Transcript Proportionality

Proportionality
Proportionality

If all other quantities are constant, two physics
quantities can be proportional to each other.
Proportionality
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
If all other quantities are constant, two physics
quantities can be proportional to each other.
The symbol that means “is proportional to” is _____?
Proportionality
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
If all other quantities are constant, two physics
quantities can be proportional to each other.
The symbol that means “is proportional to” is α
Proportionality

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If all other quantities are constant, two physics
quantities can be proportional to each other.
The symbol that means “is proportional to” is α
For example, if V α I, or “voltage” is proportional to
“current”, what does this mean?
Proportionality

If all other quantities are constant, two physics
quantities can be proportional to each other.

The symbol that means “is proportional to” is α

V α I means ...
As I increases by a multiplication factor of “ y “, then
V also increases by the same multiplication factor “y”
Or ... we can symbolize: As I ↑ y , then V ↑ y
Proportionality

If all other quantities are constant, two physics
quantities can be proportional to each other.

The symbol that means “is proportional to” is α

V α I means ...
As I increases by a multiplication factor of “ y “, then
V also increases by the same multiplication factor “y”
Or ... we can symbolize: As I ↑ y , then V ↑ y
Also ... As I divides by a factor “x”, V also divides by
the same factor “x”.
or
As I ↓ x , then V ↓ x
Graphing Proportionality
Graphing Proportionality
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If V α I , what does the graph of V vs I look like?
Graphing Proportionality
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If V α I , what does the graph of V vs I look like?
V
Linear graph
I
Graphing Proportionality

If V α I , what does the graph of V vs I look like?
V
Linear graph
I

If V α I ,What is the general equation relating V vs I ?
Graphing Proportionality

If V α I , what does the graph of V vs I look like?
V
Linear graph
I

If V α I , V = K I
Graphing Proportionality

If V α I , what does the graph of V vs I look like?
V
Linear graph
I
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If V α I , V = K I
K is just the slope of the graph. In physics, it is also
called the _____________ of ________________.
Graphing Proportionality
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If V α I , what does the graph of V vs I look like?
V
Linear graph
I
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If V α I , V = K I
K is just the slope of the graph. In physics, it is also
called the constant of proportionality.
Converting between a Proportionality
Relationship and an Equation
Converting between a Proportionality
Relationship and an Equation
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We determined that if V α I , then V = ?
Converting between a Proportionality
Relationship and an Equation

We determined that if V α I , then V = K I
Converting between a Proportionality
Relationship and an Equation
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
We determined that if V α I , then V = K I
Rule: We can change a proportionality relationship to
an equation by changing the α to an ______ sign and
inserting a __________ of _________________.
Converting between a Proportionality
Relationship and an Equation
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We determined that if V α I , then V = K I
Rule: We can change a proportionality relationship to
an equation by changing the α to an equal sign and
inserting a constant of proportionality.
Converting between a Proportionality
Relationship and an Equation

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
We determined that if V α I , then V = K I
Rule: We can change a proportionality relationship to
an equation by changing the α to an equal sign and
inserting a constant of proportionality.
Try this! Convert this proportionality relationship
|a| α |Fnet| to an equation.
Converting between a Proportionality
Relationship and an Equation
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We determined that if V α I , then V = K I
Rule: We can change a proportionality relationship to
an equation by changing the α to an equal sign and
inserting a constant of proportionality.
|a| α |Fnet|
↔
|a| =
K|Fnet|
Converting between a Proportionality
Relationship and an Equation
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
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We determined that if V α I , then V = K I
Rule: We can change a proportionality relationship to
an equation by changing the α to an equal sign and
inserting a constant of proportionality.
|a| α |Fnet|
↔
|a| =
K|Fnet|
Do you know what physics quantity K is ?
Converting between a Proportionality
Relationship and an Equation
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
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We determined that if V α I , then V = K I
Rule: We can change a proportionality relationship to
an equation by changing the α to an equal sign and
inserting a constant of proportionality.
|a| α |Fnet|
↔
|a| =
K|Fnet|
K is 1/m in the case of Newton's Second Law
Converting between a Proportionality
Relationship and an Equation
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Try this! Convert this proportionality relationship
a α v2 to an equation.
Converting between a Proportionality
Relationship and an Equation
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a α v2 ↔
a=
Kv2
Converting between a Proportionality
Relationship and an Equation
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a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?
Converting between a Proportionality
Relationship and an Equation
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a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?
As v ↑ x , v2 ↑ x2 , and a ↑ x2 or ...
Converting between a Proportionality
Relationship and an Equation
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
a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?

As v ↑ x , v2 ↑ x2 , and a ↑ x2
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If v ↑ 3 , v2 ↑ 32 , and a ↑ 32
Converting between a Proportionality
Relationship and an Equation


a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?

As v ↑ x , v2 ↑ x2 , and a ↑ x2

If v ↑ 3 , v2 ↑ 32 , and a ↑ 32 or ...
Converting between a Proportionality
Relationship and an Equation


a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?

As v ↑ x , v2 ↑ x2 , and a ↑ x2
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If v ↑ 3 , v2 ↑ 32 , and a ↑ 32 or ...
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If v ↓ 4 , then v2 ? , and then a ?
Converting between a Proportionality
Relationship and an Equation


a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?

As v ↑ x , v2 ↑ x2 , and a ↑ x2

If v ↑ 3 , v2 ↑ 32 , and a ↑ 32 or ...

If v ↓ 4 , then v2 ↓ 42 , and then a ?
Converting between a Proportionality
Relationship and an Equation


a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?

As v ↑ x , v2 ↑ x2 , and a ↑ x2

If v ↑ 3 , v2 ↑ 32 , and a ↑ 32 or ...

If v ↓ 4 , then v2 ↓ 42 , and then a ↓ 42
Converting between a Proportionality
Relationship and an Equation


a α v2 ↔
a=
Kv2
In this case, we say a is directly proportional to v2.
What does this mean?

As v ↑ x , v2 ↑ x2 , and a ↑ x2

If v ↑ 3 , v2 ↑ 32 , and a ↑ 32 or ...
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If v ↓ 4 , then v2 ↓ 42 , and then a ↓ 42
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Note: “Proportional to” or “directly proportional to” is
like “monkey-see monkey do”. In this case, if v2
changes by a factor, a changes by the same factor.
Inverse proportion
Inverse proportion
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If E is inversely proportional to r , what does this
mean?
Inverse proportion
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E inversely proportional to r means ...
As r ↑ x , E ↓ x
Inverse proportion
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E inversely proportional to r means ...
As r ↑ x , E ↓ x
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If E is inversely proportional to r, and variable r was
increased by a multiplication factor of 13, say, how
would E change ?
Inverse proportion

E inversely proportional to r means ...
As r ↑ x , E ↓ x
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If E is inversely proportional to r, and variable r was
increased by a multiplication factor of 13, say, how
would E change ? E would divide by 13
Inverse proportion

E inversely proportional to r means ...
As r ↑ x , E ↓ x
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If E is inversely proportional to r, and variable r was
increased by a multiplication factor of 13, say, how
would E change ? E would divide by 13
As r ↓ 37 , how would E change if E and r are inversely
proportional?
Inverse proportion
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E inversely proportional to r means ...
As r ↑ x , E ↓ x
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If E is inversely proportional to r, and variable r was
increased by a multiplication factor of 13, say, how
would E change ? E would divide by 13
As r ↓ 37 , E ↑ 37
Inverse proportion

E inversely proportional to r means ...
As r ↑ x , E ↓ x
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If E is inversely proportional to r, and variable r was
increased by a multiplication factor of 13, say, how
would E change ? E would divide by 13
As r ↓ 37 , E ↑ 37
How can we use symbols to state that E is “inversely
proportional to” r ?
Inverse proportion

E inversely proportional to r means ...
As r ↑ x , E ↓ x
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
If E is inversely proportional to r, and variable r was
increased by a multiplication factor of 13, say, how
would E change ? E would divide by 13
As r ↓ 37 , E ↑ 37
E is “inversely proportional to” r can be written
symbolically: E α 1/r
The inverse square law
The inverse square law
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A common proportionality relationship in physics is
called “the inverse square law”
The inverse square law


A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects.
The inverse square law


A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects. Note that the
two words “inversely” and “square” imply that the
inverse square law is being used.
The inverse square law


A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects. Note that the
two words “inversely” and “square” imply that the
inverse square law is being used. How can we write
this proportionality relationship symbolically?
The inverse square law



A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects. Note that the
two words “inversely” and “square” imply that the
inverse square law is being used.
Fg α 1/d2
The inverse square law



A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects. Note that the
two words “inversely” and “square” imply that the
inverse square law is being used.
Fg α 1/d2 What equation does this correspond to?
The inverse square law
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

A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects. Note that the
two words “inversely” and “square” imply that the
inverse square law is being used.
Fg α 1/d2
↔
Fg = C/d2
The inverse square law




A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects. Note that the
two words “inversely” and “square” imply that the
inverse square law is being used.
Fg α 1/d2
↔
Fg = C/d2
What does Fg α 1/d2 mean?
The inverse square law


A common proportionality relationship in physics is
called “the inverse square law”
For example Fg is inversely proportional to the square
of the distance d between two objects. Note that the
two words “inversely” and “square” imply that the
inverse square law is being used.

Fg α 1/d2
↔

Fg α 1/d2 means... as d ↑ x, d2 ↑ x2, and then Fg ↓ x2
Fg = C/d2
Review on Proportionality ☺
Review on Proportionality ☺

If Fg α 1/d2 , and d multiplies by 5, how does Fg
change?
Review on Proportionality ☺

If Fg α 1/d2 , and d multiplies by 5, how does Fg
change? As d ↑ 5
d2 ↑ 52 or 25
Fg ↓ 25
Fg divides by 25 !
Review on Proportionality ☺

If Fg α 1/d2 , and d multiplies by 5, how does Fg
change? As d ↑ 5
d2 ↑ 52 or 25
Fg ↓ 25

Fg divides by 25 !
If a is inversely proportional to m, write the
proportionality relationship in symbols and then write
the corresponding equation.
Review on Proportionality ☺

If Fg α 1/d2 , and d multiplies by 5, how does Fg
change? As d ↑ 5
d2 ↑ 52 or 25
Fg ↓ 25


Fg divides by 25 !
If a is inversely proportional to m, write the
proportionality relationship in symbols and then write
the corresponding equation.
a α 1/m
↔
a = k/m
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
Isolate the constant variable:
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
Isolate the constant variable: ac = v2/r
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
Isolate the constant variable: ac = v2/r
Rewrite the equation with a proportionality
constant:
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
Isolate the constant variable: ac = v2/r
Rewrite the equation with a proportionality
constant: ac = (1/r)v2 or ac = k v2 where
k=1/r
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
Isolate the constant variable: ac = v2/r
Rewrite the equation with a proportionality
constant: ac = (1/r)v2 or ac = k v2 where
k=1/r
In symbols, write the proportionality:
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
Isolate the constant variable: ac = v2/r
Rewrite the equation with a proportionality
constant: ac = (1/r)v2 or ac = k v2 where
k=1/r
In symbols, write the proportionality:
ac α v2
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
Equation with r, ac, and v ?
ac = v2/r
Isolate the constant variable: ac = v2/r
Rewrite the equation with a proportionality
constant: ac = (1/r)v2 or ac = k v2 where
k=1/r
In symbols, write the proportionality:
ac α v2 or acceleration is directly proportional to the speed squared
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
ac α v2
Now use the proportionality to find how ac changes.
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
ac α v2
Now use the proportionality to find how ac changes.
As v ↑ 7
v2 ↑ ?
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
ac α v2
Now use the proportionality to find how ac changes.
As v ↑ 7
v2 ↑ 72 or 49
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
ac α v2
Now use the proportionality to find how ac changes.
As v ↑ 7
v2 ↑ 72 or 49
and ac changes how?
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
ac α v2
Now use the proportionality to find how ac changes.
As v ↑ 7
v2 ↑ 72 or 49
and ac ↑ 49
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(a) The radius is kept constant, but the speed is
increased by a multiplication factor of 7.
ac α v2
Now use the proportionality to find how ac changes.
As v ↑ 7
v2 ↑ 72 or 49
and ac ↑ 49
Therefore, the centripetal acceleration multiplies by 49
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ?
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
Isolate the constant variable:
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
Isolate the constant variable: ac = 4π2r/ T2
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
Isolate the constant variable: ac = 4π2r/ T2
Rewrite the equation with a proportionality
constant:
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
Isolate the constant variable: ac = 4π2r/ T2
Rewrite the equation with a proportionality
constant: ac = k/ T2
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
Isolate the constant variable: ac = 4π2r/ T2
Rewrite the equation with a proportionality
constant: ac = k/ T2
Note k = 4π2r
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
Isolate the constant variable: ac = 4π2r/ T2
Rewrite the equation with a proportionality
constant: ac = k/ T2
Note k = 4π2r
In symbols, write the proportionality:
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
Equation with r, ac, and T ? ac = 4π2r/ T2
Isolate the constant variable: ac = 4π2r/ T2
Rewrite the equation with a proportionality
constant: ac = k/ T2
Note k = 4π2r
In symbols, write the proportionality: ac α 1/ T2
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
We have ac α 1/ T2
How would you state the above proportionality in
words?
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
We have ac α 1/ T2
Acceleration is inversely proportional to the
square of the period.
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
We have ac α 1/ T2
Acceleration is inversely proportional to the
square of the period.
If T ↓ 3
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
We have ac α 1/ T2
Acceleration is inversely proportional to the
square of the period.
If T ↓ 3
Then T2 will change how?
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
We have ac α 1/ T2
Acceleration is inversely proportional to the
square of the period.
If T ↓ 3
Then T2 ↓ 32 or 9
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
We have ac α 1/ T2
Acceleration is inversely proportional to the
square of the period.
If T ↓ 3
Then T2 ↓ 32 or 9
And ac will change how?
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(b) The radius is kept constant, but the period is
decreased by a factor of 3.
We have ac α 1/ T2
Acceleration is inversely proportional to the
square of the period.
If T ↓ 3
Then T2 ↓ 32 or 9
And ac ↑ 9
The acceleration multiplies by 9!
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(c) The radius is kept constant but the frequency
quadruples (multiplies by four) You try this one:
UCM proportionality example #1: How and by what factor does
the centripetal acceleration change if the following changes are
made to an object undergoing UCM?
(c) The radius is kept constant but the frequency
quadruples (multiplies by four) You try this one:
ac = 4π2r f 2
f ↑ 4
ac = 4π2r f 2
f 2 ↑ 42 or 16
ac = k f 2
ac ↑ 42 or 16
ac α f 2
The centripetal acceleration
multiplies by 16 !
Try this for Practice!

Showing all steps as learned in class, use proportionality
methods to determine how and by what factor the centripetal
acceleration changes if these changes are made...
(a) The radius is kept constant but the period quadruples
(multiplies by four) Check answer: Divides by 16
(b) The frequency is kept constant but the radius is tripled.
Check answer: multiplies by 3
(a) The speed is kept constant but the radius is halved.
Check answer: multiplies by 2
(a) The radius is kept constant, but the speed multiplies by
eight. Check answer: multiplies by 64