Here are the steps to plot the graph and calculate the gradient:
3. Draw a graph of T$^2$ against L on the graph provided below.
Step 1: Set up the axes.
X-axis (horizontal): Label this axis $L \, (\text{cm})$.
Choose a scale where $1$ major grid square (10 small squares) represents $10 \, \text{cm}$.
Mark the axis at $0, 10, 20, 30, \dots, 90 \, \text{cm}$.
Y-axis (vertical): Label this axis $T^2 \, (\text{s}^2)$.
Choose a scale where $1$ major grid square (10 small squares) represents $0.5 \, \text{s}^2$.
Mark the axis at $0, 0.5, 1.0, 1.5, \dots, 4.0 \, \text{s}^2$.
Step 2: Plot the data points from the table.
For each pair of $(L, T^2)$ values, mark a small 'x' or a dot on the graph.
$(40 \, \text{cm}, 1.61 \, \text{s}^2)$: Plot at $x=4$ major squares, $y=3$ major squares and $2.2$ small squares up.
$(50 \, \text{cm}, 2.02 \, \text{s}^2)$: Plot at $x=5$ major squares, $y=4$ major squares and $0.4$ small squares up.
$(60 \, \text{cm}, 2.43 \, \text{s}^2)$: Plot at $x=6$ major squares, $y=4$ major squares and $8.6$ small squares up.
$(70 \, \text{cm}, 2.82 \, \text{s}^2)$: Plot at $x=7$ major squares, $y=5$ major squares and $6.4$ small squares up.
$(80 \, \text{cm}, 3.24 \, \text{s}^2)$: Plot at $x=8$ major squares, $y=6$ major squares and $4.8$ small squares up.
$(90 \, \text{cm}, 3.61 \, \text{s}^2)$: Plot at $x=9$ major squares, $y=7$ major squares and $2.2$ small squares up.
Step 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 $(0,0)$ as $T^2$ is directly proportional to $L$ for a simple pendulum.
4. From the graph calculate the gradient.
Step 1: 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: $(L_1, T_1^2) = (40 \, \text{cm}, 1.61 \, \text{s}^2)$
Point 2: $(L_2, T_2^2) = (90 \, \text{cm}, 3.61 \, \text{s}^2)$
Step 2: Calculate the gradient using the formula $m = \frac{\Delta T^2}{\Delta L}$.
$$m = \frac{T_2^2 - T_1^2}{L_2 - L_1}$$
$$m = \frac{3.61 \, \text{s}^2 - 1.61 \, \text{s}^2}{90 \, \text{cm} - 40 \, \text{cm}}$$
$$m = \frac{2.00 \, \text{s}^2}{50 \, \text{cm}}$$
$$m = 0.04 \, \text{s}^2/\text{cm}$$
The gradient of the graph is $\boxed{0.04 \, \text{s}^2