Vibrate and record. Tabulate your readings.
| s/4 | u (cm) | v (cm) | m = v/4 |
|---|---|---|---|
| 1.05 | 25.00 | 96.50 | |
| 2.00 | 30.00 | 82.40 | |
| 3.00 | 25.00 | 77.50 | |
| 4.00 | 40.00 | 76.00 | |
| 5.00 | 45.00 | 77.10 | |
Answer (2)
Calculate m using m = v / u .
Calculate s using s = 4 ∗ ( s /4 ) .
Perform linear regression on u vs v to find the relationship: v = − 0.55 u + 100.10 .
Conclude a weak negative correlation between u and v .
Explanation
Analyzing the Data We are given a table of data with columns s /4 , u (in cm), v (in cm), and m = v / u . Our goal is to analyze the data and determine any relationships between the variables.
Calculating m First, let's calculate the values of m for each row using the formula m = v / u . The calculated values of m are approximately 3.86, 2.75, 3.10, 1.90, and 1.71.
Calculating s Next, we calculate the values of s using the formula s = 4 ∗ ( s /4 ) . The calculated values of s are 4.2, 8.0, 12.0, 16.0, and 20.0.
Linear Regression Analysis We can perform a linear regression analysis on the u vs v data. The equation of the best-fit line is approximately v = − 0.55 u + 100.10 , with a correlation coefficient of approximately -0.588. This indicates a weak negative correlation between u and v .
Conclusion Based on the calculations and analysis, we can see that there is a weak negative correlation between u and v . As u increases, v tends to decrease slightly. The value of m decreases as both s and u increase. The relationship is not strong, as indicated by the correlation coefficient.
Examples
Understanding the relationships between variables is crucial in many scientific and engineering applications. For example, in optics, the relationship between object distance (u) and image distance (v) is described by the lens equation. Analyzing experimental data using linear regression can help determine the focal length of a lens and assess the quality of the experimental setup. Similarly, in mechanics, analyzing the relationship between force and displacement can help determine the spring constant of a spring.
The calculations provide the values of m and s from the given data, indicating the relationships between these variables. Performing a linear regression analysis on the u vs. v data could reveal a weak negative correlation. The results suggest that as u increases, v tends to decrease slightly, which can inform further investigations in physics experiments.
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