1. Physics 30 - Graphing
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2. Physics 30 - Graphing - Variables
Manipulated variable:
The experimenter make the selection of values for this variable.
Time is usually a manipulated variable.
Must be graphed on the horizontal axis unless otherwise stated.
Responding variable:
This variable changes because of the selection of a particular manipulated value.
The measured part of the experiment.
Must be graphed on the vertical axis.
Control variables:
Variables that must be held constant in an experiment to prove a connection between the manipulated and responding variables.
The “other” variables in the pertinent formula.
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3. Physics 30 - Graphing - Variables
Examples:
Momentum is measured for several different values of velocity …
The force imparted by a collision is measured for several different values of collision time …
The formula that applies is r = mv. From the information, velocity is the manipulated variable, momentum the responding and mass is the control variable.
The formula that applies is F(t) = m(vf - vi). From this information we know that time is the manipulated variable, force is the responding variable and the mass and both velocities must be controlled.
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4. Physics 30 - Graphing - Format For Graph
The scales should be logical multiples on the major graph lines so that they are easily divisible into ten parts.
Increments should be… 0.1, , 1, 2, 5, 10, 20, 50, 100 …
Do not go up by odd numbers such as 3’s, 7’s which are not metrically divisible. (Where is 4.25 if you are increasing by 3’s?).
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5. Physics 30 - Graphing - Format For Graph
Label the axis with the responding variable on the y-axis (units in brackets) and manipulated variable on the x-axis (units in brackets)
The title for the graph should be : (use the actual names)
Responding vs. Manipulated
Responding as a function of manipulated
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6. Physics 30 - Graphing - Format For Graph
The scales should be set to use up as much of the graphing surface as possible - data must take up at least half the page.
You do not have to get the absolute best scale, just one that works well!
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7. Physics 30 - Graphing - Format For Graph
The preferred format, when there is a break in the graphing surface is to have (0,0) in the corner and indicate a break using a jagged line
When using this format be careful not to make inferences about the broken area of your graph - such as x or y intercepts.
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8. Physics 30 - Graphing - Format For Graph
Draw a smooth curve through the data – do not join the dots
If the graph is a straight line, mark two points on the line and use these to calculate the slope
Always calculate the slope of a straight line … you must include units … just take rise over run units and simplify.
Do not leave complex fractions
Include powers of ten!
The number of significant digits for the slope comes from the data.
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9. Common Graphing Errors
Errors students commonly make when graphing and curve straightening include:
• placing the manipulated and responding variables on the wrong axes
• drawing a linear graph through non-linear data points
• forgetting to include units or powers of 10 when calculating slope other averaging techniques
• forgetting to perform algebraic manipulation on units and powers when doing curve straightening
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10. Graphing Recognition - Dep
Students seem to experience difficulty in connecting the shape of a graph with the corresponding mathematical equation. Students should recognize the following basic relationships.
1) Linear
2) Partial
3) Squared
4) Square Root
5) Inverse
6) Inverse Square
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11. Graphing - Relationship from numbers
DIRECT RELATION Manipulated Responding
y a x or y = k x which is a linear relationship
Note: as x increases, so does y This type of relationship is said to be direct.
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12. Graphing - Relationship from numbers
DIRECT SQUARE RELATION
Manipulated Responding
y a x2 or y = k x2 which is a direct square relationship
Note: as x increases, so does y
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13. Graphing - Relationship from numbers
DIRECT SQUARE ROOT RELATION
Manipulated Responding
which is a direct square root relationship
Note: as x increases, so does y
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14. Graphing : Relationship from formula
PARTIAL DIRECT RELATIONSHIP
Manipulated Responding
y a kx + b or y = kx + b which is a partial relationship
Note: as x increases, so does y
It can be recognized as none of the other direct types, or if a number is subtracted from each (5) the relation becomes direct
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15. Graphing - Relationship from numbers
INVERSE RELATION
Manipulated Responding
y a 1/x or y = k /x which is a inverse relationship
Note: as x increases, y decreases
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16. Graphing - Relationship from numbers
INVERSE SQUARE RELATION
Manipulated Responding
y a 1/x2 or y = k/x2 which is a inverse square relationship
Note: as x increases, y decreases
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17. Graphing : Relationship from formula
You should also be able to predict the type of relationship between two variables if you know the formula that applies to the situation
Example: If the experiment is talking about centripetal force and we know that centripetal force (Fcent) is to be measured for different values of radius of the circle, then the applicable formula is:
Since centripetal force is the responding variable, we isolate it in the equation:
Note: Since m and v are control variables for this experiment, their values must remain constant. Giving the relationship:
This should graph as an inverse relationship
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18. Graphing : Relationship from formula
Since L, n and l must be controlled for the experiment:
And we know that shape of the graph should be:
What would the shape be for l as a function of d?
Suppose we are performing a replica of Young’s double slit experiment and we are trying to graph the first antinodal distance (x) as a function of the slit separation (d) The applicable formula is:
With our responding variable isolated:
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19. Graphing: Making use of the Slope
Relationships which are direct may use a slope technique to determine important information about an experiment.
For example suppose the ejection speed (v) is measured for different stretch distances (x) of a slingshot, and we wish to determine the spring constant of the sling. The applicable formula is:
Isolating the dependent variable gives us:
Since mass and the spring constant are to be controlled:
Which tells us the shape of the graph should be direct with a slope of:
If the mass is known, we can calculate the spring constant.
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