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
qtan-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