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# Physics 203 College Physics I Fall 2012 - PowerPoint PPT Presentation

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 203College Physics IFall 2012

S. A. Yost

Chapter 3

Motion in 2 Dimensions – Part 1

• 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.

• 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.

• 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

• Which of the following is a vector?

• A. Mass

• B. Temperature

• C. Distance

• D. Displacement

• E. Speed

→ → →

• 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.

→ →

→ →

→ →

→ →

• 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

• 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.

• Which of the following is a vector?

• A. Mass

• B. Temperature

• C. Distance

• D. Displacement

• E. Speed

• 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 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

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

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

• a2 + b2 = c2

c

b

a

• 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

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

• 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

• 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

• 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

→ →

• 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

• 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

• 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

• 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

→ → →

• 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

→ →