Graphs in physics
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Graphs in physics Presentation Transcript

  • 1. GRAPHS IN PHYSICS 1
  • 2. Linear GraphsGraphing data shows if a relationship existsbetween two quantities also called variables.If two variables show a linear relationship theyare directly proportional to each other. Examine the following graph: 2
  • 3. Linear Graphs Graph of Force vs Mass for a hanging object 45 40 35 30 Force (N) 25 20 15 10 5Dependent 0Variable 0 1 2 3 4 5 Mass (kg) 6 7 8 9 Independent Variable 3
  • 4. Linear Graphs – Slope of a LineThe slope of a line is a ratio between thechange in the y-value and the change inthe x- value.This ratio tells whether the two quantitiesare related mathematically.Calculating the slope of a line is easy! 4
  • 5. Linear Graphs – Slope of a Line y y2 Rise = Δy = y2 – y1 Rise y1 Slope = Run = Δx = x2 – x1 Run y2 – y1 x1 x x2 Slope = x2 – x1 5
  • 6. Linear Graphs – Equation of a LineOnce you know the slope then the equation of a line is very easilydetermined.Slope Intercept form for any line: y = mx + b y-intercept slope (the value of y when x =0) Of course in Physics we don’t use “x” & “y”. We could use F and m, or d and t, or F and x etc.) 6
  • 7. Linear Graphs: Area Under the Curve Graph of Applied Force vs Distance Object Travelled 45Sometimes it’s what’s 40under the line that is 35important! 30 Force (N) 25 20Work = Force x distance 15 10 W=Fxd 5 0 0 2 4 6 8 10 12 How much work was Distance (m ) done in the first 4 m? How much work was done moving the object over the last 6 m? 7
  • 8. Non Linear Relationships Not all relationships between variables are linear. Some are curves which show a squared or square root relationshipIn this course we use simple techniques to“straighten the curve” into linear relationship. 8
  • 9. Non Linear Relationships 60 60 50 50 40 40 y 30 30 y 20 20 10 10 0 0 0 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 x x-squared This is not linear. Try squaring the x-axis values to produce a straight line graph Equation of the straight line would then be: y = x2 9
  • 10. Non Linear Relationships 1.2 1.2 1 1 0.8 0.8 0.6 y y 0.6 0.4 0.4 0.2 0.2 0 0 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 x 1/x This is not linear. It is an inverse relationship. Try plotting: y vs 1/x. Equation of the straight line would then be: y = 1/x 10
  • 11. Meaning of Slope from Equations Often in Physics graphs are plotted and the calculation of and the meaning of the slope becomes an important factor. We will use the slope intercept form of the linear equation described earlier. y = mx + b 11
  • 12. Meaning of Slope from Equations Unfortunately physicists do not use the same variables as mathematicians! For example: s = ½xa x t 2 is a very common kinematic equation.where s = distance, a = acceleration and t = time 12
  • 13. Meaning of Slope from Equations Physicists may plot a graph of s vs t, but this would yield a non-linear graph:s s To straighten the curve Square the time t t2 13
  • 14. Meaning of Slope from EquationsBut what would the slope of a d vs t2 graph represent? Let’s look at the equation again: s = ½at2 {s is plotted vs t2} y = mx + b What is left over must be equal to the slope of the line! slope = ½ x a {and do not forget about units: ms-2} 14
  • 15. Meaning of Slope from EquationsNow try These. A physics equation will be given, as well as what is initially plotted. Tell me what should be plotted to straighten the graph and then state what the slope of this graph would be equal to. Example #1: a = v2/r Plot a vs v2 to straighten graph a Let’s re-write the equation a little: a = (1/r)v2 Therefore plotting a vs. v2 v would let the slope be: Slope = 1/r 15
  • 16. Meaning of Slope from Equations Example #2: F = 2md/t2 Plot F vs 1/t2 F F to straighten the graph t 1/t2 Slope = 2md Go on to the worksheet on this topic 16
  • 17. Error Bars on Graphs You already know about including errors with all measured values. These errors must be included in any graph that is created using these measured value. The errors are shown as bars both in the horizontal and vertical direction. For example: 2.3 + 0.2 (horizontal ) 15.7 + 0.5 (vertical)This would be shownlike this on the graph. Error Bars! 17
  • 18. Error Bars on Graphs Plot the following data and add in the error bars: time (s) Distance (m) (+0.2) (+0.5) 0.0 0.0 0.4 2.4 0.8 4.9 1.2 7.3 1.6 11.1 2.0 13.5 2.4 15.2 2.8 17.9 3.2 20.0 3.6 22.7 18
  • 19. Error Bars on Graphs Graph of Distance vs Time 25.0 Max. slope 20.0 Best fit line 15.0 Minimum slope Distance (m) 10.0 5.0 D = 6.3 m/s x t 0.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -5.0 Time (s) 19