Thermal Properties Of Matter

Step 1: Understanding Wave Velocity in a Wire The velocity of a transverse wave in a stretched string is given by: where: is the tension in the wire, is the mass per unit length of the wire.
Step 2: Expressing Mass Per Unit Length For a cylindrical wire of density and diameter : Since the diameter of A is twice that of B:
Step 3: Expressing Velocity Ratio Given that the tension in A is half that of B: Using the velocity formula: Thus, the ratio of velocities is:

Step 1: Understanding Wave Speed in a Stretched String The speed of a transverse wave in a stretched string is given by: where: is the tension in the string, is the mass per unit length. Since both strings are made of the same material and have the same cross-section, their linear mass densities ( ) are equal.
Step 2: Analyzing Tension in the Strings - Let the tension in string AB be . - The tension in string CD, which supports an additional load due to the pulley arrangement, is given by:
Step 3: Finding the Velocity Ratio Using the wave speed formula: Thus, the ratio is:

The B-H curve represents the magnetic behavior of a material. The curve shows the relationship between magnetic flux density (B) and magnetic field strength (H).
Step 1: The shape of the B-H curve for Sample X shows a higher saturation magnetization, indicating that it retains a higher residual magnetism after being magnetized. This is a characteristic of materials suitable for making permanent magnets.
Step 2: On the other hand, the B-H curve for Sample Y shows a lower saturation magnetization and a relatively steeper slope, indicating that it is more easily magnetized and demagnetized. This is a characteristic of materials suitable for making electromagnets, which require easy switching of magnetization.
Step 3: Therefore, Sample X is more suited for making permanent magnets, while Sample Y is more suited for making electromagnets. Thus, the correct answer is .

Step 1: Velocity Components at the Highest Point At the highest point of the projectile's motion: - The vertical component of velocity becomes zero. - The horizontal component remains: Since there are no external horizontal forces, the total momentum in the horizontal direction is conserved.
Step 2: Applying Conservation of Momentum Before the explosion: After the explosion: - The projectile splits into two equal parts of mass . - One of the fragments continues with the same velocity . - Let the velocity of the second fragment be . Applying the law of conservation of momentum in the horizontal direction:
Step 3: Solving for Thus, the velocity of the second fragment is:

Radioactive decay follows an exponential decay law, and the activity of a radioactive material at any time is given by: where is the initial activity, is the decay constant, and is the time elapsed.
Step 1: Let be the activity at time and be the activity at time . Using the decay law for both times: At time : At time :
Step 2: To find the relationship between and , divide the equation for by the equation for : Thus, the relationship between the activities is: Hence, the correct answer is .

Area mass density:  
Mass of each hole:  
Distance between center of plate and center of hole:  
 

Moment of inertia of one hole about the Z-axis:  
Moment of inertia of whole plate about the Z-axis:  
Required moment of inertia:  
Given:  
Final Calculation:

The process shown in the figure indicates an isochoric process where the volume of the gas is directly proportional to the inverse of the temperature. For an ideal gas undergoing such a process, the relation between the pressure, volume, and temperature is: This implies that the process follows a path where the slope of versus is constant. For an ideal diatomic gas, the molar specific heat is related to the slope of the curve in the diagram. From the relationship for an ideal gas: where is the molar specific heat at constant volume for a diatomic gas, which is . Since the process shown is a specific heat process that involves both heat exchange and work, the total molar specific heat will be .
Thus, the molar specific heat of the gas in the process is .

The rate of flow of liquid through a capillary tube is governed by the formula: where: - is the rate of flow, - is the radius of the capillary, - is the pressure difference, - is the viscosity of the liquid, - is the length of the capillary tube.
Step 1: If the radius is increased by 25%, the new radius will be:
Step 2: The rate of flow is proportional to the fourth power of the radius, so the new rate of flow will be:
Step 3: Simplifying: Thus, the new rate of flow will be approximately 144% of the original rate of flow. Hence, the rate of flow will change by .

Step 1: Understanding Power Calculation The power of the gun is given by the rate of change of kinetic energy: The kinetic energy of a single bullet is: where: (mass of one bullet), (velocity of bullet).
Step 2: Finding the Kinetic Energy of One Bullet
Step 3: Finding Power Output The gun fires 300 bullets per minute, i.e., 5 bullets per second: Thus, the correct answer is:

Step 1: When the container is moving, the gas molecules inside have kinetic energy due to their motion. For a monoatomic ideal gas, the total kinetic energy is given by the formula: where is the number of moles, is the universal gas constant, and is the temperature of the gas.
Step 2: When the container is suddenly stopped, the gas molecules stop moving in the direction of the container's motion, and the kinetic energy associated with the motion of the container is converted into internal energy, which causes the temperature to increase.
Step 3: The total kinetic energy of the gas molecules is related to the motion of the container. The kinetic energy of the container is , where is the mass of the gas and is the velocity of the container.
Step 4: Since the system is insulated, the change in internal energy equals the change in kinetic energy. For an ideal monoatomic gas, the change in temperature is related to the change in internal energy. The equation for the change in temperature is: where is the molar specific heat at constant volume.
Step 5: Substituting and solving for : Since (where is the molar mass), we get: Thus, the change in the temperature of the gas is .

Step 1: Understanding the Photoelectric Equation The photoelectric equation is given by: where: is the stopping potential energy, is Planck’s constant, is the frequency of incident light, is the threshold frequency.
Step 2: Determining the Slope from the Graph The given graph shows a straight-line relationship between and frequency , where the slope represents: From the graph:
Step 3: Calculating Planck’s Constant Since , we calculate as:

Given: Velocity of submarine A, m/s 
Velocity of submarine B, m/s 
Signal sent by submarine B is detected by submarine A can be shown as: 

 Frequency of the signal, Hz 
Speed of sound in water m/s 
Step 1: Calculate the frequency received by submarine A 
Substituting values:  
Step 2: Calculate the frequency received by submarine B 
Substituting values and using from Eq. (i), we get:

In Young's double-slit experiment, the shift in the central fringe occurs due to the introduction of a phase difference between the two slits. When a glass plate of thickness and refractive index is introduced, the optical path difference changes. The phase shift introduced by the glass plate is given by: where is the thickness of the glass plate and is the refractive index of the glass. Step 1: For the first slit with refractive index and thickness , the path difference is: For the second slit with refractive index and the same thickness , the path difference is: The total optical path difference between the two slits is:
Step 2: This optical path difference shifts the central maximum to the sixth fringe. The fringe width is given by: where is the wavelength of light, is the distance between the screen and the slits, and is the distance between the slits. The shift in the fringe due to the optical path difference is given by: Given that the shift is to the sixth fringe, we can equate the shift to , and solve for .
Step 3: Using and substituting into the above equation: Substitute into the equation: Thus, the thickness of the glass plates is .

Step 1: Understanding the Electric Potential of a Non-Conducting Sphere For a uniformly charged non-conducting sphere: - Inside the sphere ( ): The electric potential at a distance from the center is given by: This shows a parabolic decrease from the center to the surface. - Outside the sphere ( ): The sphere behaves like a point charge, and the potential follows: which represents an inverse relationship with distance.
Step 2: Identifying the Correct Graph - Inside the sphere ( ), follows a quadratic relation. - Outside the sphere ( ), follows an inverse relation , which shows a smooth decrease. Among the given graphs, Graph D correctly represents this behavior: - A parabolic decrease inside the sphere. - A smooth inverse decrease outside the sphere.

Step 1: Understanding radioactive decay. The decay of a radioactive material follows the exponential decay law: where: - = Initial amount of substance - = Remaining amount after time - = Half-life - = Given time
Step 2: Finding the half-life. Given that the material is reduced to of its original value in 3 days, we set up: Since , we compare powers: Thus, day (half-life is 1 day).
Step 3: Finding . After 5 days, the amount left is kg. Using the decay formula: Solving for : Final Answer:

Step 1: Maxwell's equations for time-varying conditions: One of the key equations that is valid only for time-varying conditions is Faraday's Law of Induction. It states that: This equation describes how a changing magnetic field induces an electric field.
Step 2: Understanding the options: - Option (A): This is Ampère's Law for magnetostatics, where the integral of around a closed loop is equal to the current enclosed. This equation is valid under static conditions as well. - Option (B): This is one form of the conservative electric field equation, valid for electrostatics, where the line integral of is zero for static conditions. - Option (C): This is the correct choice, as it describes Faraday's Law of Induction, which is valid only for time-varying conditions. - Option (D): This is Gauss's Law for electricity, valid for both static and dynamic conditions.
Step 3: Verifying the correct answer. The equation is the correct choice as it applies only in time-varying situations (changing magnetic field).

Step 1: {Formula for RMS speed and speed of sound}
The root mean square (RMS) speed of gas molecules is given by: The velocity of sound in a gas is: where is the adiabatic index.
Step 2: {Relation between and }
Dividing the equations:
Step 3: {Finding using degrees of freedom}
For a mixture of gases with an average degree of freedom :
Step 4: {Compute velocity of sound}
Since , we get: Thus, the correct answer is .

Step 1: {Define variables}
Let be the volume immersed in water and be the volume immersed in oil.
Step 2: {Equilibrium condition}

Step 3: {Solve for ratio}
Thus, the correct answer is 1:1.

Step 1: {Young's modulus dependency}
Step 2: {Effect of change in length and area}
Young's modulus is a material property and does not depend on length or cross-sectional area. Thus, the correct answer is .

Step 1: {Young's modulus definition}

Step 2: {Effect of temperature increase} : As temperature increases, strain increases, leading to a decrease in Young’s modulus.
Thus, the correct answer is Young’s modulus decreases.

Step 1: {Determine velocity on hitting the surface}
Using free-fall kinematics: Step 2: {Velocity after first bounce}
Step 3: {Time taken from first to second bounce}
Time of flight formula: Thus, the correct answer is 1.5 s.

Step 1: {Identifying Wheatstone Bridge}
The given network forms a balanced Wheatstone bridge. In a Wheatstone bridge, four resistors are arranged in a diamond shape, with two resistors on each arm, and a galvanometer connected across the middle of the bridge. In this case, when the resistor in the BD arm is removed, the bridge remains balanced.
Key Concept: A Wheatstone bridge is balanced when the ratio of resistances in one pair of opposite arms is equal to the ratio of resistances in the other pair. The condition for balance in the Wheatstone bridge is given by: Where and are the resistances in the four arms of the bridge. When the resistor is removed from the BD arm, it does not affect the total resistance because the bridge is balanced and the current flowing through the galvanometer is zero, meaning no current flows through the branch containing the resistor. Hence, the equivalent resistance of the bridge does not change.
Step 2: {Finding Equivalent Resistance}
Since the bridge remains balanced, the equivalent resistance of the network remains unchanged. For a balanced Wheatstone bridge, the total equivalent resistance across the bridge is determined by the resistances in the remaining arms. Let’s denote the resistances of the other arms as and . When the bridge is balanced, we know the equivalent resistance for the two parallel arms (AC and BD) can be calculated as: This is true for both the upper and lower parts of the Wheatstone bridge, and they contribute equally to the total equivalent resistance. Since the resistor is removed from the BD arm, it does not change the equivalent resistance of the network because the Wheatstone bridge was balanced and no current flows through the removed resistor. The equivalent resistance remains as calculated previously.
Step 3: {Conclusion}
Thus, the total equivalent resistance of the bridge remains unchanged when the resistor is removed.

Step 1: Understanding Huygens' Wave Theory

Huygens' wave theory describes how light propagates by treating every point on a wavefront as a secondary wave source. Using this principle, the laws of: Reflection , Refraction , Diffraction are successfully derived.

Step 2: Analyzing the Given Options

Reflection: Huygens' principle explains reflection by considering the secondary wavelets on the incident wavefront, which create the reflected wavefront.

Refraction: Huygens' principle explains refraction by stating that different parts of a wavefront move at different speeds when passing through media with different refractive indices.

Diffraction: Huygens' principle accounts for diffraction, as each point on a wavefront acts as a source of secondary wavelets, allowing light to bend around obstacles.

Step 3: Why Huygens' Principle Fails to Explain Spectra

The origin of spectral lines arises due to the emission and absorption of photons by atoms, which is best explained by quantum mechanics. Huygens' wave theory does not consider the particle nature of light or energy quantization, which are essential for understanding:

Atomic emission spectra , Blackbody radiation , Photoelectric effect

Since Huygens' theory only deals with the wave nature of light and not its quantum properties, it cannot explain the origin of spectra.

Thus, the correct answer is:

Step 1: Understanding Gravity Variation with Depth
The acceleration due to gravity at a depth d inside the Earth is given by the formula:

g' = g × (1 - d / R)

Where:

• g' is the acceleration due to gravity at depth d.
• g is the acceleration due to gravity on the surface of the Earth.
• R is the radius of the Earth.

Step 2: Explanation of the Formula
Gravity decreases linearly as we go deeper inside the Earth because only the mass enclosed within the radius R - d contributes to the gravitational force at depth d. This leads to the relation:

g' = g × (R - d) / R = g × (1 - d / R)

Thus, the correct expression is:

g' = g × (1 - d / R)

Correct Option: Option (D)

Step 1: {Understanding Vernier Caliper}
The vernier scale least count is given by: Step 2: {Applying the Given Condition}
Since main scale divisions equal vernier scale divisions, Step 3: {Solving for Least Count}
Thus, the correct answer is (B)

Step 1: {Understanding the Logic Circuit}
The given circuit consists of: - NOT gates applied to inputs and . - AND gate processing the inverted signals.
Step 2: {Deriving the Boolean Expression}
The circuit implements: Applying De-Morgan’s theorem:
Step 3: {Comparing with Output Waveforms}
Analyzing the truth table, we match the waveform and determine that the correct output is given by waveform 4. Thus, the correct answer is (D).

Step 1: {Understanding the Rydberg Formula}
The wavelength of emitted radiation during an electron transition is given by the Rydberg formula: where: - is the Rydberg constant, - is the atomic number, - and are the principal quantum numbers.
Step 2: {Applying to Hydrogen and Helium Ions}
For hydrogen ( ), the wavelength of the transition is: For the He ion ( ), the wavelength of transition is: Equating , we get: Solving for integer values, we find , satisfies the condition. Thus, the correct answer is .