If `P` is the atmospheric pressure in the last problems find the percentage increase in tension of the string after heating

Options

A. `(25)/((1 - P//P_(0)))`
B. `(25)/((1 - P//P_(0)))`
C. `25 (1 - P//P_(0))`
D. `25 (1 - P//P_(0))`

Explanation

To solve the problem of finding the percentage increase in tension of the string after heating, we can follow these steps:

### Step 1: Define the initial tension
Let the initial tension in the string before heating be denoted as \( T_0 \). According to the problem, the initial tension can be expressed as:
\[ T_0 = P_0 \cdot A \]
where \( P_0 \) is the atmospheric pressure and \( A \) is the cross-sectional area of the string.

### Step 2: Define the tension after heating
After heating, the tension in the string becomes:
\[ T = P_2 \cdot A \]
where \( P_2 \) is the new pressure after heating.

### Step 3: Relate the pressures before and after heating
From the problem, we know that:
\[ P_2 = \frac{5}{4} P_0 \]
Thus, the tension after heating can be expressed as:
\[ T = P_2 \cdot A = \frac{5}{4} P_0 \cdot A \]

### Step 4: Calculate the increase in tension
The increase in tension \( \Delta T \) can be calculated as:
\[ \Delta T = T - T_0 = \left(\frac{5}{4} P_0 \cdot A\right) - (P_0 \cdot A) \]
Simplifying this gives:
\[ \Delta T = \left(\frac{5}{4} P_0 - P_0\right) \cdot A = \left(\frac{5}{4} P_0 - \frac{4}{4} P_0\right) \cdot A = \frac{1}{4} P_0 \cdot A \]

### Step 5: Calculate the percentage increase in tension
The percentage increase in tension can be calculated using the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta T}{T_0}\right) \cdot 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \left(\frac{\frac{1}{4} P_0 \cdot A}{P_0 \cdot A}\right) \cdot 100 = \left(\frac{1}{4}\right) \cdot 100 = 25\% \]

### Final Answer
The percentage increase in tension of the string after heating is:
\[ \text{Percentage Increase} = 25\% \]