A barometer is faulty . When the true barometer reading are `73` cm and `75` cm of Hg , the faulty barometer reads `69` cm and `70` cm respectively (i) What is the total length of the barometer tube? (ii) What is the reading when the faulty barometer reads `69.5`cm? (iii) What is the faulty barometer reading when the true barometer reads `74`cm?

Explanation

To solve the problem step by step, we will break it down into three parts as per the question:

### Part (i): Finding the total length of the barometer tube

1. **Understanding the problem**:
- We have two true barometer readings: 73 cm and 75 cm.
- Corresponding faulty barometer readings are: 69 cm and 70 cm.

2. **Setting up the equations**:
- Let the height of trapped air in the barometer be \( x \).
- For the first case (true reading = 73 cm, faulty reading = 69 cm):
\[ P_0 = P_{\text{air}} + \rho g (73) \quad \text{(1)} \]
\[ P_0 = P_1 + \rho g (69) \quad \text{(2)} \]
- For the second case (true reading = 75 cm, faulty reading = 70 cm):
\[ P_0 = P_{\text{air}} + \rho g (75) \quad \text{(3)} \]
\[ P_0 = P_2 + \rho g (70) \quad \text{(4)} \]

3. **Finding the height of trapped air**:
- From equations (1) and (2):
\[ P_{\text{air}} + \rho g (73) = P_1 + \rho g (69) \]
Rearranging gives:
\[ P_1 = P_{\text{air}} + \rho g (4) \quad \text{(5)} \]
- From equations (3) and (4):
\[ P_{\text{air}} + \rho g (75) = P_2 + \rho g (70) \]
Rearranging gives:
\[ P_2 = P_{\text{air}} + \rho g (5) \quad \text{(6)} \]

4. **Using the constant temperature process**:
- Since the trapped air follows a constant temperature process, we can write:
\[ P_1 \cdot V_1 = P_2 \cdot V_2 \]
- Substituting from equations (5) and (6):
\[ \rho g (4) \cdot A \cdot x = \rho g (5) \cdot A \cdot (x - 1) \]
- Canceling \( A \) and \( \rho g \):
\[ 4x = 5(x - 1) \]
- Solving gives:
\[ 4x = 5x - 5 \implies x = 5 \text{ cm} \]

5. **Calculating the total length of the barometer tube**:
- The total length of the barometer tube is the height of the mercury column plus the height of the trapped air:
\[ \text{Total Length} = 69 \text{ cm} + 5 \text{ cm} = 74 \text{ cm} \]

### Part (ii): Finding the reading when the faulty barometer reads 69.5 cm

1. **Using the constant temperature process**:
- Let \( h \) be the true reading when the faulty barometer reads 69.5 cm.
- The height of trapped air will be \( x - 0.5 \) cm since the faulty reading is 0.5 cm less than 69 cm.
- Using the equation:
\[ P_1 \cdot V_1 = P_3 \cdot V_3 \]
- Substituting:
\[ \rho g (4) \cdot A \cdot x = P_3 \cdot A \cdot (x - 0.5) \]
- Canceling \( A \) and \( \rho g \):
\[ 4x = P_3 (x - 0.5) \]
- Substituting \( x = 5 \):
\[ 4(5) = P_3 (5 - 0.5) \implies 20 = P_3 (4.5) \]
- Solving gives:
\[ P_3 = \frac{20}{4.5} \approx 4.44 \text{ cm} \]
- The actual height:
\[ \text{Actual Reading} = 69.5 + 4.44 = 73.94 \text{ cm} \]

### Part (iii): Finding the faulty barometer reading when the true barometer reads 74 cm

1. **Using the constant temperature process**:
- Let \( y \) be the faulty reading when the true barometer reads 74 cm.
- The height of trapped air will be \( x \) cm.
- Using the equation:
\[ P_1 \cdot V_1 = P_2 \cdot V_2 \]
- Substituting:
\[ \rho g (4) \cdot A \cdot x = \rho g (74 - y) \cdot A \cdot (74 - x) \]
- Canceling \( A \) and \( \rho g \):
\[ 4x = (74 - y)(74 - x) \]
- Rearranging gives:
\[ 74 - y = \frac{4x}{74 - x} \]
- Substituting \( x = 5 \):
\[ 74 - y = \frac{20}{69} \implies y = 74 - \frac{20}{69} \approx 69.528 \text{ cm} \]

### Final Answers:
1. Total length of the barometer tube: **74 cm**
2. Actual reading when the faulty barometer reads 69.5 cm: **73.94 cm**
3. Faulty barometer reading when the true barometer reads 74 cm: **69.528 cm**