A student measures the time for 20 oscillations of a pendulum. The experiment is repeated four times.
The measurements are: 10.45 s, 10.30 s, 10.70 s, 10.55 s
What is the best estimate of the average time and the uncertainty in the average time for 20
oscillations?

To determine the best estimate of the average time and the uncertainty in the average time for 20 oscillations, follow these steps:

### 1. Calculate the Average Time
The average time \(\bar{T}\) is calculated by taking the mean of the four measurements.

\[
\bar{T} = \frac{T_1 + T_2 + T_3 + T_4}{4}
\]

Substituting the given values:

\[
\bar{T} = \frac{10.45 \, \text{s} + 10.30 \, \text{s} + 10.70 \, \text{s} + 10.55 \, \text{s}}{4} = \frac{42.00 \, \text{s}}{4} = 10.50 \, \text{s}
\]

So, the average time is \( \bar{T} = 10.50 \, \text{s} \).

### 2. Calculate the Uncertainty in the Average Time
The uncertainty in the average time is typically estimated using the standard deviation of the measurements divided by the square root of the number of measurements (which is 4 in this case).

First, calculate the deviations of each measurement from the mean, then square them:

\[
(T_1 - \bar{T})^2 = (10.45 - 10.50)^2 = (-0.05)^2 = 0.0025
\]
\[
(T_2 - \bar{T})^2 = (10.30 - 10.50)^2 = (-0.20)^2 = 0.0400
\]
\[
(T_3 - \bar{T})^2 = (10.70 - 10.50)^2 = (0.20)^2 = 0.0400
\]
\[
(T_4 - \bar{T})^2 = (10.55 - 10.50)^2 = (0.05)^2 = 0.0025
\]

Next, calculate the variance:

\[
\text{Variance} = \frac{0.0025 + 0.0400 + 0.0400 + 0.0025}{4} = \frac{0.0850}{4} = 0.02125
\]

The standard deviation \(\sigma\) is the square root of the variance:

\[
\sigma = \sqrt{0.02125} \approx 0.146 \, \text{s}
\]

Finally, the uncertainty in the average time \(\Delta \bar{T}\) is:

\[
\Delta \bar{T} = \frac{\sigma}{\sqrt{4}} = \frac{0.146}{2} \approx 0.073 \, \text{s}
\]

### 3. Conclusion
The best estimate of the average time for 20 oscillations is \(10.50 \, \text{s}\) with an uncertainty of \( \pm 0.07 \, \text{s} \).

### Summary:
- **Best Estimate of Average Time:** \(10.50 \, \text{s}\)
- **Uncertainty in the Average Time:** \( \pm 0.07 \, \text{s} \)

Would you like more details on any of these steps or have any questions?

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Here are 5 related questions you might find useful:
1. How do you calculate the standard deviation for a different set of measurements?
2. What is the significance of uncertainty in experimental measurements?
3. How does the number of trials affect the uncertainty in the average measurement?
4. What are some methods to reduce uncertainty in experimental physics?
5. How would the calculation change if there were an additional measurement?

**Tip:** When calculating uncertainties, always ensure you're using the correct formula for the number of measurements to avoid underestimating or overestimating the uncertainty.

Learn how to calculate the average time and uncertainty in experimental physics using measurements of a pendulum's oscillation. Step-by-step solution includes averaging measurements and determining uncertainty. Suitable for high school physics students.

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