Proportionality

1
Proportionality
2
Proportionality

• If all other quantities are constant, two physics
  quantities can be proportional to each other.

3
Proportionality

• If all other quantities are constant, two physics
  quantities can be proportional to each other.
• The symbol that means is proportional to is
  _____?

4
Proportionality

• If all other quantities are constant, two physics
  quantities can be proportional to each other.
• The symbol that means is proportional to is a

5
Proportionality

• If all other quantities are constant, two physics
  quantities can be proportional to each other.
• The symbol that means is proportional to is a
• For example, if V a I, or voltage is
  proportional to current, what does this mean?

6
Proportionality

• If all other quantities are constant, two physics
  quantities can be proportional to each other.
• The symbol that means is proportional to is a
• V a 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

7
Proportionality

• If all other quantities are constant, two physics
  quantities can be proportional to each other.
• The symbol that means is proportional to is a
• V a 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

8
Graphing Proportionality
9
Graphing Proportionality

• If V a I , what does the graph of V vs I look
  like?

10
Graphing Proportionality

• If V a I , what does the graph of V vs I look
  like?

V
Linear graph
I
11
Graphing Proportionality

• If V a I , what does the graph of V vs I look
  like?
• If V a I ,What is the general equation relating V
  vs I ?

V
Linear graph
I
12
Graphing Proportionality

• If V a I , what does the graph of V vs I look
  like?
• If V a I , V K I

V
Linear graph
I
13
Graphing Proportionality

• If V a I , what does the graph of V vs I look
  like?
• If V a I , V K I
• K is just the slope of the graph. In physics, it
  is also called the _____________ of
  ________________.

V
Linear graph
I
14
Graphing Proportionality

• If V a I , what does the graph of V vs I look
  like?
• If V a I , V K I
• K is just the slope of the graph. In physics, it
  is also called the constant of proportionality.

V
Linear graph
I
15
Converting between a Proportionality Relationship
and an Equation
16
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V ?

17
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V K I

18
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V K I
• Rule We can change a proportionality
  relationship to an equation by changing the a to
  an ______ sign and inserting a __________ of
  _________________.

19
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V K I
• Rule We can change a proportionality
  relationship to an equation by changing the a to
  an equal sign and inserting a constant of
  proportionality.

20
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V K I
• Rule We can change a proportionality
  relationship to an equation by changing the a to
  an equal sign and inserting a constant of
  proportionality.
• Try this! Convert this proportionality
  relationship
• a a Fnet to an equation.

21
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V K I
• Rule We can change a proportionality
  relationship to an equation by changing the a to
  an equal sign and inserting a constant of
  proportionality.
• a a Fnet ? a KFnet

22
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V K I
• Rule We can change a proportionality
  relationship to an equation by changing the a to
  an equal sign and inserting a constant of
  proportionality.
• a a Fnet ? a KFnet
• Do you know what physics quantity K is ?

23
Converting between a Proportionality Relationship
and an Equation

• We determined that if V a I , then V K I
• Rule We can change a proportionality
  relationship to an equation by changing the a to
  an equal sign and inserting a constant of
  proportionality.
• a a Fnet ? a KFnet
• K is 1/m in the case of Newton's Second Law

24
Converting between a Proportionality Relationship
and an Equation

• Try this! Convert this proportionality
  relationship
• a a v2 to an equation.

25
Converting between a Proportionality Relationship
and an Equation

• a a v2 ? a Kv2

26
Converting between a Proportionality Relationship
and an Equation

• a a v2 ? a Kv2
• In this case, we say a is directly proportional
  to v2. What does this mean?

27
Converting between a Proportionality Relationship
and an Equation

• a a 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 ...

28
Converting between a Proportionality Relationship
and an Equation

• a a 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

29
Converting between a Proportionality Relationship
and an Equation

• a a 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 ...

30
Converting between a Proportionality Relationship
and an Equation

• a a 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 ? , and then a ?

31
Converting between a Proportionality Relationship
and an Equation

• a a 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 ?

32
Converting between a Proportionality Relationship
and an Equation

• a a 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

33
Converting between a Proportionality Relationship
and an Equation

• a a 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
• 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.

34
Inverse proportion
35
Inverse proportion

• If E is inversely proportional to r , what does
  this mean?

36
Inverse proportion

• E inversely proportional to r means ...
• As r ? x , E ? x

37
Inverse proportion

• E inversely proportional to r means ...
• As r ? x , E ? x
• If E is inversely proportional to r, and variable
  r was increased by a multiplication factor of 13,
  say, how would E change ?

38
Inverse proportion

• E inversely proportional to r means ...
• As r ? x , E ? x
• 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

39
Inverse proportion

• E inversely proportional to r means ...
• As r ? x , E ? x
• 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?

40
Inverse proportion

• E inversely proportional to r means ...
• As r ? x , E ? x
• 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

41
Inverse proportion

• E inversely proportional to r means ...
• As r ? x , E ? x
• 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 ?

42
Inverse proportion

• E inversely proportional to r means ...
• As r ? x , E ? x
• 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 a 1/r

43
The inverse square law
44
The inverse square law

• A common proportionality relationship in physics
  is called the inverse square law

45
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.

46
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.

47
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?

48
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 a 1/d2

49
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 a 1/d2 What equation does this correspond
  to?

50
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 a 1/d2 ? Fg C/d2

51
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 a 1/d2 ? Fg C/d2
• What does Fg a 1/d2 mean?

52
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 a 1/d2 ? Fg C/d2
• Fg a 1/d2 means... as d ? x, d2 ? x2, and
  then Fg ? x2

53
Review on Proportionality ?
54
Review on Proportionality ?

• If Fg a 1/d2 , and d multiplies by 5, how does
  Fg change?

55
Review on Proportionality ?

• If Fg a 1/d2 , and d multiplies by 5, how does
  Fg change? As d ? 5
• d2 ? 52 or 25
• Fg ? 25 Fg
  divides by 25 !

56
Review on Proportionality ?

• If Fg a 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.

57
Review on Proportionality ?

• If Fg a 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 a 1/m ? a k/m

58
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?

1. The radius is kept constant, but the speed is
   increased by a multiplication factor of 7.

59
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?

• The radius is kept constant, but the speed is
  increased by a multiplication factor of 7.
• Equation with r, ac, and v ?

60
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?

• 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

61
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?

• 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

62
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?

• 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

63
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?

• 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

64
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?

• 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 k1/r

65
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?

• 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 k1/r
• In symbols, write the proportionality

66
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?

• 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 k1/r
• In symbols, write the proportionality
• ac a v2

67
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?

• 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 k1/r
• In symbols, write the proportionality
• ac a v2 or acceleration is directly
  proportional to the speed squared

68
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?

• The radius is kept constant, but the speed is
  increased by a multiplication factor of 7.
• ac a v2
• Now use the proportionality to find how ac
  changes.

69
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?

• The radius is kept constant, but the speed is
  increased by a multiplication factor of 7.
• ac a v2
• Now use the proportionality to find how ac
  changes.
• As v ? 7
• v2 ? ?

70
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?

• The radius is kept constant, but the speed is
  increased by a multiplication factor of 7.
• ac a v2
• Now use the proportionality to find how ac
  changes.
• As v ? 7
• v2 ? 72 or 49

71
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?

• The radius is kept constant, but the speed is
  increased by a multiplication factor of 7.
• ac a v2
• Now use the proportionality to find how ac
  changes.
• As v ? 7
• v2 ? 72 or 49
• and ac changes how?

72
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?

• The radius is kept constant, but the speed is
  increased by a multiplication factor of 7.
• ac a v2
• Now use the proportionality to find how ac
  changes.
• As v ? 7
• v2 ? 72 or 49
• and ac ? 49

73
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?

• The radius is kept constant, but the speed is
  increased by a multiplication factor of 7.
• ac a 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

74
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.

75
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ?

76
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2

77
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2
• Isolate the constant variable

78
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2
• Isolate the constant variable ac 4p2r/ T2
  

79
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2
• Isolate the constant variable ac 4p2r/ T2
  
• Rewrite the equation with a
  proportionality constant

80
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2
• Isolate the constant variable ac 4p2r/ T2
  
• Rewrite the equation with a
  proportionality constant ac k/
  T2

81
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2
• Isolate the constant variable ac 4p2r/ T2
  
• Rewrite the equation with a
  proportionality constant ac k/
  T2 Note k 4p2r

82
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2
• Isolate the constant variable ac 4p2r/ T2
  
• Rewrite the equation with a
  proportionality constant ac k/
  T2 Note k 4p2r
• In symbols, write the proportionality

83
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• Equation with r, ac, and T ? ac 4p2r/
  T2
• Isolate the constant variable ac 4p2r/ T2
  
• Rewrite the equation with a
  proportionality constant ac k/
  T2 Note k 4p2r
• In symbols, write the proportionality
  ac a 1/ T2

84
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• We have ac a 1/ T2
• How would you state the above
  proportionality in words?

85
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• We have ac a 1/ T2
• Acceleration is inversely
  proportional to the square of the
  period.

86
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• We have ac a 1/ T2
• Acceleration is inversely
  proportional to the square of the
  period.
• If T ? 3

87
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• We have ac a 1/ T2
• Acceleration is inversely
  proportional to the square of the period.
• If T ? 3
• Then T2 will change how?

88
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• We have ac a 1/ T2
• Acceleration is inversely
  proportional to the square of the period.
• If T ? 3
• Then T2 ? 32 or 9

89
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• We have ac a 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?

90
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?

• The radius is kept constant, but the period is
  decreased by a factor of 3.
• We have ac a 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!

91
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?

1. The radius is kept constant but the frequency
   quadruples (multiplies by four) You try
   this one

92
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?

• The radius is kept constant but the frequency
  quadruples (multiplies by four) You try
  this one
• ac 4p2r f 2 f ? 4
• ac 4p2r f 2 f 2 ? 42 or
  16
• ac k f 2 ac ? 42 or 16
• ac a f 2 The centripetal
  acceleration
• 
  multiplies by 16 !

93
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...
• The radius is kept constant but the period
  quadruples (multiplies by four) Check
  answer Divides by 16
• The frequency is kept constant but the radius
  is tripled.
• Check answer multiplies by 3
• The speed is kept constant but the radius is
  halved.
• Check answer multiplies by 2
• The radius is kept constant, but the speed
  multiplies by eight. Check answer
  multiplies by 64