ch 03b motion in a plane
TRANSCRIPT
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 1
Motion in a Plane
Chapter 3
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 2
Motion in a Plane
• Vector Addition
• Velocity
• Acceleration
• Projectile motion
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 3
Graphical Addition and Subtraction of Vectors
A vector is a quantity that has both a magnitude and a direction. Position is an example of a vector quantity.
A scalar is a quantity with no direction. The mass of an object is an example of a scalar quantity.
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 4
Notation
Vector: FF
or
The magnitude of a vector: .or or FF
F
Scalar: m (not bold face; no arrow)
The direction of vector might be “35 south of east”; “20 above the +x-axis”; or….
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 5
To add vectors graphically they must be placed “tip to tail”. The result (F1 + F2) points from the tail of the first vector to the tip of the second vector.
This is sometimes called the resultant vector R
F1
F2
R
Graphical Addition of Vectors
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 6
Vector Simulation
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 7
Examples
• Trig Table
• Vector Components
• Unit Vectors
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 8
Types of Vectors
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 9
Relative Displacement Vectors
C = A + B
C - A = B
Vector Addition
Vector Subtraction
B
is a relative displacement vector of point P3 relative to P2
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 10
Vector Addition via Parallelogram
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 11
Graphical Method of Vector Addition
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 12
Think of vector subtraction A B as A+(B), where the vector B has the same magnitude as B but points in the opposite direction.
Graphical Subtraction of Vectors
Vectors may be moved any way you please (to place them tip to tail) provided that you do not change their length nor rotate them.
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 13
Vector Components
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 14
Vector Components
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 15
Graphical Method of Vector Addition
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 16
Unit Vectors in Rectangular Coordinates
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 17
Vector Components in Rectangular Coordinates
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 18
x y z A = A i + A j+ A kˆ ˆ ˆ
x y z B = B i +B j + B kˆ ˆ ˆ
Vectors with Rectangular Unit Vectors
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 19
Dot Product - Scalar
The dot product multiplies the portion of A that is parallel to B with B
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 20
Dot Product - Scalar
The dot product multiplies the portion of A that is parallel to B with B
In 2 dimensions
In any number of dimensions
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 21
Cross Product - Vector
The cross product multpilies the portion of A that is perpendicular to B with B
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 22
x y z
x y z
i j kA A AB B B
ˆ ˆ ˆ
y z z y
x z z x
x y x y
= (A B - A B )i
+ (A B - A B ) j
+ (A B - A B ) k
ˆˆˆ
A B = A B sin( )
In 2 dimensions
In any number of dimensions
Cross Product - Vector
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 23
Velocity
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 24
y
x
ri rf
t
rvav Points in the direction of r
r
vi
The instantaneous velocity points tangent to the path.vf
A particle moves along the curved path as shown. At time t1 its position is ri and at time t2 its position is rf.
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 25
tt
rv lim0
velocityousInstantane
The instantaneous velocity is represented by the slope of a line tangent to the curve on the graph of an object’s position versus time.
t
rvav velocityAverage
txv x,av :be wouldcomponent - xThe
A displacement over an interval of time is a velocity
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 26
Acceleration
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 27
y
x
vi
ri rf
vf
A particle moves along the curved path as shown. At time t1 its position is r0 and at time t2 its position is rf.
v
Points in the direction of v.t
vaav
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 28
t
vaavonaccelerati Average
A nonzero acceleration changes an object’s state of motion
Δt 0
ΔvInstantaneous acceleration = a = limΔt
These have interpretations similar to vav and v.
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 29
Motion in a Plane with Constant Acceleration - Projectile
What is the motion of a struck baseball? Once it leaves the bat (if air resistance is negligible) only the force of gravity acts on the baseball.
Acceleration due to gravity has a constant value near the surface of the earth. We call it g = 9.8 m/s2
Only the vertical motion is affected by gravity
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 30
The baseball has ax = 0 and ay = g, it moves with constant velocity along the x-axis and with a changing velocity along the y-axis.
Projectile Motion
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 31
Example: An object is projected from the origin. The initial velocity components are vix = 7.07 m/s, and viy = 7.07 m/s.
Determine the x and y position of the object at 0.2 second intervals for 1.4 seconds. Also plot the results.
2f i iy y
f i ix
1Δy = y - y = v Δt + a Δt2
Δx = x - x = v Δt
Since the object starts from the origin, y and x will represent the location of the object at time t.
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 32
t (sec) x (meters) y (meters)0 0 0
0.2 1.41 1.220.4 2.83 2.040.6 4.24 2.480.8 5.66 2.521.0 7.07 2.171.2 8.48 1.431.4 9.89 0.29
Example continued:
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 33
0
2
4
6
8
10
12
0 0.5 1 1.5
t (sec)
x,y
(m)
This is a plot of the x position (black points) and y position (red points) of the object as a function of time.
Example continued:
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 34
Example continued:
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10
x (m)
y (m
)
This is a plot of the y position versus x position for the object (its trajectory). The object’s path is a parabola.
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 35
Example (text problem 3.50): An arrow is shot into the air with = 60° and vi = 20.0 m/s.
(a) What are vx and vy of the arrow when t = 3 sec?
The components of the initial velocity are:
m/s 3.17sinm/s 0.10cos
iiy
iix
vvvv
At t = 3 sec:m/s 1.12
m/s 0.10
tgvtavv
vtavv
iyyiyfy
ixxixfx
x
y
60°
vi
CONSTANT
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 36
(b) What are the x and y components of the displacement of the arrow during the 3.0 sec interval?
y
x
r
2x f i ix x ix
2 2y f i iy y iy
1Δr = Δx = x - x = v Δt + a Δt = v Δt + 0 = 30.0 m21 1Δr = Δy = y - y = v Δt + a Δt = v Δt - gΔt = 7.80 m2 2
Example continued:
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 37
Example: How far does the arrow in the previous example land from where it is released?
The arrow lands when y = 0. 021 2 tgtvy iy
Solving for t: sec 53.32
gv
t iy
The distance traveled is: ixΔx = v Δt = 35.3 m
iy1Δy = (v - gΔt) t = 02
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 38
Summary
• Adding and subtracting vectors (graphical method & component method)
• Velocity
• Acceleration
• Projectile motion (here ax = 0 and ay = g)
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MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 39
Projectiles Examples
• Problem solving strategy
• Symmetry of the motion
• Dropped from a plane
• The home run