The period of oscillation of a simple pendulum is given by `T=2pisqrt((l)/(g))` where l is about 100 cm and is known to have 1 mm accuracy. The period is about 2 s. The time of 100 oscillation is measrued by a stop watch of least count 0.1 s. The percentage error is g is

Options

A. `0.1 %`
B. `1%`
C. `0.2%`
D. `0.8%`

Explanation

To solve the problem, we need to determine the percentage error in the value of \( g \) using the formula for the period of a simple pendulum. Let's break down the steps:

### Step 1: Write the formula for the period of a simple pendulum
The period \( T \) of a simple pendulum is given by:
\[ T = 2\pi\sqrt{\frac{L}{g}} \]

### Step 2: Rearrange the formula to find \( g \)
We can rearrange the formula to express \( g \) in terms of \( T \) and \( L \):
\[ g = \frac{4\pi^2 L}{T^2} \]

### Step 3: Identify the variables and their uncertainties
From the problem:
- Length \( L = 100 \, \text{cm} = 10000 \, \text{mm} \) with an uncertainty \( \Delta L = 1 \, \text{mm} \)
- Period \( T \approx 2 \, \text{s} \) with a measurement of 100 oscillations, so the total time for 100 oscillations is measured with a stopwatch of least count \( \Delta t = 0.1 \, \text{s} \).

### Step 4: Calculate the percentage error in \( L \)
The percentage error in \( L \) is given by:
\[ \text{Percentage error in } L = \frac{\Delta L}{L} \times 100 = \frac{1 \, \text{mm}}{10000 \, \text{mm}} \times 100 = 0.01\% \]

### Step 5: Calculate the percentage error in \( T \)
Since the period \( T \) is measured for 100 oscillations, the total time \( T_{100} \) is:
\[ T_{100} = 100 \times T \]
The percentage error in \( T \) can be calculated as:
\[ \text{Percentage error in } T = \frac{\Delta t}{T_{100}} \times 100 = \frac{0.1 \, \text{s}}{100 \times 2 \, \text{s}} \times 100 = 0.05\% \]

### Step 6: Calculate the total percentage error in \( g \)
The formula for the percentage error in \( g \) is:
\[ \text{Percentage error in } g = \text{Percentage error in } L + 2 \times \text{Percentage error in } T \]
Substituting the values we calculated:
\[ \text{Percentage error in } g = 0.01\% + 2 \times 0.05\% = 0.01\% + 0.1\% = 0.11\% \]

### Step 7: Final result
Thus, the percentage error in \( g \) is approximately \( 0.11\% \).