In a certain obervation we got,`l = 23.2cm,r =1.32cm` and time taken for 10 oscillations was 10.0 s. Find, maximum percentage error in determinaton of 'g'.

Explanation

To find the maximum percentage error in the determination of 'g', we will follow these steps:

### Step 1: Understand the given values
- Length (L) = 23.2 cm
- Radius (R) = 1.32 cm
- Time for 10 oscillations (T_total) = 10.0 s

### Step 2: Identify the errors in measurements
- Maximum error in L (ΔL) = 0.1 cm (as estimated from the reading)
- Maximum error in R (ΔR) = 0.01 cm (as estimated from the reading)
- Maximum error in time (ΔT) = 0.1 s (as estimated from the reading)

### Step 3: Calculate the effective length
The effective length (L + R) is:
\[ L + R = 23.2 \, \text{cm} + 1.32 \, \text{cm} = 24.52 \, \text{cm} \]

### Step 4: Convert time for one oscillation
Time for one oscillation (T) is:
\[ T = \frac{T_{\text{total}}}{10} = \frac{10.0 \, \text{s}}{10} = 1.0 \, \text{s} \]

### Step 5: Write the formula for g
The formula for g in terms of T and L is:
\[ T^2 = \frac{4\pi^2 (L + R)}{g} \]
Rearranging gives:
\[ g = \frac{4\pi^2 (L + R)}{T^2} \]

### Step 6: Calculate the relative error in g
Using the formula for the relative error:
\[ \frac{\Delta g}{g} = \frac{\Delta L}{L + R} + \frac{\Delta R}{L + R} + 2 \frac{\Delta T}{T} \]

Substituting the values:
- \( \Delta L = 0.1 \, \text{cm} \)
- \( \Delta R = 0.01 \, \text{cm} \)
- \( L + R = 24.52 \, \text{cm} \)
- \( \Delta T = 0.1 \, \text{s} \)
- \( T = 1.0 \, \text{s} \)

Calculating each term:
1. \( \frac{\Delta L}{L + R} = \frac{0.1}{24.52} \approx 0.00407 \)
2. \( \frac{\Delta R}{L + R} = \frac{0.01}{24.52} \approx 0.000407 \)
3. \( 2 \frac{\Delta T}{T} = 2 \times \frac{0.1}{1.0} = 0.2 \)

Adding these together:
\[ \frac{\Delta g}{g} \approx 0.00407 + 0.000407 + 0.2 \approx 0.204477 \]

### Step 7: Calculate the percentage error
To find the maximum percentage error:
\[ \text{Percentage Error} = \frac{\Delta g}{g} \times 100\% \approx 0.204477 \times 100\% \approx 20.45\% \]

### Final Answer
The maximum percentage error in the determination of 'g' is approximately **20.45%**.

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