Thermal Properties Of Matter

The correct option is (A) : K₁A₁ = K₂A₂

Use

Key Idea: Rate of Heat Flow

The rate of heat flow is directly proportional to the following:

Consider a rod maintained at a temperature difference of

between its ends. The rate of heat conduction is given by:

In this case, we are given the following parameters:

The rod with the above parameters will conduct the most heat because the rate of conduction depends on the ratio of . Therefore, the rate of heat conduction increases as the cross-sectional area increases and as the length decreases.

Change in Length for Two Rods

When two rods are subjected to the same change in temperature, the change in length for both rods should be the same:

The formula for the change in length is given by the expression:

Since the temperature change is the same for both rods, it cancels out from both sides:

Conclusion:

This equation indicates that the product of the initial length and the coefficient of linear expansion for each rod must be equal for both rods. Therefore, the relationship between the two rods is:

Height Calculation from Given Formula

The given equation is:

Where: - is the mass, - is the acceleration due to gravity, - is the height, - is a given length.

Step-by-Step Calculation:

Rearranging the equation to solve for :

Substituting the given values into the formula:

Simplifying:

Conclusion:

The calculated height is .

Thermodynamics Calculation: Finding

We are given a thermodynamic process involving temperature changes. Let’s break down the problem step by step:

Step 1: Temperature Relationships

We start with the following temperature relationships:

Step 2: Formulating the First Equation

We are given the temperature changes over a time interval and can use the following relationship between the initial and final temperatures:

The first equation is based on the temperature change from to over 10 minutes:

....(\)

This equation is based on the fact that heat exchange rate is proportional to the temperature difference.

Step 3: Formulating the Second Equation

The second equation is based on the temperature change from to over 10 minutes:

....(i\)

Step 4: Solving the Equations

Next, we need to take the ratio of the two equations (i) and (ii) to solve for the final temperature :

Step 5: Simplifying the Equation

After simplifying the equation, we get:

Step 6: Solving for

Rearranging this equation to solve for , we get:

Now, simplify further:

Finally, we find the value of :

Conclusion:

The final temperature is:

Radiation Emission and Wavelength

We are dealing with the wavelength corresponding to the maximum amount of emitted radiation, which can be derived using **Wien's Displacement Law**:

Step 1: Calculate the Wavelength

Where: - is the wavelength corresponding to the maximum emitted radiation, - (Wien's constant), - (temperature).

Substituting the values into the formula:

Step 2: Graph Interpretation

From the graph, we can see that the values of energy (represented as , , and ) follow the relationship:

Conclusion:

The maximum wavelength corresponding to the maximum emitted radiation is , and the energy values follow the relationship .

This question involves the concept of Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference between its own temperature and the ambient temperature. The formula used for Newton's Law of Cooling is:

where:

• is the temperature of the object at time
• is the ambient (room) temperature
• is a constant

The solution proceeds in two stages:

1. First cooling from 90°C to 80°C.
2. Then cooling from 80°C to 60°C.

Step 1: Cooling from 90°C to 80°C

Given that the coffee initially cools from 90°C to 80°C in minutes. Applying the formula for Newton's Law of Cooling, the temperature falls from to .

The average temperature during this time is:\)

Hence, the differential cooling equation becomes:

Simplifying gives:

Step 2: Cooling from 80°C to 60°C

Given the same room temperature, we calculate similarly where the temperature falls from to . The average temperature is:\)

Using the differential cooling equation:

Solving gives:

k = \frac{1}{6.5t}, we find:

t' gives:

Thus, the time taken by a similar cup of coffee to cool from 80°C to 60°C, while at the room temperature of 20°C, is tim\)

The correct option is: .

To find the energy radiated by a 100 kW transmitter in 1 hour, we use the formula for energy: E = P × t, where P is power and t is time.

Step 1: Identify the given values.

• Power, P = 100 kW = 100,000 watts
• Time, t = 1 hour = 3600 seconds

Step 2: Substitute the values into the formula.

E = 100,000 W × 3600 s

E = 360,000,000 J

Step 3: Convert the energy into scientific notation.

E = 3.6 × 10⁸ J

Step 4: Match with provided options.

The correct answer is 36 × 10⁷ J, which is equivalent to 3.6 × 10⁸ J.