Saturday, March 14, 2015

Physics Graphs Basics 1 – Coordinates of Points

  • Physics Graphs Basics 1 – Coordinates of Points


Understanding and interpreting graphs are very important in physics. In terms of exams, graphs are not only part of practical questions, but also in MCQs and long questions.

Data from graphs can be as follows:
·         Data that can be directly read from the graphs (e.g. coordinates of points, nature of the gradient, …)
·         Data that can be calculated from the graphs (e.g. gradient, area under graph, …)



Finding coordinates of points
To find the coordinates of any point on a graph, first the minimum value that can be read on each axis should be identified. This can be done as follows:


  • Find the difference between 2 consecutive labeled values on the graph. In the example above, on the x-axis, the difference in consecutive labels is (0.6 – 0.4 =) 0.2 and on the y-axis, the difference is (27.0 – 26.5 =) 0.5. It’s also a good practice, in graphs such as the above, to include labels at the intermediate steps. That is, in this graph, labels are given after 10 squares. So, additional labels could be added after 5 squares.
  • Identify the number of smallest squares representing this difference. For the x-axis, 0.2 is represented by 10 squares and on the y-axis, 0.5 is represented by 10 squares.
  • Calculate the smallest value that can be read on the axis by dividing the ‘difference’ by the number of squares. For the x-axis, the smallest readable value is (0.2 / 10 =) 0.02 and on the y-axis, the smallest readable value is (0.5 / 10 =) 0.05. Note that half a square may also be read on the graph {this would be 0.01 on x-axis and 0.025 on y-axis}, but any smaller value would be inappropriate {that’s the purpose of including uncertainties in answers}.




Now, let’s try to find the coordinates of some points on the graph below.
 


Point A
To find the coordinates of point A, identify by how many squares it is separated from the closest labels on both axes (both the x- and y-axes).
On the x-axis, it is separated by 4 squares from (after) the 0.4 label. So, the value of the x-coordinate is 0.4 + 4(0.02) = 0.4 + 0.08 = 0.48.
On the y-axis, it is separated by 4 squares from (after) the 27.0 label. So, the value of the y-coordinate is 27.0 + 4(0.05) = 27.0 + 0.20 = 27.20.

Coordinates of point A are (0.48, 27.20)


Point B
On the x-axis, it is separated by 3 squares from (before) the 0.8 label. So, the value of the x-coordinate is 0.8 – 3(0.02) = 0.48 + 0.06 = 0.74.
On the y-axis, it is separated by 3 squares from (before) the 27.0 label. So, the value of the y-coordinate is 27.0 – 3(0.05) = 27.0 – 0.15 = 26.85.

Coordinates of point B are (0.74, 26.85)


Point C
On the x-axis, it is separated by 2 squares from (after) the 0.8 label. So, the value of the x-coordinate is 0.8 + 2(0.02) = 0.8 + 0.04 = 0.84.
On the y-axis, it is separated by 2.5 squares from (before) the 27.5 label. So, the value of the y-coordinate is 27.5 – 2.5(0.05) = 27.5 – 0.125 = 27.375.

Coordinates of point C are (0.84, 27.375)




Next, let’s try to plot some given points on the graph.
Consider the points X: (0.45, 27.3), Y: (0.7, 26.6) and Z: (0.76, 26.8)

Point X (0.45, 27.3)
On x-axis, 0.45 is closest to the 0.4 label. 1 smallest square represents 0.02 (as calculated previously). Number of smallest squares required = (0.45 – 0.4) / 0.02 = 2.5. So, on the x-axis, point X should be 2.5 squares after the 0.4 label.
On y-axis, 27.3 is closest to the 27.5 label. 1 smallest square represents 0.05 (as calculated previously). Number of smallest squares required = (27.5 – 27.3) / 0.05 = 4. So, on the y-axis, point X should be 4 squares before the 27.5 label.


Point Y (0.7, 26.6)
On x-axis, 0.7 is midway between the 0.6 and 0.8 labels. Number of smallest squares required = (0.8 – 0.7) / 0.02 = 5. So, on the x-axis, point Y should be 5 squares after the 0.6 label or equivalently, 5 squares before the 0.8 label.
On y-axis, 26.6 is closest to the 26.5 label. Number of smallest squares required = (26.6 – 26.5) / 0.05 = 2. So, on the y-axis, point Y should be 2 squares after the 26.5 label.


Point Z (0.76, 26.8)
On x-axis, 0.76 is closest to the 0.8 label. Number of smallest squares required = (0.8 – 0.76) / 0.02 = 2. So, on the x-axis, point Z should be 2 squares before the 0.8 label.
On y-axis, 26.8 is closest to the 27.0 label. Number of smallest squares required = (27.0 – 26.8) / 0.05 = 4. So, on the y-axis, point Z should be 4 squares before the 27.0 label.



With practice, these methods would be performed quicker and more easily.


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