Thermodynamics

To solve the problem, we need to determine which wire among the four shows the maximum increase in length when subjected to the same force (load), and all are made of the same material.

1. Formula for Elongation in a Wire:
The extension (increase in length) in a wire under tension is given by:

Where:
- = force applied (same for all),
- = original length of the wire,
- = cross-sectional area of the wire,
- = Young’s modulus (same material ⇒ constant).

So,

2. Cross-Sectional Area:
Since the wire is cylindrical,
So,

3. Calculating for Each Option:

• Option 1:
• Option 2:
• Option 3:
• Option 4:

4. Conclusion:
Since elongation is directly proportional to , the maximum value is 200 for Option 4.

Final Answer:
The wire with Length 50 cm, Diameter 0.5 mm has the maximum increase in length.

To solve the problem, we need to find the temperature at which the Fahrenheit and Kelvin scales show the same numerical value.

1. Understanding the Relation Between Fahrenheit and Kelvin:
The conversion formulas between Fahrenheit (°F), Celsius (°C), and Kelvin (K) are:

2. Set Kelvin Equal to Fahrenheit:
We are told that Fahrenheit and Kelvin temperatures are numerically equal, i.e.,

Using the conversion of °F to °C and then to K:
Let be the Celsius temperature.
Then and

3. Set the Equations Equal:

4. Solve for :

5. Convert to Fahrenheit or Kelvin:

Final Answer:
The Fahrenheit and Kelvin scales show the same reading at 574.6°F.

To solve the problem, we need to determine the ratio of heat energies required per unit volume for two substances so that both experience the same rise in temperature.

1. Understanding the Formula:
The heat energy required per unit volume for a temperature change is given by:

Where:
- = heat energy per unit volume
- = density of the substance
- = specific heat capacity
- = temperature rise (same for both)

2. Given:
- Ratio of densities:
- Ratio of specific heat capacities:
- is same for both

3. Apply the Formula:


So the ratio of heat energies per unit volume is:

4. Substitute the Values:

Final Answer:
The ratio of heat energies required per unit volume is 1 : 2.

Step 1: Expression for and From the first law of thermodynamics, the change in internal energy is related to heat added at constant volume and work done: The relation between internal energy and pressure-volume work is given by: Now, considering the general thermodynamic relation: where is the universal gas constant.

Step 2: Finding the ratio We can use the thermodynamic identity for the ratio of specific heat capacities: From the given equation , and the fact that is related to the equation , we substitute the values and derive the relation: Thus, the correct answer is:

Step 1: Use the Ideal Gas Equation From the ideal gas equation: where: - is pressure, - is volume, - is the number of moles, - is the universal gas constant, - is temperature. Since the volume remains constant, we can use: where: - , , , - , , .

Step 2: Solve for Cancel and solving for : Thus, the correct answer is:

Step 1: The efficiency of a Carnot engine is given by: where and are the temperatures of the source and sink, respectively.

Step 2: The initial temperatures are and . The initial efficiency is:

Step 3: After the sink temperature decreases by 10\%, the new temperature of the sink is . The new efficiency is:

Step 4: The change in efficiency is:

Step 5: The efficiency increases by 7.5\%, so Option (4) is correct.

Step 1: Relation Between Time Period and Length of a Pendulum The time period of a pendulum is given by: where: - is the length of the pendulum, - is the acceleration due to gravity. Since the pendulum length increases with temperature, we consider the fractional change in length: where: - is the coefficient of linear expansion, - is the temperature change. \vspace{0.5cm}

Step 2: Calculate Given that the pendulum loses 10.8 seconds per day at and gains 10.8 seconds per day at , the total time change per day is: Since time lost or gained per day is given by: Using: Solving for :

Step 1: Understanding the Relationship Between Heat and Work
For an ideal gas undergoing an isoic process, the first law of thermodynamics states: For a monatomic gas: Thus, the heat supplied: For a diatomic gas: Thus, the heat supplied:

Step 2: Ratio Calculation

Step 3: Conclusion
Thus, the ratio is:

Step 1: Understanding heat exchange
When steam condenses into water at , it releases latent heat. This heat is absorbed by the cooler water at , raising its temperature until thermal equilibrium is reached.

Step 2: Heat released by steam during condensation
The heat released when g of steam condenses into water at is: Given that cal/g, the total heat released by condensation is:

Step 3: Heat lost by condensed water cooling from to equilibrium temperature
Let the final equilibrium temperature be . The heat lost by the condensed water when it cools from to is:

Step 4: Heat gained by water at
The heat gained by g of water to reach is:

Step 5: Applying heat conservation
By the principle of conservation of energy:

Step 6: Solve for the mass ratio
For equilibrium, solving for , we approximate . Substituting: Thus, the final mass ratio of steam to water in equilibrium is:

Step 1: Formula for Newton's Law of Cooling:
For small temperature differences or linear approximation, we use: where is the surrounding temperature.
Step 2: Case 1 ( in 10 min):

Step 3: Case 2 ( in 15 min):

Step 4: Solve for :
Divide equation (1) by (2):
Step 5: Find :
From (1): .
Step 6: Case 3 ( in min):

Step 1: Understand the Gas Law:
For a closed vessel, Volume ( ) is constant. According to Gay-Lussac's Law, Pressure ( ) is directly proportional to Absolute Temperature ( ).
Step 2: Substitute Values:
Given: Percentage increase in pressure, . Change in temperature (Change in Celsius is same as Kelvin). Let initial temperature be Kelvin.
Step 3: Convert to Celsius: