Transcript Physics 1422 - Introduction

Physics 203 – College Physics I
Department of Physics – The Citadel
Physics 203
College Physics I
Fall 2012
S. A. Yost
Chapter 3
Motion in 2 Dimensions – Part 1
Physics 203 – College Physics I
Department of Physics – The Citadel
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.
Physics 203 – College Physics I
Department of Physics – The Citadel
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.
Physics 203 – College Physics I
Department of Physics – The Citadel
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
→
→
→
→
→
B
A
C. B = A + C
→
→
→
D. C = A + B
E. None of these
→
C
Physics 203 – College Physics I
Quiz: Question 1
Which of the following is a vector?
A. Mass
B. Temperature
C. Distance
D. Displacement
E. Speed
Department of Physics – The Citadel
Physics 203 – College Physics I
Department of Physics – The Citadel
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.
Physics 203 – College Physics I
Department of Physics – The Citadel
Quiz: Question 4
→
Vector A has a magnitude of 10 and a direction
angle θ = 60o measured 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 = 60o
x
Physics 203 – College Physics I
Department of Physics – The Citadel
Vectors and Scalars
Scalars are quantities described entirely by a
number, with no need to specify a direction –
the temperature, for example.
Vectors require 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.
Physics 203 – College Physics I
Quiz: Question 1
Which of the following is a vector?
A. Mass
B. Temperature
C. Distance
D. Displacement
E. Speed
Department of Physics – The Citadel
Physics 203 – College Physics I
Department of Physics – The Citadel
Displacement Vectors
The position of a point B relative to →
a point A is
given by a displacement vector D pointing
from A to B.
B
This vector tells you how to get from
point A to point B.
→
D
A
Physics 203 – College Physics I
Department of Physics – The Citadel
Cartesian Components
Cartesian coordinates are used to label points in
a plane.
y
The lengths of a
vector along the
two axes are
called its
Cartesian
components.
Dx = 2, Dy = 5.
Dy
0
→
D
Dx
x
Physics 203 – College Physics I
Department of Physics – The Citadel
Polar Coordinates
A vector can also be specified by giving its
magnitude and direction.
The magnitude is
the length of →
the
vector: D = |D|.
y
→
D
The direction can
θ
be given by an
angle relative to
x
0
an axis. The angle
in polar coordinates
is measured counterclockwise from the x axis.
Physics 203 – College Physics I
Department of Physics – The Citadel
Mathematical Review: Right Triangle
The sides of a right triangle satisfy the
Pythagorean Theorem:
a2 + b2 = c2
c
b
a
Physics 203 – College Physics I
Department of Physics – The Citadel
Mathematical Review: Trigonometry
The ratios of sides of a right triangle define the
trigonometric functions.
sin θ = b/c
cos θ = a/c
tan θ = b/a
csc θ = c/b
sec θ = c/a
cot θ = a/b
c
b
θ
a
Inverses: θ = asin (b/c) = acos(a/c) = atan(b/a)
Physics 203 – College Physics I
Department of Physics – The Citadel
Polar Coordinates
→
Find the magnitude and direction of D.
Dx = 2, Dy = 5
y
D = √ Dx2 + Dy2
= √29 = 5.4
→
tan θ = 5/2 = 2.5
θ
θ = tan-1 (2.5)
= 68o
D
0
x
Physics 203 – College Physics I
Department of Physics – The Citadel
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:
→
→
→
C = A +B
→
→
C
B
→
A
Cx = Ax + Bx
Cy = Ay + By
Physics 203 – College Physics I
Department of Physics – The Citadel
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
→
→
→
→
→
B
A
C. B = A + C
→
→
→
D. C = A + B
E. None of these
→
C
Physics 203 – College Physics I
Department of Physics – The Citadel
Vectors
→
→
Two vectors, A and B, of length 5 and 3
respectively, lie in a plane, but the directions
are unspecified.
→
→
What is the maximum magnitude of A + B?
→
→
A
B
→
→ →
|A+B| = 8
→ →
C
What is the minimum magnitude of A + B?
→ →
→
A
→
C
|A+ B|=2
→
B
Physics 203 – College Physics I
Department of Physics – The Citadel
Scalar Multiple
Vectors can be multiplied by scalars (numbers).
Multiplying by a positive number changes the length,
→
not the direction:
A
→
2A
Multiplying by a negative number also changes the
direction by 180o:
→
→
–A
A
Physics 203 – College Physics I
Department of Physics – The Citadel
Quiz: Question 4
→
Vector A has a magnitude of 10 and a direction
angle θ = 60o measured 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 = 60o
x
Physics 203 – College Physics I
Department of Physics – The Citadel
Quiz: Question 4
→
Vector A has a magnitude of 10 and a direction
angle θ = 60o measured counter-clockwise from
the +x axis. What are the
magnitude and direction
→
angle of the vector – 2A?
A. – 20,
60o
y
θ = 240o
20,
C.
20, –
E. – 20, –
x
30o
D. – 20, 240o
30o
A
θ = 60o
240o
B.
→
→
– 2A
Physics 203 – College Physics I
Department of Physics – The Citadel
Vector Difference
→
→
The vector difference
A – B can
be formed by adding
→
→
the vector – B to the vector A.
→
A
→
→
A–B
→
→
→
B
A – B can be interpreted
as the displacement that
→
→
→
takes you from
– B B to A.
Physics 203 – College Physics I
Department of Physics – The Citadel
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?
B
A
→
A. Always
C
→
→
→
→
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.