We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to the 2.63 sec, 2.56 sec, 2.42 se, 2.71 sec, 2.80 sec

Options

A. The absolute errors in the measurements are 0.01 sec, 0.06 sec, 0.20 sec, 0.20 sec, 0.09 sec, 0.18 sec
B. The mean absolute error is 0.11 sec
C. Then mean time period can be reported as `(2.62+0.11)` sec.
D. The time period can actually reported as `(2.6+0.1)` sec, with a percentatge error of 4%

Answer

To solve the problem of measuring the period of oscillation of a simple pendulum based on the given readings, we will follow these steps:

### Step 1: List the Measurements
The measurements of the period of oscillation are:
- 2.63 seconds
- 2.56 seconds
- 2.42 seconds
- 2.71 seconds
- 2.80 seconds

### Step 2: Calculate the Mean Period
To find the mean period of oscillation, we sum all the measurements and divide by the number of measurements.

\[ \text{Mean} = \frac{\text{Sum of measurements}}{\text{Number of measurements}} \]

Calculating the sum:
\[ 2.63 + 2.56 + 2.42 + 2.71 + 2.80 = 13.12 \text{ seconds} \]

Now, divide by the number of measurements (5):
\[ \text{Mean} = \frac{13.12}{5} = 2.624 \text{ seconds} \]

Rounding to two decimal places, we get:
\[ \text{Mean} \approx 2.62 \text{ seconds} \]

### Step 3: Calculate Absolute Errors
The absolute error for each measurement is calculated as the absolute difference between each measurement and the mean period.

1. For 2.63 seconds:
\[ |2.63 - 2.62| = 0.01 \text{ seconds} \]
2. For 2.56 seconds:
\[ |2.56 - 2.62| = 0.06 \text{ seconds} \]
3. For 2.42 seconds:
\[ |2.42 - 2.62| = 0.20 \text{ seconds} \]
4. For 2.71 seconds:
\[ |2.71 - 2.62| = 0.09 \text{ seconds} \]
5. For 2.80 seconds:
\[ |2.80 - 2.62| = 0.18 \text{ seconds} \]

### Step 4: List the Absolute Errors
The absolute errors are:
- 0.01 seconds
- 0.06 seconds
- 0.20 seconds
- 0.09 seconds
- 0.18 seconds

### Step 5: Calculate Mean Absolute Error
To find the mean absolute error, we sum the absolute errors and divide by the number of measurements.

\[ \text{Mean Absolute Error} = \frac{0.01 + 0.06 + 0.20 + 0.09 + 0.18}{5} = \frac{0.54}{5} = 0.108 \text{ seconds} \]

Rounding to two decimal places, we get:
\[ \text{Mean Absolute Error} \approx 0.11 \text{ seconds} \]

### Step 6: Report the Mean Period with Uncertainty
The mean period can be reported as:
\[ 2.62 \pm 0.11 \text{ seconds} \]

### Step 7: Calculate Percentage Error
To find the percentage error, we use the formula:
\[ \text{Percentage Error} = \left(\frac{\text{Mean Absolute Error}}{\text{Mean Period}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Error} = \left(\frac{0.11}{2.62}\right) \times 100 \approx 4.19\% \]

### Summary of Results
- Mean Period: \(2.62 \text{ seconds}\)
- Absolute Errors: \(0.01, 0.06, 0.20, 0.09, 0.18 \text{ seconds}\)
- Mean Absolute Error: \(0.11 \text{ seconds}\)
- Percentage Error: \(4.19\%\)