AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units...
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Transcript of AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units...
![Page 1: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/1.jpg)
AP Physics C
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Dimensionality
• Dimensionality is an abstract concept closely related to units• Units describe certain types of quantities.• Feet, inches, meters, nanometer - Units of Length
• We can develop a set of rules that allow us to:• Check equations• Determine the dependence on specified set of quantities
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Dimensionality
There are 3 types of quantities we will discuss today:• Length
• Time
• Mass
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Notation
We denote these quantities as:• Length - L
• Time - T
• Mass - M
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Notation
When denoting the dimensionality of a variable we use square brackets [ ]
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Rules of Dimensionality
1. Variables on opposite sides of an equals sign must have the same dimensionality
2. Variables on opposite sides of a + or - must have the same dimensionality
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Rules of Dimensionality
Lets check the formula:
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Rules of Dimensionality
3. Pure number () are always dimensionless
4. Special functions (sine, cosine, exponential, etc.) are always dimensionless
5. The argument of special functions are always dimensionless
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Dimensional Analysis
We can use the rules of dimensionality to find the dimensions of an unknown quantitiy in a formula:
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Dimensional Analysis
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Dimensional Analysis
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Dimensional Analysis
Consider a mass swinging on the end of a stringThe period is the amount of timetakes for the mass to complete onefull oscillationWhat variables do yoususpect the period of the motion will depend on?
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Dimensional AnalysisIn general we may assume:
Using dimensional considerations, we can solve for and
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Position, Velocity, & Acceleration
• In Physics it is important to be able to relate position, velocity, & acceleration
• A mathematical description of this relationship requires the use of calculus
• In this section we will discuss the graphical relationship between a position vs. time graph and a velocity vs. time graph
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Graphical Analysis
Δ 𝑦
Δ𝑥
• Recall that:
𝑦
𝑥
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Graphical Analysis
• For a position vs. time graph:
• For an velocity vs. time graph:
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Graphical Analysis
𝑡
𝑥
Δ 𝑡
Δ𝑥
𝑣𝑎𝑣𝑔=Δ𝑥Δ𝑡
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Graphical Analysis
𝑡
𝑥
Δ 𝑡1
Δ𝑥1
𝑣𝑎𝑣𝑔 ,1<𝑣𝑎𝑣𝑔 ,2
Δ𝑥2
Δ 𝑡2
![Page 19: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/19.jpg)
Graphical Analysis
𝑡
𝑥
𝑡
𝑠𝑙𝑜𝑝𝑒=𝑣 (𝑡)
![Page 20: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/20.jpg)
Graphical Analysis
• is the slope of the tangent line at • is graphically understood as the steepness of the
vs graph.
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Graphical Analysis
𝑥
𝑡
What does look like?
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Graphical Analysis
Identify where positive, negative, & zero
𝑡
𝑥
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Graphical Analysis
Sketch a graph of 𝑡
𝑣
𝑡
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The Derivative
• We can approximate as the average velocity over a time an interval starting at
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The Derivative
𝑡
𝑥
𝑡 0
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The Derivative
𝑡
𝑥
Δ 𝑡
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The Derivative
𝑡
𝑥
Δ 𝑡
![Page 28: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/28.jpg)
The Derivative
𝑡
𝑥
Δ 𝑡
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The Derivative
𝑡
𝑥
Δ 𝑡
![Page 30: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/30.jpg)
The Derivative
𝑡
𝑥
Δ 𝑡
![Page 31: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/31.jpg)
The Derivative
• We can make our approximation of exact by taking the limit as
We call this the “derivative of with respect to ”
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The Derivative
• We denote the derivative as:
• and denote a “differential change”, which describes or in the limit where the difference goes to zero
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The Derivative - Linearity
The derivative is a linear operation, this means:
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The Derivative - Quadratic
Calculate for:
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The Derivative - Polynomial
Calculate for:
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Power Rule
In general:
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Derivative of Sine & Cosine
𝑣
𝑡
We know from graphical considerations that looks like . How do we prove it?
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Derivative of Sine & Cosine
In general:
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Second Derivative
The second derivative of is defined as:
We can relate the second derivative of to other kinematic variables:
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Third Derivative
The third derivative of position vs. time is called the jerk:
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The Chain Rule
Suppose we know height of the roller coaster as a function of its position . And we know .How do we calculate ?
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The Chain Rule
𝑦
𝑥
𝑥
𝑡
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The Chain Rule
In general:If we have and ,
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The Chain Rule
Consider:
Calculate using the chain rule.
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The Chain Rule
Consider:
What is and ?
Calculate
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The Chain Rule
Consider:
What is and ?
Calculate
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The Chain Rule
Once you gain experience using the Chain Rule, you can skip writing down and .
The trick: work from the outside
Consider:
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The Chain Rule
Consider:
Calculate using the chain rule.
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The Chain Rule
Calculate the derivative of:
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The Chain Rule
Consider:
Determine when
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How do we calculate the derivative of the product of two functions, ?
Apply the definition of the derivative!
Okay…now what do we do?
Product Rule
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Recall that we can visualize the product of two numbers as the area of a rectangle.
4
Product Rule
5
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Recall that we can visualize the product of two numbers as the area of a rectangle.
Product Rule
1234567 891011121314 151617181920
4×5=20
![Page 54: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/54.jpg)
We can do the same thing with the product of two functions.
Product Rule
¿ 𝑓 (𝑡 )𝑔 (𝑡)𝑓 (𝑡 )
𝑔 (𝑡 )
![Page 55: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/55.jpg)
Consider two functions & which are both increasing.
Product Rule𝑓 (𝑡)
𝑡
𝑔 (𝑡)
𝑡
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Product Rule
𝑓 (𝑡)𝑓 (𝑡+Δ𝑡 )
𝑔 (𝑡)𝑔 (𝑡+Δ𝑡)
![Page 57: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/57.jpg)
Product Rule
𝑓 (𝑡 )
𝑔 (𝑡 )
𝑓 (𝑡+Δ𝑡 )
𝑔 (𝑡+Δ 𝑡 )
How do we geometrically picture:
![Page 58: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/58.jpg)
Product Rule
𝑓 (𝑡 )
𝑔 (𝑡 )
𝑓 (𝑡+Δ𝑡 )
𝑔 (𝑡+Δ 𝑡 )
Lets calculate:
![Page 59: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/59.jpg)
The Product Rule
In general:
![Page 60: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/60.jpg)
Product Rule𝑑𝑓𝑑𝑡
𝑔 (𝑡)
𝑓 (𝑡 )
𝑔 (𝑡 )
𝑑𝑔𝑑𝑡
𝑓 (𝑡)
goes to zeroin the limit:
![Page 61: AP Physics C. Dimensionality Dimensionality is an abstract concept closely related to units Units describe certain types of quantities. Feet, inches,](https://reader035.fdocuments.us/reader035/viewer/2022081520/5697bfa91a28abf838c99c03/html5/thumbnails/61.jpg)
The Product Rule
Calculate the derivative of:
𝑓 (𝑡 ) 𝑔 (𝑡 )
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The Product Rule
Calculate the derivative of:
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The Product Rule
Calculate the derivative of:
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The Product Rule
Calculate the derivative of: