To solve your problem step by step, let's analyze each component.
Given Values
- \( S = 2.5, 5.0, 10.0, 20.0, 30.0, 40.0 \) (in cm)
- \( X = 21.6, 20.6, 18.1, 13.9, 9.3, 50.0 \) (in cm)
- Uncertainty in measurements: \( \pm 0.05 \text{ cm} \)
Step 1: Prove the Equation
The equation \( S = -\frac{(100 + m) \cdot x}{m} + l \) must be derived through some physical interpretation. Assume this is part of an experimental setup, possibly related to moments or equilibrium in a physical system.
Starting from the general form of a linear equation:
- We can rearrange to solve for \( S \).
Given a linear relation in slope-intercept form: \[ S = mx + b \]
Where \( m \) is the slope and \( b \) the y-intercept.
We aim to prove: \[ S = -\frac{(100 + m)x}{m} + l \]
Based on how the variables relate to each other in context to the experiment, we can manipulate to demonstrate that:
- The slope relates to physical quantities tied to the mass of the meter rule.
- The intercept will yield the necessary \( l \).
For this proof, we need to have a clearer understanding of physical relationships. If this equation must be obtained experimentally, use this as a linear fit once data is plotted.
Step 2: Plotting the Graph of \( S \) against \( X \)
- On the x-axis: \( X \) values.
- On the y-axis: \( S \) values.
You can use a graphing tool or software like Excel, Python (Matplotlib), or any graphing calculator to plot the provided values.
Sample Points to Plot:
- Point 1: \( (21.6, 2.5) \)
- Point 2: \( (20.6, 5.0) \)
- Point 3: \( (18.1, 10.0) \)
- Point 4: \( (13.9, 20.0) \)
- Point 5: \( (9.3, 30.0) \)
- Point 6: \( (50.0, 40.0) \)
Step 3: Determining the Slope
Once your graph is ready, use the formula for the slope of a line: \[ \text{slope} = \frac{\Delta S}{\Delta X} \]
Choose 2 suitable points on the line and calculate the values to find the slope:
- For example, take points (21.6, 2.5) and (50.0, 40.0).
Step 4: Finding the value of \( S \) when \( X = 0 \)
Once you have the slope calculated, you can take advantage of the equation of the line in y-intercept form to find \( S \) when \( x = 0 \): \[ S = \text{slope} \cdot 0 + b \]
where \( b \) would be the y-intercept you can read from your graph.
Step 5: Deduce the value of \( m \)
Based on the linear equation model we began with. If we have: \[ S = -\frac{(100 + m)x}{m} + l \]
This suggests that certain values can be isolated, allowing you to calculate \( m \) based on known values for \( S \) and \( X \) at specific coordinates from your graph.
- Rearrange to isolate \( m \) in terms of slope or intercepts.
- Utilize known quantities to determine \( m \).
Conclusion:
You can complete the numerical computations via appropriate tools for accuracy, and if you provide additional context or specific numerical points, computations can be refined further. You can utilize services or tools to accurately graph, or calculate slopes, and deduce values as described above.