Physics 207: Lecture 2 Notes
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Transcript Physics 207: Lecture 2 Notes
Lecture 4
Today:
Ch. 3 (all) & Ch. 4 (start)
Perform vector algebra (addition and subtraction)
Interconvert between Cartesian and Polar coordinates
Work with 2D motion
• Deconstruct motion into x & y or parallel & perpendicular
• Obtain velocities
• Obtain accelerations
Deduce components parallel and
perpendicular to the trajectory path
Physics 207: Lecture 4, Pg 1
Coordinate Systems and vectors
In 1 dimension, only 1 kind of system,
Linear Coordinates
(x)
+/-
In 2 dimensions there are two commonly used systems,
Cartesian Coordinates
(x,y)
Circular Coordinates
(r,q)
In 3 dimensions there are three commonly used systems,
Cartesian Coordinates
(x,y,z)
Cylindrical Coordinates (r,q,z)
Spherical Coordinates
(r,q,f)
Physics 207: Lecture 4, Pg 5
Vectors look like...
There are two common ways of indicating that
something is a vector quantity:
Boldface notation: A
A or A
“Arrow” notation:
A
Physics 207: Lecture 4, Pg 7
Vectors act like…
Vectors have both magnitude and a direction
Vectors: position, displacement, velocity, acceleration
Magnitude of a vector A ≡ |A|
For vector addition or subtraction we can shift vector
position at will (NO ROTATION)
Two vectors are equal if their directions, magnitudes & units
match.
A=C
A
B
C
A ≠B, B ≠ C
Physics 207: Lecture 4, Pg 8
Scalars
A scalar is an ordinary number.
A magnitude ( + or - ), but no direction
May have units (e.g. kg) but can be just a number
No bold face and no arrow on top.
The product of a vector and a scalar is another vector
in the same “direction” but with modified magnitude
A = -0.75 B
A
B
Physics 207: Lecture 4, Pg 9
Vectors and 2D vector addition
The sum of two vectors is another vector.
B
C
A =B+C
B
A
C
Physics 207: Lecture 4, Pg 11
2D Vector subtraction
Vector subtraction can be defined in terms of addition.
B-C
= B + (-1)C
B
C
Physics 207: Lecture 4, Pg 12
2D Vector subtraction
Vector subtraction can be defined in terms of addition.
B-C
= B + (-1)C
B
B-C
-C
-C
B
B+C
Different direction
and magnitude !
Physics 207: Lecture 4, Pg 13
Unit Vectors
A Unit Vector points : a length 1 and no units
Gives a direction.
Unit vector u points in the direction of U
Often denoted with a “hat”: u = û
U = |U| û
û
Useful examples are the cartesian
unit vectors [ i, j, k ] or
Point in the direction of the [ xˆ , yˆ , zˆ ]
x, y and z axes.
R = rx i + ry j + rz k
k
or
z
R=xi+yj+zk
y
j
i
x
Physics 207: Lecture 4, Pg 14
Vector addition using components:
Consider, in 2D, C = A + B.
(a) C = (Ax i + Ay j ) + (Bx i + By j ) = (Ax + Bx )i + (Ay + By )
(b) C = (Cx i + Cy j )
Comparing components of (a) and (b):
Cx = Ax + Bx
Cy = Ay + By
|C| =[ (Cx)2+ (Cy)2 ]1/2
C
B
A
Ay
By
Bx
Ax
Physics 207: Lecture 4, Pg 15
Example
Vector Addition
Vector A = {0,2,1}
Vector B = {3,0,2}
Vector C = {1,-4,2}
What is the resultant vector, D, from adding A+B+C?
A.
B.
C.
D.
{3,-4,2}
{4,-2,5}
{5,-2,4}
None of the above
Physics 207: Lecture 4, Pg 16
Converting Coordinate Systems (Decomposing vectors)
In polar coordinates the vector R = (r,q)
In Cartesian the vector R = (rx,ry) = (x,y)
We can convert between the two as follows:
rx x r cos q
y
ry y r sin q
r x ˆi y ˆj
r x y
2
(x,y)
ry
q
rx
2
qtan-1 ( y / x )
• In 3D
r
r x y z
2
2
x
2
Physics 207: Lecture 4, Pg 17
Decomposing vectors into components
A mass on a frictionless inclined plane
A block of mass m slides down a frictionless ramp
that makes angle q with respect to horizontal.
What is its acceleration a ?
m
a
q
Physics 207: Lecture 4, Pg 18
Decomposing vectors into components
A mass on a frictionless inclined plane
A block of mass m slides down a frictionless ramp
that makes angle q with respect to horizontal.
What is its acceleration a ?
yˆ
m
g sin q
q
g=-gj q
-g cos q
xˆ
Physics 207: Lecture 4, Pg 19
Motion in 2 or 3 dimensions
Position
ri , ti and rf , t f
Displacement
r rf ri
r
Velocity (avg.)
v avg.
t
v
Acceleration (avg.)
a avg.
t
Physics 207: Lecture 4, Pg 20
Instantaneous Velocity
But how we think about requires knowledge of the path.
The direction of the instantaneous velocity is along a line
that is tangent to the path of the particle’s direction of
motion.
v
Physics 207: Lecture 4, Pg 22
Average Acceleration
The average acceleration of particle motion reflects
changes in the instantaneous velocity vector (divided
by the time interval during which that change occurs).
Instantaneous
acceleration
a
Physics 207: Lecture 4, Pg 23
Tangential ║ & radial acceleration
aT a||
ar a
v
a
2
2
a ar aT
a = a + a
Acceleration outcomes:
If parallel changes in the magnitude of
(speeding up/ slowing down)
Perpendicular changes in the direction of
(turn left or right)
v
v
Physics 207: Lecture 4, Pg 25
Kinematics
In 2-dim. position, velocity, and acceleration of a particle:
r= xi +y j
v = vx i + vy j (i , j unit vectors )
a = ax i + ay j
x x(t )
y y (t )
dx
vx
dt
d 2x
ax 2
dt
dy
vy
dt
d2y
ay 2
dt
with, if constant x accel. : x(t ) x0 v x t 12 ax t 2
0
with, if constant y accel. : y (t ) y0 v y t 12 a y t 2
0
All this complexity is hidden away in
r = r(t)
v = dr / dt
a = d2r / dt2
Physics 207: Lecture 4, Pg 26
Special Case
Throwing an object with x along the
horizontal and y along the vertical.
x and y motion both coexist and t is common to both
Let g act in the –y direction, v0x= v0 and v0y= 0
x vs t
x
y
y vs t
t=0
y
0
4
t
0
4
t
x vs y
4
x
Physics 207: Lecture 4, Pg 28
Another trajectory
Can you identify the dynamics in this picture?
How many distinct regimes are there?
Are vx or vy = 0 ? Is vx >,< or = vy ?
t=0
x vs y
y
t =10
x
Physics 207: Lecture 4, Pg 29
Another trajectory
Can you identify the dynamics in this picture?
How many distinct regimes are there?
0<t<3
I.
3<t<7
7 < t < 10
vx = constant = v0 ; vy = 0
II. vx = vy = v0
t=0
III. vx = 0 ; vy = constant < v0
x vs y
What can you say about the
acceleration?
y
t =10
x
Physics 207: Lecture 4, Pg 30
Exercises 1 & 2
Trajectories with acceleration
A rocket is drifting sideways (from left to right) in deep
space, with its engine off, from A to B. It is not near any
stars or planets or other outside forces.
Its “constant thrust” engine (i.e., acceleration is constant) is
fired at point B and left on for 2 seconds in which time the
rocket travels from point B to some point C
Sketch the shape of the path
from B to C.
At point C the engine is turned off.
Sketch the shape of the path
after point C (Note: a = 0)
Physics 207: Lecture 4, Pg 31
Exercise 1
Trajectories with acceleration
B
From B to C ?
A.
B.
C.
D.
E.
A
B
C
D
None of these
A
B
C
B
B
C
C
B
C
D
C
Physics 207: Lecture 4, Pg 32
Exercise 2
Trajectories with acceleration
After C ?
A.
B.
C.
D.
E.
A
B
C
D
None of these
C
C
A
B
C
C
C
D
Physics 207: Lecture 4, Pg 33
Exercise 2
Trajectories with acceleration
After C ?
A.
B.
C.
D.
E.
A
B
C
D
None of these
C
C
A
B
C
C
C
D
Physics 207: Lecture 4, Pg 34
Lecture 4
Assignment: Read all of Chapter 4, Ch. 5.1-5.3
Physics 207: Lecture 4, Pg 35