To find the average absolute error of the recorded periods of oscillation of a simple pendulum, we will follow these steps:
### Step 1: Calculate the Mean Value
First, we need to find the mean (average) of the recorded periods.
The recorded periods are:
- \( t_1 = 2.63 \, s \)
- \( t_2 = 2.56 \, s \)
- \( t_3 = 2.42 \, s \)
- \( t_4 = 2.71 \, s \)
- \( t_5 = 2.80 \, s \)
The mean value \( \bar{t} \) is calculated as follows:
\[
\bar{t} = \frac{t_1 + t_2 + t_3 + t_4 + t_5}{5} = \frac{2.63 + 2.56 + 2.42 + 2.71 + 2.80}{5}
\]
Calculating the sum:
\[
2.63 + 2.56 + 2.42 + 2.71 + 2.80 = 13.12
\]
Now, divide by 5:
\[
\bar{t} = \frac{13.12}{5} = 2.624 \, s
\]
### Step 2: Calculate the Absolute Errors
Next, we calculate the absolute error for each recorded period:
1. \( \Delta t_1 = |t_1 - \bar{t}| = |2.63 - 2.624| = 0.006 \, s \)
2. \( \Delta t_2 = |t_2 - \bar{t}| = |2.56 - 2.624| = 0.064 \, s \)
3. \( \Delta t_3 = |t_3 - \bar{t}| = |2.42 - 2.624| = 0.204 \, s \)
4. \( \Delta t_4 = |t_4 - \bar{t}| = |2.71 - 2.624| = 0.086 \, s \)
5. \( \Delta t_5 = |t_5 - \bar{t}| = |2.80 - 2.624| = 0.176 \, s \)
### Step 3: Calculate the Average Absolute Error
Now, we find the average of these absolute errors:
\[
\text{Average Absolute Error} = \frac{\Delta t_1 + \Delta t_2 + \Delta t_3 + \Delta t_4 + \Delta t_5}{5}
\]
Calculating the sum of absolute errors:
\[
0.006 + 0.064 + 0.204 + 0.086 + 0.176 = 0.536
\]
Now, divide by 5:
\[
\text{Average Absolute Error} = \frac{0.536}{5} = 0.1072 \, s
\]
### Final Result
Rounding to two decimal places, the average absolute error is approximately:
\[
\text{Average Absolute Error} \approx 0.11 \, s
\]
