lecture 4 physics i -...
Post on 31-May-2020
8 Views
Preview:
TRANSCRIPT
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Lecture 4
Chapter 3
Introduction to vectors
Physics I
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Today we are going to discuss:
Chapter 3:
Vectors and Scalars: Section 3.1 Addition/ Subtraction of Vectors/ Multiplication of a
Vector by a Scalar : Section 3.2 Vector Components: Section 3.3 Unit Vectors: Section 3.4
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Vector and Scalar
e.g. distance, speed, temperature, mass, time, density, volume
r, v , a
e.g. displacement, velocity, acceleration, force, momentum
has only magnitude (no need in direction)
Vector quantity Scalar quantity
r
has both direction and magnitude
V=10 L
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Addition of Vectors (1D)
If the vectors are in opposite directions
If the vectors are in the same direction
For vectors in one dimension, simple addition and subtraction are all that is needed. Easy!!!!
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Addition of Vectors (2D). Graphical MethodsTriangle method.
“Tail-to-Tip” method Draw first vector Draw second vector, placing its tail at the tip of the first vector Resultant: Arrow from the tail of 1st vector to the tip of 2nd vector
The situation is somewhat more complicated in a case of two dimensions.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Addition of Vectors (2D). Graphical MethodsParallelogram method.
The two vectors, , are drawn as the sides of the parallelogram The resultant, , is its diagonal
Commutative property of vectors
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Subtraction of vectors
is a vector with the same magnitude as but in the oppositedirection. So we can rewrite subtraction as additionB
BA
=A
AB
B
B
B
So, we add the negative vector.
)( BA
ConcepTest 1 Vector Addition
You are adding vectors of length
20 and 40 units. What is the only
possible resultant magnitude that
you can obtain out of the
following choices?
A) 0
B) 18
C) 37
D) 64
E) 100
The minimum resultant occurs when the vectorsare opposite, giving 20 units. The maximum resultant
occurs when the vectors are aligned, giving 60 units.
Anything in between is also possible for angles
between 0° and 180°.
4020
Min=40‐20=20 Max=40+20=60
Resultant is between 20 and 60
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Multiplication of a Vector by a Scalar
A vector can be multiplied by a scalar b(positive); the result is a vector that has the same direction but a magnitude .
If b is negative, the resultant vector points in the opposite direction.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Addition of three or more vectorsCan use “tip to tail” for more than 2 vectors
+ + =
Order of addition does not matter
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Vector componentsThe graphical addition of vectors is not an especially good way to find
quantitative results.
Here we will need a coordinate system to resolve a vector into components along
mutually perpendicular directions.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Determining vector components (Vx, Vy)If we know magnitude, V and direction, θ
Assume that θ is measured counterclockwise from the positive x-axis In 2D, we can always write any vector as the sum of a vector in the x-
direction, and one in the y-direction.V
Vx
Vy
cos VVx V
Vx
Vy
sinVVy
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
x
y
VV
tan
Given Vx and Vy , we can find V, θ Vx, Vy are the legs of the right triangle Vector as the hypotenuse. So, the magnitudes of the vectors satisfy the Pythagorean Theorem.
V
Vx
Vy
Vy
222yx VVV
22yx VVV
x
y
VV1tan so
x
y
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Determining the direction, θ of a vectorWe can describe vector’s direction by its angle relative to some reference direction.
Assume that θ is measured from the positive x-axis Θ is positive if measured counterclockwise from +x
V
Vx
Vy
0 x
y
0x
y Vy
Vx
V
V
Vx
Vy
0xy0
V
Vx
Vy
xy
0yV
0xV
0yV
0xV
0yV
0xV
0yV
0xV
I IIIII IV
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Alternative ways of determining θWe can describe vector’s direction by its angle relative
to some other reference direction.
x
y Vy
Vx
V
0 V
Vx
Vy
xy
θ is measured clockwise from the positive x‐axis (Θ is negative)
θ is measured clockwise from the negative x‐axis
(you can say: ‐30 deg or 30 deg below +x axis)
(you can say: 30 deg above –x axis)
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Example A vector is given by its vector components:
Write the vector in terms of magnitude and direction.4,2 yx VV
47.42042 22 V
x
y
‐1
4
2
‐2
4yV
2xV
22
yx VVV
x
y
VV
tan 224tan
magnitude
axisxabove 11763180180
632tan 1
axisxfrom
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Adding vectors by components
Given and , how can we find ?
21 VVV
V1
V2V
V1x
V1y
V2x
V2y
V1
V2
V1
V2
x
y
=
= yx VVV ,
yx VVV 111 ,
yx VVV 222 ,
yyxx VVVV 2121 ,
Adding corresponding components
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Unit Vectors
As we said before, a vector has both magnitude and direction. Now, it’s time to simplify a notation of direction: Let’s introduce unit vectors
vectorsunitasknownkji ˆ,ˆ,ˆ
x
y
z
• They point along major axes of our coordinate system
= = =1 • Unit vector has a magnitude of 1
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Writing a vector with unit vectorsWriting a vector with unit vectors is equivalent to
multiplying each unit vector by a scalar
• If a vector has components:
• In unit vector notation, we write
3,4 yx VV
jiV ˆ3ˆ4
yx VVV
x
y
xV
),( yx VV
)3,4(iVxˆ
jViV yxˆˆ
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Example: a hiker on a trail.
D1 (2500m)i (500m) jD2 (500m)i (700m) j (700m)kD3 (600m) jD4 (500m)k
D (3000m)i (1800m) j (200m)k
D
D1
D2
D3
D4
(2500m)i (500m) j
(500m)i (700m) j (700m)k
(600m) j
(500m)k
What is the hiker’s total displacement?
The first leg is a flat hike to the foot of the mountain: ----------------------------On the second leg, he climbs the mountain:----------------------------------
On the third, he walks along a plateau: ---
Then, he falls off a cliff: ---------------------
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Thank youSee you on Wednesday
Department of Physics and Applied PhysicsPHYS.1410 Lecture 4 Danylov
Example 1
A vector is given by its magnitude and direction (V,) What is the x, y-component
of the vector?
mV 10axisxabove 30
top related