Thermal Properties Of Matter

Step 1:
The rate of heat emission from a black body is proportional to the fourth power of its temperature:
Step 2:
Let the initial temperature be , and the final temperature be .
Step 3:
Using the Stefan-Boltzmann law:

Step 1:
The energy lost by the metal block will be used to melt the ice. The energy lost by the metal block is given by: where , , and .
Step 2:
The energy required to melt the ice is: where is the latent heat of fusion and is the mass of the ice melted.
Step 3:
Since all the energy from the metal is used to melt the ice:

Step 1: Find the temperature of the perfect radiator using Wien's Displacement Law.
Wien's Law states that , where b is Wien's constant.
We are given the wavelength of maximum intensity, Å.
Convert this wavelength to meters: .
Wien's constant is mK.
Now, solve for the temperature T.
K.
Step 2: Find the intensity of radiation using the Stefan-Boltzmann Law.
The Stefan-Boltzmann Law states that the total intensity (or emissive power) I of a perfect radiator (black body) is given by , where is the Stefan-Boltzmann constant.
We are given Wm K .
We found K.
Substitute these values into the formula.
.
Wm .

Step 1: Apply Archimedes' principle to the cube floating alone.
The buoyant force equals the weight of the object. Let be the weight of the cube and be the total volume of the cube. The weight of the displaced water is , where is the density of water.

Step 2: Apply Archimedes' principle to the cube with the disc.
Let be the weight of the disc. The total weight is . The new volume of displaced water is . The buoyant force is .

Step 3: Solve for the weight of the disc.
Substitute the expression for from Step 1 into the equation from Step 2.

Step 4: Calculate the mass of the disc.
The mass of the disc is . We are given the side of the cube is cm. The volume is . The density of water is . Converting to kilograms:

Step 1: Identify the given parameters.
The scale is a "metre scale", so its original length at the calibration temperature is m mm. The calibration temperature is C. The maximum allowed error in measurement is the change in length, mm. The coefficient of linear expansion is / C.

Step 2: Use the formula for linear thermal expansion.
The change in length due to a change in temperature is given by: We are interested in the magnitude of the temperature change, so we use:

Step 3: Set up the inequality for the allowed temperature range.
We are given that the error must be at most 0.5 mm. Substitute the given values:

Step 4: Solve the inequality for T.
This inequality can be written as: Add 25 to all parts of the inequality: The range of temperatures is from -25 C to +75 C.

Step 1: State the formula for the period of a simple pendulum.
The period of a simple pendulum is given by , where is its length.

Step 2: Find the fractional change in the period due to thermal expansion.
From the formula, . The fractional change in the period is given by: The fractional change in length due to a temperature change is given by , where is the coefficient of linear expansion. Combining these, we get:

Step 3: Calculate the fractional change in the period.
Given values: / C. The change in temperature is .

Step 4: Calculate the total time lost or gained in one day.
The total time change in a day is the fractional change multiplied by the total number of seconds in a day. Number of seconds in a day = . This is approximately 7.8 s. Since the temperature increased, the length increased, and the period also increased. A longer period means the clock runs slower, so it loses time. The clock is 7.8 s slow per day.

Step 1: Analyze the change in volume.
Let 'r' be the radius of each small drop and 'R' be the radius of the big drop.
The total volume is conserved when the drops combine.
Volume of 'n' small drops = Volume of one big drop.
.
This gives the relationship between the radii: .
Step 2: Analyze the change in surface area.
Total initial surface area of 'n' small drops: .
Final surface area of the big drop: .
Substitute into the final area equation.
.
Let's compare and . We can look at their ratio.
.
Since n>1 for combination, , and therefore .
This means . So, the surface area decreases.
Step 3: Analyze the change in energy.
Surface energy is given by , where T is the surface tension and A is the surface area.
Since the surface area decreases, the total surface energy of the system also decreases.
The change in energy is , which is negative.
By conservation of energy, this loss in potential energy must be converted into another form, which is released as heat.
Therefore, heat is released.
Combining the results, the surface area decreases and heat is released.

The work done in stretching a wire is equal to the elastic potential energy stored in it.
The energy stored per unit volume (energy density) is given by .
Young's modulus is defined as , so Stress = .
Substituting this into the energy density formula: .
The total work done (total energy stored) is .
We are given: Strain = and Nm .
First, let's find the volume of the wire.
Mass kg.
Density . We must convert this to SI units (kg/m ).
kg/m .
Volume m .
Now, calculate the total work done.
.
.
J.
The question asks for the answer in kJ.
.

According to Archimedes' principle, for an object to float, the weight of the object must be equal to the weight of the fluid it displaces.
Let litre = 1000 cm be the total outer volume of the block.
Let be the volume of the wood material and be the volume of the cavity.
Then .
The volume of water displaced is half the total volume: cm .
The weight of the displaced water is . (Density of water g/cm ).
The weight of the wooden block is the weight of the wood material only (the cavity is empty and has no weight).
Weight of the block .
The specific gravity of wood is given as 3/4, so g/cm .
.
Equating the weight of the block and the weight of the displaced water:
.
cm .
Finally, find the volume of the cavity.
cm .
cm or 333.3 ml.

Step 1: Calculate the rate of heat flow into the box.
The rate of heat conduction ( ) through the walls is given by:
.
Where:
= thermal conductivity = W/(m K).
= total surface area = 1000 cm = m = m .
= temperature difference = K.
= thickness of the walls = 2 mm = m.
J/s.
Step 2: Calculate the total heat that flows into the box in the given time.
Time minutes = seconds = 9600 s.
Total heat J.
Step 3: Calculate the mass of ice that melts due to this heat.
The heat required to melt a mass of ice is , where is the latent heat of fusion.
J/kg.
.
kg.
Step 4: Calculate the mass of ice remaining.
Initial mass of ice kg.
Mass remaining = kg.

Step 1: Define variables and formulas.
Let the circular rod be rod 1 and the square rod be rod 2. Elongation formula: , where is tension, is original length, is cross-sectional area, and is Young's modulus. Mass formula: , where is density.

Step 2: Determine the properties of the rods.
Rod 1 (circular): diameter cm, radius cm. Area cm . Rod 2 (square): side cm. Area cm . Both rods are steel (same , same ) and have equal mass ( ) and are under the same tension ( ).

Step 3: Relate the lengths of the rods.
Since the masses are equal: .

Step 4: Calculate the ratio of elongations.
We need to find the ratio . Since and , this simplifies to: From Step 3, we know both these ratios are equal to .

The work done in increasing the surface area of a liquid film is given by:
, where T is the surface tension and is the total change in surface area.
A thin film has two surfaces (top and bottom), so the total surface area is twice the geometric area.
Let the length of the wires be cm = m.
Initial separation cm = m.
Final separation cm = m.
Initial geometric area of the film: .
Final geometric area of the film: .
The change in geometric area is .
.
The total change in surface area, considering both surfaces, is:
.
Now, calculate the work done.
The surface tension is given as N/m.
.
J.
J.
Since J, the work done is J.

Let F be the temperature reading on the Fahrenheit scale and C be the temperature reading on the Celsius scale.
The problem states that the Fahrenheit reading is 90% more than the Celsius reading.
This can be written as an equation: .
The standard conversion formula between Fahrenheit and Celsius is:
.
We have a system of two equations with two variables. Substitute the first equation into the second one:
.
Cross-multiply to solve for C:
.
.
.
.
So, the temperature is 320 C.
The question asks for the temperature reading on the Fahrenheit scale. We use the relation .
.
.
So, the temperature is 608 F.

Step 1: Understanding the Concept:

Work done in blowing a soap bubble is stored as surface potential energy. A soap bubble has two free surfaces (inner and outer). Work . Total Area increase .
Step 2: Key Formula or Approach:

. Given Diameter cm cm = m. .
Step 3: Detailed Explanation:

Substitute values: (Since ) Let's simplify differently: . . . Joules. J. .
Step 4: Final Answer:

The work done is .

Step 1: Find the correct temperature in Celsius.
The correct Fahrenheit temperature is °F.
The formula to convert Fahrenheit to Celsius is .
°C.
So, the true temperature of the body is 50 °C.
Step 2: Compare the correct temperature with the reading from the faulty thermometer.
The faulty Celsius thermometer reads °C.
The error in the reading is .
Error = 50 °C - 49 °C = 1 °C.
Step 3: Determine the correction to be applied.
The correction is what needs to be *added* to the faulty reading to get the true reading.
Correction = True Value - Faulty Reading.
Correction = 50 °C - 49 °C = +1 °C.
A correction of +1 °C must be applied to the faulty thermometer's reading.

Step 1: Understanding the Concept:
Power is the rate at which energy is transferred. In this case, it's the rate at which the refrigerator removes heat from the water. The total heat removed consists of two parts: the heat removed to cool the water from 30 °C to 0 °C (sensible heat), and the heat removed to convert the water at 0 °C into ice at 0 °C (latent heat).
Step 2: Key Formula or Approach:
1. Heat removed to cool water: , where is the specific heat capacity of water.
2. Heat removed to freeze water: , where is the latent heat of fusion of ice.
3. Total heat removed: .
4. Power: .
Standard values to be used: , .
Step 3: Detailed Explanation:
Given values:
- Mass of water, kg.
- Initial temperature = 30 °C.
- Final temperature = 0 °C.
- Time, hour = seconds.
Calculate the heat removed to cool the water ( ):
Calculate the heat removed to freeze the water ( ):
Calculate the total heat removed ( ):
Calculate the power of the refrigerator ( ):
Step 4: Final Answer:
The power of the refrigerator is 1925 W. Therefore, option (B) is correct.

Step 1: Understanding the Concept:
This problem deals with linear thermal expansion, which describes how the length of an object changes with a change in temperature. The change in length is proportional to the original length, the change in temperature, and the coefficient of linear expansion.
Step 2: Key Formula or Approach:
The formula for linear expansion is:
where is the change in length, is the original length, is the coefficient of linear expansion, and is the change in temperature.
The fractional change in length is .
The percentage change in length is .
Step 3: Detailed Explanation:
We are given that the percentage increase in length is 0.4%.
This means the fractional increase in length is:
We are also given the coefficient of linear expansion, °C⁻¹.
Now, we can use the formula to find the required change in temperature, .
The required temperature increase is 200 °C.
A change in temperature has the same magnitude in both Celsius and Kelvin scales. That is, .
Therefore, the temperature of the rod must be increased by 200 K.
Step 4: Final Answer:
The required increase in temperature is 200 K. Option (C) is correct.

Step 1: Understanding the Concept:
Since the temperature of the water is uniform, the process of the air bubble rising is considered isothermal. Therefore, we can apply Boyle's Law, which states that for a fixed mass of gas at constant temperature, the product of pressure and volume is constant ( ). We also need to relate the bubble's volume to its surface area and use the formula for pressure at a certain depth in a fluid.
Step 2: Key Formula or Approach:
Let the conditions at the bottom of the tank be denoted by subscript 1 and at the top by subscript 2.
1. Boyle's Law: .
2. Pressure at depth : , where is the atmospheric pressure.
3. Pressure at the top (surface): .
4. For a spherical bubble, surface area and volume . From these, we can establish the relationship .
Step 3: Detailed Explanation:
First, let's establish the relationship between the surface areas and volumes.
The surface area at the top ( ) is 125% more than the surface area at the bottom ( ).
Since the volume of a sphere is proportional to the power of its surface area ( ), the ratio of the volumes is:
Now, applying Boyle's Law for the isothermal process:
Let the depth of the tank be . We are given that the atmospheric pressure is equivalent to a 10 m water column. Let this height be m.
The pressure at the top (surface) is .
The pressure at the bottom (depth ) is .
Now, we substitute these pressure expressions into our ratio:
The term cancels out:
Step 4: Final Answer:
The depth of the water in the tank is 23.75 m. Therefore, option (D) is the correct answer.

Step 1: Understanding the Concept:
Since the temperature of the water is uniform, the process is isothermal. We can apply Boyle's Law ( ) to the air bubble. We also need to relate the surface area of the bubble to its volume and understand how pressure varies with depth in a fluid.
Step 2: Key Formula or Approach:
Let the conditions at the bottom be denoted by subscript 1 and at the top by subscript 2.
- Boyle's Law:
- Pressure at depth : , where is the atmospheric pressure.
- Pressure at the top: .
- For a spherical bubble, surface area and volume . From this, we can deduce that .
Step 3: Detailed Explanation:
First, let's find the relationship between the volumes.
We are given that the surface area at the top ( ) is 125% more than at the bottom ( ).
Since , the ratio of the volumes is:
Now, let's apply Boyle's Law:
Let the depth of the tank be . We are given that atmospheric pressure is equivalent to a 10 m water column. Let m.
The pressure at the bottom is:
Substituting these into the pressure ratio:
Now, solve for :
Step 4: Final Answer:
The depth of the water in the tank is 23.75 m. Therefore, option (D) is correct.

Step 1: Understanding the Concept:

We need to calculate the total heat energy required for a multi-step process involving temperature changes and phase changes: 1. Ice ( C) Ice ( C) 2. Ice ( C) Water ( C) (Fusion) 3. Water ( C) Water ( C) 4. Water ( C) Steam ( C) (Vaporization)
Step 3: Detailed Explanation:

Mass . 1. Heat to raise ice temp ( ): 2. Heat to melt ice ( ): 3. Heat to raise water temp ( ): 4. Heat to vaporize water ( ): Total Heat in Joules: Convert Joules to calories (using ):
Step 4: Final Answer:

The heat required is 5840 cal.

Step 1: Understanding the Concept:

The heat conducted through the metal rod is used to melt the ice. We need to equate the rate of heat flow through the rod to the rate of heat absorption by the ice for fusion.
Step 2: Key Formula or Approach:

1. Rate of heat flow: 2. Heat for phase change:
Step 3: Detailed Explanation:

Given: Length Area Temperature difference Time Thermal Conductivity Latent Heat First, calculate the rate of heat flow: Total heat transferred in 420 seconds: Now, find the mass of ice melted using : Converting to grams:
Step 4: Final Answer:

The mass of ice melted is 22.5 g.

Step 1: Understanding the Concept:

Terminal velocity is proportional to the square of the radius ( ). Since mass (for same density), we can relate velocity to mass.
Step 2: Key Formula or Approach:

1. Mass . 2. Terminal Velocity . 3. Combine: .
Step 3: Detailed Explanation:

Let sphere 1 have mass g and velocity cm/s. Let sphere 2 have mass g and velocity . Ratio: Substitute values:
Step 4: Final Answer:

The terminal velocity is .

Step 1: Understanding the Concept:

When submerged, the sphere experiences uniform pressure , causing a volume compression. We relate pressure, bulk modulus ( ), and volume strain ( ). Then we relate volume strain to linear strain ( ).
Step 2: Key Formula or Approach:

1. Bulk Modulus: . 2. Relation between Volume and Radius change: . 3. Combine: .
Step 3: Detailed Explanation:

Given: Radius cm. Pressure Pa. Bulk Modulus GPa = Pa. Substitute values into : Wait, let's recheck the calculation. . . Result: cm. This matches Option (B). However, the Answer Key provided in the prompt indicates Correct Answer is (D) . Let's check the units again. Pa. GPa = Pa. cm. Maybe pressure is or is different? No, sticking to text. Let's re-read the option mark in image. In the image for Q92, Option 2 is marked with a green check: cm. Option 4 is cm. The text extraction might have listed options in a specific order in the prompt block (e.g., A, B, C, D). Looking at the screenshot: Option 1: Option 2: (Green Check) Option 3: Option 4: So the correct answer is indeed Option 2: cm. The "Correct Answer" in the prompt text above said (D), but based on my calculation and the visual tick mark, it should be (B). I will follow the visual evidence and calculation.
Step 4: Final Answer:

The change in radius is cm.

1. Heat lost by water and calorimeter = Heat gained by ice to reach 0°C + heat to melt + remaining ice mass
2. Let = mass of ice melted.
3. (in Joules)
4. Solve for → g

1. Heat conducted through the steel shell per unit time: , where = thermal conductivity, = area, = temperature difference, = thickness.
2. Surface area of sphere:
3. Thickness of steel:
4.
5. Heat conducted per second:
6. Mass of ice: , (Check units: )
7. Heat required to melt ice:
8. Time required:

• Convert masses to kilograms: metal , water .
• Initial temperatures: , . Final equilibrium temperature: .
• Heat lost by metal .
• Heat gained by water where .
• Equate heat lost and gained (neglecting losses to environment): • Substitute numbers: • Hence the specific heat capacity of the metal is }.