Physics 207, Lecture 4, Sept. 15

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Physics 207, Lecture 4, Sept. 15

• Goals for Chapts. 3 4
• Perform vector algebra (addition subtraction)
  graphically or by x,y z components
• Interconvert between Cartesian and Polar
  coordinates
• Distinguish position-time graphs from particle
  trajectory plots
• Obtain particle velocities and accelerations
  from trajectory plots
• Determine both magnitude and acceleration
  parallel and perpendicular to the trajectory
  path
• Solve problems with multiple accelerations in
  2-dimensions (including linear, projectile and
  circular motion)
• Discern different reference frames and
  understand how they relate to particle motion in
  stationary and moving frames

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Physics 207, Lecture 4, Sept. 15

• Assignment Read Chapter 5 (sections 1- 4
  carefully)
• MP Problem Set 2 due Wednesday (should have
  started)
• MP Problem Set 3, Chapters 4 and 5 (available
  today)

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Vector addition

• The sum of two vectors is another vector.

A B C
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Vector subtraction

• Vector subtraction can be defined in terms of
  addition.

B (-1)C
B - C
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Unit Vectors

• A Unit Vector is a vector having length 1 and no
  units
• It is used to specify a direction.
• Unit vector u points in the direction of U
• Often denoted with a hat u û

• Useful examples are the cartesian unit vectors
  i, j, k
• Point in the direction of the x, y and z axes.
• R rx i ry j rz k

y
j
x
i
k
z
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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

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Exercise 1Vector Addition

• Vector A 0,2,1
• Vector B 3,0,2
• Vector C 1,-4,2

What is the resultant vector, D, from adding
ABC?

1. 3,-4,2
2. 4,-2,5
3. 5,-2,4
4. None of the above

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Converting Coordinate Systems

• 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

y
(x,y)
r
ry
?
rx
x

• In 3D cylindrical coordinates (r,q,z), r is the
  same as the magnitude of the vector in the x-y
  plane sqrt(x2 y2)

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Exercise Frictionless inclined plane

• A block of mass m slides down a frictionless ramp
  that makes angle ? with respect to horizontal.
  What is its acceleration a ?

m
a
?
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Resolving vectors, little g the inclined plane

• g (bold face, vector) can be resolved into its
  x,y or x,y components
• g - g j
• g - g cos q j g sin q i
• The bigger the tilt the faster the
  acceleration..
• along the incline

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Dynamics II Motion along a line but with a
twist(2D dimensional motion, magnitude and
directions)

• Particle motions involve a path or trajectory
• Recall instantaneous velocity and acceleration

• These are vector expressions reflecting x, y z
  motion
• r r(t) v dr / dt a d2r / dt2

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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
  particles direction of motion.

• The magnitude of the instantaneous velocity
  vector is the speed, s. (Knight uses v)
• s (vx2 vy2 vz )1/2

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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).

• The average acceleration is a vector quantity
  directed along ?v
• ( a vector! )

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Instantaneous Acceleration

• The instantaneous acceleration is the limit of
  the average acceleration as ?v/?t approaches zero

• The instantaneous acceleration is a vector with
  components parallel (tangential) and/or
  perpendicular (radial) to the tangent of the path

• Changes in a particles path may produce an
  acceleration
• The magnitude of the velocity vector may change
• The direction of the velocity vector may
  change
• (Even if the magnitude remains constant)
• Both may change simultaneously (depends path vs
  time)

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Motion along a path ( displacement, velocity,
acceleration )

• 2 or 3D Kinematics vector equations
• r r(Dt) v dr / dt a d2r / dt2

Velocity
y
path
vav ?r / ?t
v dr / dt
Acceleration
aav ?v / ?t
x
a dv / dt
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Generalized motion with non-zero acceleration
need both path time
Two possible options
Change in the magnitude of
Change in the direction of
Animation
Q1 What is the time sequence in this particles
motion? Q2 Is the particle speeding up or
slowing down?
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Examples of motion Chemotaxis
An example..

• Q1 How do single cell animals locate their
  food ?
• Possible mechanism Tumble and run paradigm.
• Q2 How does a fly find its meal?
• Possible mechanism If you smell it fly into the
  wind, if you dont fly across the wind.

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Kinematics

• The position, velocity, and acceleration of a
  particle in
• 3-dimensions can be expressed as
• r x i y j z k
• v vx i vy j vz k (i , j , k
  unit vectors )
• a ax i ay j az k

• All this complexity is hidden away in
• r r(Dt) v dr / dt a d2r / dt2

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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 y
y vs t
t 0
y
4
y
x
t
4
0
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Another trajectory
Can you identify the dynamics in this
picture? How many distinct regimes are there? Are
vx or vy 0 ? Is vx gt,lt or vy ?
x vs y
t 0
y
t 10
x
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Another trajectory

• Can you identify the dynamics in this picture?
• How many distinct regimes are there?
• 0 lt t lt 3 3 lt t lt 7 7 lt t lt 10
• I. vx constant v0 vy 0
• II. vx vy v0
• III. vx 0 vy constant lt v0

x vs y
t 0
What can you say about the acceleration?
y
t 10
x
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Exercise 2 3Trajectories 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

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Exercise 2Trajectories with acceleration
From B to C ?
A
B

1. A
2. B
3. C
4. D
5. None of these

C
D
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Exercise 3Trajectories with acceleration
C
C
After C ?
A
B

1. A
2. B
3. C
4. D
5. None of these

C
C
C
D
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Trajectory with constant acceleration along the
vertical
How does the trajectory appear to different
observers ? What if the observer is moving with
the same x velocity (i.e. running in parallel)?
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Trajectory with constant acceleration along the
vertical
This observer will only see the y motion
x
In an inertial reference frame all see the same
acceleration
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Trajectory with constant acceleration along the
vertical
What do the velocity and acceleration vectors
look like? Velocity vector is always tangent
to the curve! Acceleration may or may not be!
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Home Exercise The Pendulum

• Which statement best describes the
• motion of the pendulum bob at the
• instant of time drawn ?
• when the bob is at the top of its swing.
• which quantities are non-zero ?

1
q 30
C) vr 0 ar ? 0 vT 0 aT ? 0
A) vr 0 ar 0 vT 0 aT ? 0
B) vr 0 ar ? 0 vT 0 aT 0
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Home Exercise The Pendulum Solution
O
NOT uniform circular motion If circular motion
then ar not zero, If speed is increasing so aT
not zero
q 30
T
However, at the top of the swing the bob
temporarily comes to rest, so v 0 and the net
tangential force is mg sin q
a
mg
aT
C) vr 0 ar 0 vT 0 aT ? 0
Everywhere else the bob has a non-zero velocity
and so then (except at the bottom of the swing)
vr 0 ar ? 0 vT ? 0 aT ? 0
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Physics 207, Lecture 4, Sept. 15

• Assignment Read Chapter 5 (sections 1- 4
  carefully)
• MP Problem Set 2 due Wednesday (should have
  started)
• MP Problem Set 3, Chapters 4 and 5 (available
  today)