Proportionality

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

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

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

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 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

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

Graphing Proportionality If V α I, what does the graph of V vs I look like? V I Linear graph

Graphing Proportionality If V α I, what does the graph of V vs I look like? If V α I,What is the general equation relating V vs I ? V I Linear graph

Graphing Proportionality If V α I, what does the graph of V vs I look like? If V α I, V = K I V I Linear graph

Graphing Proportionality If V α I, what does the graph of V vs I look like? If V α I, V = K I K is just the slope of the graph. In physics, it is also called the _____________ of ________________. V I Linear graph

Graphing Proportionality If V α I, what does the graph of V vs I look like? 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. V I Linear graph

Converting between a Proportionality Relationship and an Equation

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 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 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 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 |F net | to an equation.

Converting between a Proportionality Relationship and an Equation 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 |F net | ↔ |a| = K |F net |

Converting between a Proportionality Relationship and an Equation 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 |F net | ↔ |a| = K |F net | Do you know what physics quantity K is ?

Converting between a Proportionality Relationship and an Equation 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 |F net | ↔ |a| = K |F net | K is 1/m in the case of Newton's Second Law

Converting between a Proportionality Relationship and an Equation Try this! Convert this proportionality relationship a α v 2 to an equation.

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean?

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean? As v ↑ x, v 2 ↑ x 2, and a ↑ x 2 or...

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean? As v ↑ x, v 2 ↑ x 2, and a ↑ x 2 If v ↑ 3, v 2 ↑ 3 2, and a ↑ 3 2

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean? As v ↑ x, v 2 ↑ x 2, and a ↑ x 2 If v ↑ 3, v 2 ↑ 3 2, and a ↑ 3 2 or...

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean? As v ↑ x, v 2 ↑ x 2, and a ↑ x 2 If v ↑ 3, v 2 ↑ 3 2, and a ↑ 3 2 or... If v ↓ 4, then v 2 ?, and then a ?

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean? As v ↑ x, v 2 ↑ x 2, and a ↑ x 2 If v ↑ 3, v 2 ↑ 3 2, and a ↑ 3 2 or... If v ↓ 4, then v 2 ↓ 4 2, and then a ?

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean? As v ↑ x, v 2 ↑ x 2, and a ↑ x 2 If v ↑ 3, v 2 ↑ 3 2, and a ↑ 3 2 or... If v ↓ 4, then v 2 ↓ 4 2, and then a ↓ 4 2

Converting between a Proportionality Relationship and an Equation a α v 2 ↔ a = K v 2 In this case, we say a is directly proportional to v 2. What does this mean? As v ↑ x, v 2 ↑ x 2, and a ↑ x 2 If v ↑ 3, v 2 ↑ 3 2, and a ↑ 3 2 or... If v ↓ 4, then v 2 ↓ 4 2, and then a ↓ 4 2 Note: “Proportional to” or “directly proportional to” is like “monkey-see monkey do”. In this case, if v 2 changes by a factor, a changes by the same factor.

Inverse proportion

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

Inverse proportion E inversely proportional to r means... As r ↑ x, E ↓ x

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 ?

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

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?

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

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 ?

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 α 1/r

The inverse square law

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 F g 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 F g 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 F g 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 F g 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. F g α 1/d 2

The inverse square law A common proportionality relationship in physics is called “the inverse square law” For example F g 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. F g α 1/d 2 What equation does this correspond to?

The inverse square law A common proportionality relationship in physics is called “the inverse square law” For example F g 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. F g α 1/d 2 ↔ F g = C/d 2

The inverse square law A common proportionality relationship in physics is called “the inverse square law” For example F g 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. F g α 1/d 2 ↔ F g = C/d 2 What does F g α 1/d 2 mean?

The inverse square law A common proportionality relationship in physics is called “the inverse square law” For example F g 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. F g α 1/d 2 ↔ F g = C/d 2 F g α 1/d 2 means... as d ↑ x, d 2 ↑ x 2, and then F g ↓ x 2

Review on Proportionality ☺

If F g α 1/d 2, and d multiplies by 5, how does F g change?

Review on Proportionality ☺ If F g α 1/d 2, and d multiplies by 5, how does F g change? As d ↑ 5 d 2 ↑ 5 2 or 25 F g ↓ 25 F g divides by 25 !

Review on Proportionality ☺ If F g α 1/d 2, and d multiplies by 5, how does F g change? As d ↑ 5 d 2 ↑ 5 2 or 25 F g ↓ 25 F g 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 F g α 1/d 2, and d multiplies by 5, how does F g change? As d ↑ 5 d 2 ↑ 5 2 or 25 F g ↓ 25 F g 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, a c, 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, a c, and v ? a c = v 2 /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, a c, and v ? a c = v 2 /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, a c, and v ? a c = v 2 /r Isolate the constant variable: a c = v 2 /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, a c, and v ? a c = v 2 /r Isolate the constant variable: a c = v 2 /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, a c, and v ? a c = v 2 /r Isolate the constant variable: a c = v 2 /r Rewrite the equation with a proportionality constant: a c = (1/r)v 2 or a c = k v 2 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, a c, and v ? a c = v 2 /r Isolate the constant variable: a c = v 2 /r Rewrite the equation with a proportionality constant: a c = (1/r)v 2 or a c = k v 2 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, a c, and v ? a c = v 2 /r Isolate the constant variable: a c = v 2 /r Rewrite the equation with a proportionality constant: a c = (1/r)v 2 or a c = k v 2 where k=1/r In symbols, write the proportionality: a c α v 2

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, a c, and v ? a c = v 2 /r Isolate the constant variable: a c = v 2 /r Rewrite the equation with a proportionality constant: a c = (1/r)v 2 or a c = k v 2 where k=1/r In symbols, write the proportionality: a c α v 2 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. a c α v 2 Now use the proportionality to find how a c 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. a c α v 2 Now use the proportionality to find how a c changes. As v ↑ 7 v 2 ↑ ?

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. a c α v 2 Now use the proportionality to find how a c changes. As v ↑ 7 v 2 ↑ 7 2 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. a c α v 2 Now use the proportionality to find how a c changes. As v ↑ 7 v 2 ↑ 7 2 or 49 and a c 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. a c α v 2 Now use the proportionality to find how a c changes. As v ↑ 7 v 2 ↑ 7 2 or 49 and a c ↑ 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. a c α v 2 Now use the proportionality to find how a c changes. As v ↑ 7 v 2 ↑ 7 2 or 49 and a c ↑ 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, a c, 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, a c, and T ? a c = 4π 2 r/ T 2

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, a c, and T ? a c = 4π 2 r/ T 2 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, a c, and T ? a c = 4π 2 r/ T 2 Isolate the constant variable: a c = 4π 2 r/ T 2

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, a c, and T ? a c = 4π 2 r/ T 2 Isolate the constant variable: a c = 4π 2 r/ T 2 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, a c, and T ? a c = 4π 2 r/ T 2 Isolate the constant variable: a c = 4π 2 r/ T 2 Rewrite the equation with a proportionality constant: a c = k/ T 2

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, a c, and T ? a c = 4π 2 r/ T 2 Isolate the constant variable: a c = 4π 2 r/ T 2 Rewrite the equation with a proportionality constant: a c = k/ T 2 Note k = 4π 2 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? (b) The radius is kept constant, but the period is decreased by a factor of 3. Equation with r, a c, and T ? a c = 4π 2 r/ T 2 Isolate the constant variable: a c = 4π 2 r/ T 2 Rewrite the equation with a proportionality constant: a c = k/ T 2 Note k = 4π 2 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? (b) The radius is kept constant, but the period is decreased by a factor of 3. Equation with r, a c, and T ? a c = 4π 2 r/ T 2 Isolate the constant variable: a c = 4π 2 r/ T 2 Rewrite the equation with a proportionality constant: a c = k/ T 2 Note k = 4π 2 r In symbols, write the proportionality: a c α 1/ T 2

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 a c α 1/ T 2 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 a c α 1/ T 2 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 a c α 1/ T 2 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 a c α 1/ T 2 Acceleration is inversely proportional to the square of the period. If T ↓ 3 Then T 2 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 a c α 1/ T 2 Acceleration is inversely proportional to the square of the period. If T ↓ 3 Then T 2 ↓ 3 2 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 a c α 1/ T 2 Acceleration is inversely proportional to the square of the period. If T ↓ 3 Then T 2 ↓ 3 2 or 9 And a c 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 a c α 1/ T 2 Acceleration is inversely proportional to the square of the period. If T ↓ 3 Then T 2 ↓ 3 2 or 9 And a c ↑ 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: a c = 4π 2 r f 2 f ↑ 4 a c = 4π 2 r f 2 f 2 ↑ 4 2 or 16 a c = k f 2 a c ↑ 4 2 or 16 a c 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