displacement

  • E. Speed


Displacement vectors

Displacement Vectors

  • The position of a point Brelative to a point Ais given by a displacement vector Dpointing from AtoB.

  • This vector tells you how to get from point Ato point B.

B

D

A


Cartesian components

Cartesian Components

Cartesian coordinates are used to label points in a plane.

The lengths of a vector along the two axes are called its Cartesian components.

Dx = 2, Dy = 5.

y

Dy

D

x

0

Dx


Polar coordinates

Polar Coordinates

A vector can also be specified by giving its magnitudeanddirection.

The magnitude is

the length of the

vector: D = |D|.

The direction can

be given by an

angle relative to

an axis. The angle in polar coordinates is measured counterclockwise from the xaxis.

y

D

θ

x

0


Mathematical review right triangle

Mathematical Review: Right Triangle

  • The sides of a right triangle satisfy the Pythagorean Theorem:

  • a2 + b2 = c2

c

b

a


Mathematical review trigonometry

Mathematical Review: Trigonometry

  • The ratios of sides of a right triangle define the trigonometric functions.

  • sin θ = b/c cosθ = a/c tan θ = b/a

  • cscθ = c/b sec θ = c/a cot θ = a/b

  • Inverses: θ = asin (b/c) = acos(a/c) = atan(b/a)

c

b

θ

a


Polar coordinates1

Polar Coordinates

Find the magnitude and direction of D.

Dx = 2, Dy = 5

D = √ Dx2 + Dy2

= √29 = 5.4

tan θ = 5/2 = 2.5

θ = tan-1 (2.5)

= 68o

y

D

θ

x

0


Vector addition

Vector Addition

  • Geometrically, two vectors are added by following one to the end, then following the second from that point, and finding the net displacement.

  • Components:

  • = +

Cx = Ax + Bx

Cy = Ay + By

A

B

C

C

B

A


Quiz question 21

Quiz: Question 2

  • Which of the equations gives the correct relation between the vectors in the figure?

  • A.A + B + C = 0

  • B. A = B + C

  • C. B = A + C

  • D. C = A + B

  • E. None of these

→ → →

→ → →

A

B

→ → →

→ → →

C


Vectors

Vectors

  • Two vectors, AandB, of length 5and3respectively, lie in a plane, but the directions are unspecified.

  • What is the maximum magnitude of A + B?

  • |A+B| = 8

  • What is the minimum magnitude of A + B?

  • |A+ B|=2

→ →

→ →

→ →

→ →

C

B

A

A

B

C

→ →


Scalar multiple

Scalar Multiple

  • Vectors can be multiplied by scalars (numbers).

  • Multiplying by a positive number changes the length, not the direction:

  • Multiplying by a negative number also changes the direction by 180o:

A

2A

A

– A


Quiz question 41

Quiz: Question 4

  • Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A?

  • A. – 20, 60o

  • B. 20, 240o

  • C. 20, – 30o

  • D. – 20, 240o

  • E. – 20, – 30o

A

y

θ = 60o

x


Quiz question 42

Quiz: Question 4

  • Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A?

  • A. – 20, 60o

  • B. 20, 240o

  • C. 20, – 30o

  • D. – 20, 240o

  • E. – 20, – 30o

y

A

θ = 240o

10

θ = 60o

x

20

– 2A


Vector difference

Vector Difference

  • The vector difference A–B can be formed by adding the vector – B to the vector A.

  • A–B can be interpreted as the displacement that takes you from B to A.

A – B

A

B

– B


Quiz question 31

Quiz: Question 3

→ → →

  • Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B?

  • A. Always

  • B. When A and B point in opposite directions.

  • C. Never

  • D. When A and B are parallel.

  • E. When A and B are perpendicular.

→ →

→ →

→ →

C

A

B

→ →


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