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 find the percentage error in the acceleration due to gravity \( g \) using the formula for the period of a simple pendulum, we can follow these steps:

### Step 1: Write the formula for the period of a simple pendulum
The period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{l}{g}} \]
Where:
- \( T \) is the period of oscillation
- \( l \) is the length of the pendulum
- \( g \) is the acceleration due to gravity

### Step 2: Rearrange the formula to express \( g \)
We can rearrange the formula to express \( g \):
\[ g = \frac{4\pi^2 l}{T^2} \]

### Step 3: Determine the errors in \( l \) and \( T \)
Given:
- \( l = 100 \, \text{cm} = 1 \, \text{m} \) with an accuracy of \( \Delta l = 1 \, \text{mm} = 0.1 \, \text{cm} = 0.001 \, \text{m} \)
- The period \( T \) is about \( 2 \, \text{s} \)
- The time for 100 oscillations is measured with a stopwatch of least count \( 0.1 \, \text{s} \)

The period for 100 oscillations is:
\[ T_{100} = 100 \times T \]
Thus, the error in \( T \) can be calculated as:
\[ \Delta T = \frac{0.1 \, \text{s}}{100} = 0.001 \, \text{s} \]

### Step 4: Calculate the percentage error in \( g \)
The formula for the percentage error in \( g \) is given by:
\[ \frac{\Delta g}{g} = \frac{\Delta l}{l} + 2 \frac{\Delta T}{T} \]
Where:
- \( \Delta g \) is the error in \( g \)
- \( \Delta l \) is the error in \( l \)
- \( \Delta T \) is the error in \( T \)

Substituting the values:
- \( l = 1 \, \text{m} \) and \( \Delta l = 0.001 \, \text{m} \)
- \( T = 2 \, \text{s} \) and \( \Delta T = 0.001 \, \text{s} \)

Calculating the individual errors:
\[ \frac{\Delta l}{l} = \frac{0.001}{1} = 0.001 \]
\[ \frac{\Delta T}{T} = \frac{0.001}{2} = 0.0005 \]

Now substituting these into the percentage error formula:
\[ \frac{\Delta g}{g} = 0.001 + 2 \times 0.0005 = 0.001 + 0.001 = 0.002 \]

### Step 5: Convert to percentage
To convert to percentage:
\[ \text{Percentage error in } g = 0.002 \times 100\% = 0.2\% \]

### Final Answer
The percentage error in \( g \) is \( 0.2\% \).

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