To transform the formula
\[ T = 2\pi \sqrt{\frac{H - h}{g}} \]
into a linear form, we first square both sides:
\[ T^2 = 4\pi^2 \frac{H - h}{g} \]
Now, rearranging for \(H - h\):
\[ H - h = \frac{g T^2}{4\pi^2} \]
Next, we can express this as:
\[ H = h + \frac{g T^2}{4\pi^2} \]
This is in the form \(H = mh + b\) where:
- The gradient (slope) \(m = \frac{g}{4\pi^2}\)
- The intercept \(b = \frac{g T^2}{4\pi^2}\)
1. Plotting the graph
To plot the graph, we will use the provided data:
Values for \(h\): [5, 10, 15, 20, 25, 30, 35, 40, 45]
Corresponding values for \(T^2\): [13.0321, 12.6025, 12.006225, 12.1801, 11.9716, 11.79925, 11.56, 11.189925, 10.9561]
Data to plot
- x-axis: h
- y-axis: T^2
You can use a graphing software or a programming language with plotting capabilities like Python or Excel.
2. Determine the slope of the graph
Assuming you plot the points (h, T²) and fit a straight line, the slope \(m\) can be calculated using two points or a linear regression method. The formula is:
\[ m = \frac{\Delta y}{\Delta x} = \frac{(T^2_{2} - T^2_{1})}{(h_{2} - h_{1})} \]
3. Intercept on the h-axis
To find the intercept on the h-axis, substitute \(T^2 = 0\) into the linear equation derived from the plot.
Using the equation \(H = mh + b\), set \(T^2 = 0\) and rearrange:
When \(T^2 = 0\): \[ 0 = m \cdot h + b \implies h = -\frac{b}{m} \]
4. Determining the value of g from the graph
From the slope \(m\) calculated from the graph, we can find \(g\):
\[ g = 4\pi^2 m \]
5. Determining the value of H from the graph
To find \(H\), we can use the y-intercept \(b\):
\[ H = b + h_{\text{intercept}} \]
Where \(h_{\text{intercept}}\) is the value derived from step 3.
6. Estimate the error in g
To estimate the error in \(g\), you would need to consider the standard deviation of the slopes from repeated measurements or variations in your data. This can be calculated using the formula:
\[ \text{Error in } g = \frac{4\pi^2}{n} \sqrt{\sum \left(m_i - \bar{m}\right)^2} \]
Where \(n\) is the number of measurements and \(\bar{m}\) is the calculated mean slope.
Final Note
For actual calculations and plotting, you would typically use software tools to fit the data points accurately and derive the equation parameters, as hand calculating each measurement would be prone to human error.