https://file.aiquickdraw.com/mathbot/file/1726332793748xhfni2ix.jpg

From the image, I can extract the following information:

### Experimental Description:
- The experiment is aimed at measuring the spring constant (\(k\)) by measuring the oscillation period (\(T\)) of a spring. A mass of 200 grams is attached to the spring, and a force is applied by pulling the spring down by a displacement \(\Delta x\), which varies for each experiment. The time for 10 oscillations (\(f\)) is recorded.

### Data from the experiment:
| No  | \(\Delta x\) (cm) | \(f\) (s) |
|-----|------------------|----------|
| 1   | 1.5              | 9.0      |
| 2   | 2.0              | 10.2     |
| 3   | 2.15             | 11.0     |
| 4   | 3.0              | 12.1     |
| 5   | 3.5              | 12.9     |
| 6   | 4.0              | 13.5     |

### Tasks:
1. **Determine the final spring constant (\(k\)) and its relative error.**
2. **Calculate the relative percentage error with respect to the average spring constant (\(k\)), and then find the percentage precision of the measurement.**

I will start by solving the first part: calculating the spring constant \(k\) using the given data.

Let me proceed with the calculation.The calculated average spring constant (\(k\)) is approximately **6.39 N/m**.

The relative error is **29.05%**, indicating the variability in the measurements.

Would you like to proceed with calculating the percentage precision, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. How is the spring constant (\(k\)) related to the oscillation period (\(T\)) in this experiment?
2. Why is the relative error significant in evaluating the accuracy of the measurements?
3. How does the displacement (\(\Delta x\)) affect the period of oscillation in a spring system?
4. What would happen to the spring constant if a different mass were used?
5. How would using a more precise time-measuring device affect the experiment’s results?

### Tip:
When calculating experimental errors, both systematic errors (consistent bias) and random errors (unpredictable fluctuations) should be considered to improve the reliability of results.

You are conducting an experiment to measure the spring constant (k) by determining the oscillation period (T) of a spring. A 200-gram mass is attached to the spring, and a varying displacement (Δx) is applied. You record the time for 10 oscillations (f). The task includes calculating the final spring constant (k) and its relative error, and then finding the percentage precision of the measurement.

In this physics experiment, measure the spring constant (k) by calculating the oscillation period (T) with a given displacement (Δx) and mass. The task involves determining the spring constant and relative error based on recorded data. Suitable for students in Grades 11-12.

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