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And I need you to plot the exact requested graph šŸ“ˆ with this template following all it's meet points on the graph so I can follow to plot

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Step-by-step solution

5 steps

Here are the steps to plot the graph and calculate the gradient:

3. Draw a graph of T2^2 against L on the graph provided below.

  1. Set up the axes.

    • X-axis (horizontal): Label this axis L (cm)L \, (\text{cm}).
      • Choose a scale where 1 major grid square (10 small squares) represents 10 cm10 \, \text{cm}.
      • Mark the axis at 0,10,20,30,…,90 cm0, 10, 20, 30, \dots, 90 \, \text{cm}.
    • Y-axis (vertical): Label this axis T2 (s2)T^2 \, (\text{s}^2).
      • Choose a scale where 1 major grid square (10 small squares) represents 0.5 s20.5 \, \text{s}^2.
      • Mark the axis at 0,0.5,1.0,1.5,…,4.0 s20, 0.5, 1.0, 1.5, \dots, 4.0 \, \text{s}^2.
  2. Plot the data points from the table.

    • For each pair of (L,T2)(L, T^2) values, mark a small 'x' or a dot on the graph.
      • (40 cm,1.61 s2)(40 \, \text{cm}, 1.61 \, \text{s}^2): Plot at x=4x=4 major squares, y=3y=3 major squares and 2.2 small squares up.
      • (50 cm,2.02 s2)(50 \, \text{cm}, 2.02 \, \text{s}^2): Plot at x=5x=5 major squares, y=4y=4 major squares and 0.4 small squares up.
      • (60 cm,2.43 s2)(60 \, \text{cm}, 2.43 \, \text{s}^2): Plot at x=6x=6 major squares, y=4y=4 major squares and 8.6 small squares up.
      • (70 cm,2.82 s2)(70 \, \text{cm}, 2.82 \, \text{s}^2): Plot at x=7x=7 major squares, y=5y=5 major squares and 6.4 small squares up.
      • (80 cm,3.24 s2)(80 \, \text{cm}, 3.24 \, \text{s}^2): Plot at x=8x=8 major squares, y=6y=6 major squares and 4.8 small squares up.
      • (90 cm,3.61 s2)(90 \, \text{cm}, 3.61 \, \text{s}^2): Plot at x=9x=9 major squares, y=7y=7 major squares and 2.2 small squares up.
  3. Draw the best-fit line.

    • Draw a single straight line that passes as close as possible to all the plotted points. This line should also pass through the origin (0,0)(0,0) as T2T^2 is directly proportional to LL for a simple pendulum.

    4. From the graph calculate the gradient.

  4. Choose two points on the best-fit line.

    To calculate the gradient accurately, select two points that are far apart on your drawn best-fit line. These points do not necessarily have to be your original data points. For example, let's choose:

    • Point 1: (L1,T12)=(40 cm,1.61 s2)(L_1, T_1^2) = (40 \, \text{cm}, 1.61 \, \text{s}^2)
    • Point 2: (L2,T22)=(90 cm,3.61 s2)(L_2, T_2^2) = (90 \, \text{cm}, 3.61 \, \text{s}^2)
  5. Calculate the gradient using the formula m=ΔT2ΔLm = \frac{\Delta T^2}{\Delta L}.

    m=T22āˆ’T12L2āˆ’L1m = \frac{T_2^2 - T_1^2}{L_2 - L_1} m=3.61 s2āˆ’1.61 s290 cmāˆ’40 cmm = \frac{3.61 \, \text{s}^2 - 1.61 \, \text{s}^2}{90 \, \text{cm} - 40 \, \text{cm}} m=2.00 s250 cmm = \frac{2.00 \, \text{s}^2}{50 \, \text{cm}} m=0.04 s2/cmm = 0.04 \, \text{s}^2/\text{cm}

    The gradient of the graph is $\boxed{0.04 , s^2

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