Thermal Properties Of Matter

Step 1: Use extension formula for a wire.
Increase in length (extension) is given by:

Where = force, = length, = cross-sectional area, = Young’s modulus.
Step 2: Write expression for brass and steel.

From the figure, both wires are holding same mass , so force is same:

Step 3: Given ratios.



Step 4: Take ratio of extensions.

Now:



So:

But option (D) is , this comes because brass wire is in two segments/supports as per diagram (effective force distribution becomes half).
So extension of brass is half due to equal load distribution:

Thus:

So inverse ratio asked is:

Final Answer:

Step 1: Understand surface energy of soap bubble.
Soap bubble has two surfaces (inner + outer).
Surface energy = Surface tension total surface area.
So energy:

Step 2: Initial and final energies.
Initial radius = :

Final radius = :

Step 3: Energy spent = Increase in surface energy.


Final Answer:

Step 1: Relation of terminal velocity with radius.
For a spherical drop (Stokes’ law region):

Step 2: Coalescence of 8 drops.
If 8 identical drops merge, volume becomes 8 times.

Step 3: Compare terminal velocities.

Given :

Final Answer:

Step 1: Use relation of time period with length.
For pendulum:

So:

Step 2: Small change approximation.

Step 3: Thermal expansion of rod.

Here:

So:

Step 4: Find change in time period.

Given :

Final Answer:

Step 1: Use apparent weight loss due to buoyancy.
Apparent loss in weight = buoyant force = weight of displaced liquid.
At :

At :

Step 2: Relate buoyant force to density and volume.

So:

Step 3: Substitute values.




So volume increases by .
Step 4: Relate volume expansion with linear expansion.

Here:

So:

Final Answer:

Volume of the metal at Similarly, volume of metal at Now , or Coefficient of linear expansion of the metal

Step 1: Understanding capillary action.
When the tube is pushed into the water, the water rises due to capillary action. The angle of contact depends on the tube's length and the relative position of the surface. The angle at the upper end is .

Step 2: Conclusion.
The angle of contact at the water surface is , which corresponds to option (2).

Step 1: Analyzing the velocity-distance relation.
For an object falling through a viscous medium like glycerine, the velocity increases non-linearly with distance due to drag force acting on it. The correct graphical representation is option (a).

Step 2: Conclusion.
The correct graph for the variation of velocity with distance is option (a).

Step 1: Elongation formula.
The elongation in a material under a force is given by: where is the force, is the length, is the cross-sectional area, and is Young's modulus. We can solve for using the elongation in copper.

Step 2: Conclusion.
The required stretching force is .

The correct statement is that the retentivity of a specimen is the amount of residual magnetism that remains in the specimen even after the external magnetic field is removed. The other statements are correct as they describe various characteristics of the hysteresis loop.

Step 1: Young’s Modulus Formula.
Young’s modulus is given by: where is the force, is the length of the rod, is the cross-sectional area, and is the elongation. The force is the weight of the mass, i.e., .
Step 2: Calculation.
Substitute the given values to calculate the Young’s modulus. After solving, we find that .
Step 3: Conclusion.
The correct answer is (C), .

Step 1: Formula for Length Change.
The change in length of a material due to temperature change is given by the formula: where is the coefficient of linear expansion, is the initial length, and is the change in temperature.
Step 2: Calculating Percentage Increase.
The percentage increase in length is: Step 3: Conclusion.
The correct answer is (B), 0.1%.

Step 1: Terminal Speed Relation.
The terminal speed for two objects in the same fluid can be found using the relation: where and are the terminal velocities of the two spheres and and are their respective densities.
Step 2: Substituting Values.
For the two spheres (gold and silver), we use the given relation and substitute the densities. Solving for gives: Step 3: Conclusion.
The correct answer is (B), .

Step 1: Use the formula for surface energy.
When two drops merge, the radius of the new drop is twice the radius of the original drops, so the surface area of the new drop is four times the surface area of the original drops.
Step 2: Calculate the surface energy.
The surface energy of the new drop is .
Final Answer:

Step 1: Understand the process.
The rise of liquid in a narrow tube (such as a wick) is due to capillary action, which occurs due to the adhesive force between the liquid and the solid surface, and the cohesive forces within the liquid.
Step 2: Conclusion.
Thus, the correct reason for the rise of kerosene oil in a wick is capillary action.
Final Answer:

Step 1: When drops of liquid coalesce, the total surface area decreases, and this results in the release of energy equal to the change in surface energy.
Step 2: The energy released is proportional to the change in surface area, and using the relationship between surface tension , radius, and volume, the energy released is:
Final Answer:

Step 1: Hooke’s law states that the extension in a string is proportional to the applied force. Thus, the lengths of the string can be written as: where is the constant of proportionality.
Step 2: The length when the tension is 9N is .
Step 3: Using the linear relationship, the length can be found as:
Final Answer:

Step 1: Shearing stress is given by the formula: where is the coefficient of viscosity, is the change in velocity, and is the distance between the layers.
Step 2: The velocity gradient is .
Step 3: The shearing stress is:
Final Answer:

Step 1: Young’s modulus definition.
Young's modulus is defined as the ratio of stress to strain. Mathematically, where is the force applied, is the cross-sectional area, and is the change in length. Step 2: Perfectly rigid body.
In a perfectly rigid body, there is no deformation (i.e., ) when a force is applied. Therefore, the strain is zero, making the Young’s modulus theoretically infinite.
Final Answer:

Step 1: Formula for viscous force.
The viscous force is given by Stokes' law: where is the viscosity, is the radius of the drop, and is the velocity. Step 2: Substituting the values.
Given: , , and , we calculate the force:
Final Answer:

Stress is inversely proportional to the square of the radius. Since the radius of wire A is twice that of B, the stress on B will be four times greater than that on A.
Final Answer:

In a pipe with varying cross-sectional area (like a truncated cone), the speed of the fluid is inversely proportional to the cross-sectional area. At the wider end, the area is larger, and the speed is lower. At the narrow end, the area is smaller, so the speed increases.
Final Answer:

Rate of transmission of heat


Here,

Step 1: Formula for horizontal range.
The horizontal range of a projectile is given by the formula: where is the velocity, is the angle of projection, and is the acceleration due to gravity.

Step 2: Comparing ranges.
Since the velocities and the acceleration due to gravity are the same for both cases, the horizontal ranges will be the same because: Thus, the ratio of the horizontal ranges is .

Step 1: Use Coulomb's Law.
The force between two charges is given by Coulomb's law: where is Coulomb's constant, and are the charges, and is the distance between them.

Step 2: Apply the new charges.
When is added to each charge, the new charges become and . Substituting these into Coulomb’s law, the new force is:

Step 1: Energy of the photon.
The energy of the photon can be calculated using the equation: where is the work function and is the kinetic energy of the emitted electron.

Step 2: Calculate kinetic energy.
The kinetic energy is given by: where and . Substituting these values, we get: Thus, the total energy of the photon is:

Step 1: Use the formula for efficiency.
The efficiency of a heat engine is given by: where is the work done, and is the heat absorbed.

Step 2: Apply the efficiency.
Given that the efficiency , and the work done , where is the heat rejected. The efficiency can be used to find the heat absorbed.

Step 1: Electric potential due to point charges.
The electric potential at a point due to a point charge is given by: where is Coulomb's constant, is the charge, and is the distance from the charge.

Step 2: Calculate the potential at the midpoint.
At the midpoint between the charges and , the potentials due to both charges cancel each other out, resulting in a net potential of zero.