What is the percentage error in the measurement of time period of a pendulum if maximum errors in the measurement of `l` ands `g` are 2% and 4% respectively?

Options

A. `6%`
B. `4%`
C. `3%`
D. `5%`

Answer

To find the percentage error in the measurement of the time period \( T \) of a pendulum, we start with the formula for the time period:

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

### Step 1: Identify the formula for the time period
The time period \( T \) is given by:

\[ T = 2\pi \sqrt{\frac{L}{g}} \]

### Step 2: Determine the errors in measurements
We are given:
- The maximum error in the measurement of \( L \) is 2%
- The maximum error in the measurement of \( g \) is 4%

### Step 3: Calculate the percentage error in \( T \)
To find the percentage error in \( T \), we can use the formula for the propagation of errors. The percentage error in \( T \) can be calculated using the following relationship:

\[ \frac{\Delta T}{T} = \frac{1}{2} \frac{\Delta L}{L} + \frac{1}{2} \frac{\Delta g}{g} \]

Where:
- \( \Delta T \) is the absolute error in \( T \)
- \( \Delta L \) is the absolute error in \( L \)
- \( \Delta g \) is the absolute error in \( g \)

### Step 4: Substitute the given percentage errors
Substituting the given percentage errors into the equation:

\[ \frac{\Delta T}{T} = \frac{1}{2} \times 2\% + \frac{1}{2} \times 4\% \]

### Step 5: Perform the calculations
Calculating the right-hand side:

\[ \frac{\Delta T}{T} = \frac{1}{2} \times 2 + \frac{1}{2} \times 4 = 1 + 2 = 3\% \]

### Step 6: Conclusion
Thus, the percentage error in the measurement of the time period \( T \) is:

\[ \text{Percentage Error in } T = 3\% \]

### Final Answer
The percentage error in the measurement of the time period of the pendulum is **3%**.

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