$$ T=2 \pi \sqrt{(1 / g)} $$, where all symbols carry their usual meaning. If the clock losses 3 seconds in 5 minutes, determine the error in measuring $$ g $$ given that, $$ T-2.22 \mathrm{\sec}, l=121.6 \mathrm{~cm}, \Delta T \mathrm{Ti}=0.01 \mathrm{\sec} $$, and $$ \Delta l= \pm 0.05 $$.

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To determine the error in measuring $$ g $$ based on the given information, we will follow these steps:

1. **Identify the known values:**
   - $$ T = 2.22 \, \text{sec} $$
   - $$ l = 121.6 \, \text{cm} = 1.216 \, \text{m} $$ (convert to meters)
   - $$ \Delta T = 0.01 \, \text{sec} $$
   - $$ \Delta l = 0.05 \, \text{cm} = 0.0005 \, \text{m} $$ (convert to meters)
   - The clock loses 3 seconds in 5 minutes, which we will use to find the error in $$ T $$.

2. **Calculate the actual period $$ T $$ based on the clock error:**
   - The clock loses 3 seconds in 5 minutes (300 seconds), so the effective period $$ T' $$ can be calculated as:
     $$
     T' = T + \text{error in time}
     $$
   - The error in time for 5 minutes is $$ \frac{3}{300} \times T $$.

3. **Use the formula for $$ g $$:**
   - The formula for $$ g $$ derived from the period $$ T $$ is:
     $$
     g = \frac{4 \pi^2 l}{T^2}
     $$

4. **Calculate the error in $$ g $$ using the propagation of uncertainty:**
   - The error in $$ g $$ can be calculated using the formula:
     $$
     \frac{\Delta g}{g} = 2 \frac{\Delta T}{T} + \frac{\Delta l}{l}
     $$
   - Rearranging gives:
     $$
     \Delta g = g \left( 2 \frac{\Delta T}{T} + \frac{\Delta l}{l} \right)
     $$

Now, let's perform the calculations step by step.

### Step 1: Calculate the effective period $$ T' $$

The effective period $$ T' $$ can be calculated as follows:
$$
T' = T + \frac{3}{300} \times T
$$

### Step 2: Calculate $$ g $$

Using the formula for $$ g $$:
$$
g = \frac{4 \pi^2 l}{T^2}
$$

### Step 3: Calculate the error in $$ g $$

Using the error propagation formula:
$$
\Delta g = g \left( 2 \frac{\Delta T}{T} + \frac{\Delta l}{l} \right)
$$

Now, let's compute these values.

First, we will calculate $$ T' $$ and then $$ g $$ and finally $$ \Delta g $$.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$2.22+\left(\frac{3}{300}\right)\times 2.22$$
- step1: Reduce the fraction:
  $$2.22+\frac{1}{100}\times 2.22$$
- step2: Multiply the numbers:
  $$2.22+\frac{111}{5000}$$
- step3: Convert the expressions:
  $$\frac{111}{50}+\frac{111}{5000}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{111\times 100}{50\times 100}+\frac{111}{5000}$$
- step5: Multiply the numbers:
  $$\frac{111\times 100}{5000}+\frac{111}{5000}$$
- step6: Transform the expression:
  $$\frac{111\times 100+111}{5000}$$
- step7: Multiply the numbers:
  $$\frac{11100+111}{5000}$$
- step8: Add the numbers:
  $$\frac{11211}{5000}$$
Calculate or simplify the expression $$ (4 * \pi^2 * 1.216) / (2.22^2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(4\pi ^{2}\times 1.216\right)}{2.22^{2}}$$
- step1: Remove the parentheses:
  $$\frac{4\pi ^{2}\times 1.216}{2.22^{2}}$$
- step2: Convert the expressions:
  $$\frac{4\pi ^{2}\times 1.216}{\left(\frac{111}{50}\right)^{2}}$$
- step3: Multiply:
  $$\frac{\frac{608\pi ^{2}}{125}}{\left(\frac{111}{50}\right)^{2}}$$
- step4: Evaluate the power:
  $$\frac{\frac{608\pi ^{2}}{125}}{\frac{12321}{2500}}$$
- step5: Multiply by the reciprocal:
  $$\frac{608\pi ^{2}}{125}\times \frac{2500}{12321}$$
- step6: Reduce the numbers:
  $$608\pi ^{2}\times \frac{20}{12321}$$
- step7: Multiply:
  $$\frac{608\pi ^{2}\times 20}{12321}$$
- step8: Multiply:
  $$\frac{12160\pi ^{2}}{12321}$$
Calculate or simplify the expression $$ (2 * 0.01 / 2.22) + (0.0005 / 1.216) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{2\times 0.01}{2.22}\right)+\left(\frac{0.0005}{1.216}\right)$$
- step1: Multiply the numbers:
  $$\left(\frac{0.02}{2.22}\right)+\left(\frac{0.0005}{1.216}\right)$$
- step2: Divide the terms:
  $$\frac{1}{111}+\left(\frac{0.0005}{1.216}\right)$$
- step3: Divide the terms:
  $$\frac{1}{111}+\frac{1}{2432}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{2432}{111\times 2432}+\frac{111}{2432\times 111}$$
- step5: Multiply the numbers:
  $$\frac{2432}{269952}+\frac{111}{2432\times 111}$$
- step6: Multiply the numbers:
  $$\frac{2432}{269952}+\frac{111}{269952}$$
- step7: Transform the expression:
  $$\frac{2432+111}{269952}$$
- step8: Add the numbers:
  $$\frac{2543}{269952}$$
Calculate or simplify the expression $$ 9.740637 * (2 * 0.00942) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$9.740637\left(2\times 0.00942\right)$$
- step1: Remove the parentheses:
  $$9.740637\times 2\times 0.00942$$
- step2: Multiply the terms:
  $$19.481274\times 0.00942$$
- step3: Convert the expressions:
  $$\frac{9740637}{500000}\times \frac{471}{50000}$$
- step4: Multiply the fractions:
  $$\frac{9740637\times 471}{500000\times 50000}$$
- step5: Multiply the terms:
  $$\frac{4587840027}{500000\times 50000}$$
- step6: Multiply the terms:
  $$\frac{4587840027}{25000000000}$$
Now, let's summarize the calculations step by step:

1. **Effective Period $$ T' $$**:
   $$
   T' = 2.22 + \frac{3}{300} \times 2.22 \approx 2.2422 \, \text{sec}
   $$

2. **Calculate $$ g $$**:
   $$
   g = \frac{4 \pi^2 \cdot 1.216}{(2.22)^2} \approx 9.740637 \, \text{m/s}^2
   $$

3. **Calculate the relative error in $$ g $$**:
   $$
   \Delta g = g \left( 2 \frac{\Delta T}{T} + \frac{\Delta l}{l} \right) \approx 9.740637 \left( 2 \cdot 0.00942 \right) \approx 0.183514 \, \text{m/s}^2
   $$

### Final Result:
The error in measuring $$ g $$ is approximately $$ 0.1835 \, \text{m/s}^2 $$.
The error in measuring $$ g $$ is approximately $$ 0.1835 \, \text{m/s}^2 $$.

, where all symbols carry their usual meaning. If the clock losses 3 seconds in 5 minutes, determine the error in measuring given that, , and .

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