Transcript Proportionality - msamandakeller

Proportionality
SPH4U
Introduction
In physics, we are often interested in how one
variable affects another.
Introduction
In physics, we are often interested in how one
variable affects another.
Example:
If you double the voltage supplied to a circuit,
what happens to the current?
Voltage and Current
Given:
V
I
R
Voltage and Current
Given:
V
I
R
We can see that
if this doubles
Voltage and Current
Given:
V
I
R
this also doubles
We can see that
if this doubles
The “Fishy”
We can say that:
I V
“Current is directly proportional to voltage.”
The “Fishy”
We can say that:
I V
To turn an equation
into a proportionality,
set everything else in
the equation = 1.
“Current is directly proportional to voltage.”
Let’s show this algebraically:
Initial:
V1
I1 
R
Final:
V2 2V1
 V1 
I2 

 2   2 I1
R
R
R
Let’s show this algebraically:
Initial:
Final:
V1
I1 
R
Note that resistance
is constant.
It is our controlled
V2 2V1variable.
 V1 
I2 

 2   2 I1
R
R
R
Let’s show this algebraically:
Initial:
V1
I1 
R
Substitute for
what’s changed
Final:
V2 2V1
 V1 
I2 

 2   2 I1
R
R
R
Let’s show this algebraically:
Rearrange the
expression
until a part of
it equals the
original
Initial:
V1
I1 
R
Final:
V2 2V1
 V1 
I2 

 2   2 I1
R
R
R
Let’s show this algebraically:
Initial:
V1
I1 
R
Final:
V2 2V1
 V1 
I2 

 2   2 I1
R
R
R
Experimentally
How can you answer the question:
If you double the voltage supplied to a circuit,
what happens to the current?
experimentally?
The Graph
Set up your circuit,
change your voltage
(independent variable),
measure your current
(dependent variable),
and graph your data.
The Graph
Set up your circuit,
change your voltage
(independent variable),
measure your current
(dependent variable),
and graph your data.
The Graph
Set up your circuit,
change your voltage
(independent variable),
measure your current
(dependent variable),
and graph your data.
What is the slope
of this line?
Slope
Until we determine the relationship between
current and all other variables, we just call it k,
the “proportionality constant.”
I  V  I  kV
Slope
Until we determine the relationship between
current and all other variables, we just call it k,
the “proportionality constant.”
I  V  I  kV
1
(In this example, k should equal .)
R
An Inverse Relationship
Consider now the question:
If you double the resistance of a circuit, what
happens to the current?
V
I
R
The “Fishy”
We can say that:
1
I
R
“Current is inversely proportional to resistance.”
An Inverse Relationship
Initial:
Final:
V
I1 
R1
V
V
I2 

R2 2 R1
Note that this
time, voltage is
our controlled


V
variable.
 1  1
2
R 
 1
2
I1
An Inverse Relationship
Initial:
Final:
V
I1 
R1


V
V
V
I2 

 12    12 I1
R2 2 R1
 R1 
An Inverse Relationship
Initial:
Final:
V
I1 
R1


V
V
V
I2 

 12    12 I1
R2 2 R1
 R1 
An Inverse Relationship
Initial:
Final:
V
I1 
R1


V
V
V
I2 

 12    12 I1
R2 2 R1
 R1 
If the resistance is doubled,
the current is halved.
The Graph
The graph of this relationship would look like:
The Graph
The graph of this relationship would look like:
How do we know this is actually 1/R and
not some similar relationship like 1/R2?
And what’s our proportionality constant k?
The Revised Graph
We graph I vs. 1/R and see if it’s a straight line:
1 k
I  k  
( where k  V )
R R
Inverse Square
An inverse square relationship looks similar.
1
FG  2
r
You might also see powers
For example, if you were measuring the distance
Dd an object travels while accelerating from
rest in time Dt :
Slope = ½ a
You might also see roots
For example, if you were measuring the work W
required to accelerate a mass to a speed v: