Elasticity

Step 1: Understanding the force on the rope.
The maximum force acting on the rope due to the lift is the sum of the gravitational force and the additional force due to acceleration . Thus, the total force is .
Step 2: Using the stress formula.
The stress is given by: where is the cross-sectional area of the rope. The area is related to the diameter by .
Step 3: Solving for the diameter.
Substitute the expression for into the stress formula and solve for : Step 4: Conclusion.
The correct answer is (C), .

Step 1: Understanding isotones.
Isotones are atoms of different elements that have the same number of neutrons but different atomic numbers. The number of neutrons is calculated as:

Step 2: Checking each option.
(A) and :
Neutrons in K = , neutrons in Ar = .
Since the number of neutrons is different, they are not isotones.
(B) and :
Neutrons in first N = , neutrons in second N = .
These are isotopes, not isotones.
(C) and :
Neutrons in Cl = , neutrons in K = .
Neutron numbers are different, so they are not isotones.
(D) and :
Neutrons in Hg = , neutrons in Au = .
Both have the same number of neutrons but different atomic numbers, so they are isotones.
Step 3: Conclusion.
The correct pair of isotones is (D) and .

Step 1: Relation between time period and Young’s modulus.
For a spring-mass system, the time period of oscillation is given by: The spring constant for a wire is proportional to Young’s modulus .
Step 2: Express time period in terms of Young’s modulus.
Since and , we get: Step 3: Use the given ratio of Young’s moduli.
Step 4: Final ratio.

Step 1: Properties of photons in different media.
When a photon enters a different medium, such as from air to glass, its velocity and wavelength change, but its energy and momentum remain constant. The energy of a photon depends on its frequency, and the frequency of the photon does not change when it enters the new medium.
Step 2: Conclusion.
Thus, the correct answer is (B) Energy.

Step 1: Understanding the motion.
When the distance is proportional to the square of time, the motion is uniformly accelerated. The acceleration is constant because the body’s velocity increases at a constant rate.
Step 2: Conclusion.
Thus, the correct answer is (B) constant but not zero.

Step 1: Using the formula for longitudinal strain.
The longitudinal strain is given by the formula: where is the force and is the cross-sectional area. The change in radius is related to the strain and Poisson's ratio .
Step 2: Using Poisson's ratio.
The decrease in the radius is given by the formula: Substituting the values, we get the result: Step 3: Conclusion.
Thus, the decrease in the radius of the wire is , which corresponds to option (B).

Step 1: Understanding Poisson's ratio.
Poisson's ratio is the ratio of lateral strain to longitudinal strain. The longitudinal strain is given by the change in length per original length, and the lateral strain is given by the change in radius per original radius. Step 2: Expression for Poisson's ratio.
In this case, we can express Poisson's ratio in terms of the material properties of the wire, its dimensions, and the changes in radius and length. Using the given data, the Poisson's ratio can be written as: Step 3: Conclusion.
Thus, Poisson's ratio is , which corresponds to option (A).

Step 1: Understanding the elongation of the wire.
The elongation of a wire due to a force is given by Hooke's Law, which is: where is the force applied, is the length of the wire, is the cross-sectional area, and is Young’s modulus. The force acting on the sphere at the lowest point of its path is the centripetal force and the gravitational force. This force is given by: where is the angular velocity and is the acceleration due to gravity. Step 2: Calculating the cross-sectional area.
The cross-sectional area of the wire is , where is the diameter of the wire. Step 3: Substituting into the formula for elongation.
The elongation of the wire is then: Thus, the correct answer is , which corresponds to option (C).

Step 1: Understanding the energy density.
The energy density in a stretched wire is given by: where is the stress and is the strain. Stress is proportional to the square of the diameter, and strain is inversely proportional to the square of the diameter. Therefore, the energy density will vary as the square of the diameter. Step 2: Applying the ratio of diameters.
Since the diameters are in the ratio 1 : 3, the energy density ratio will be the square of the diameter ratio: Step 3: Conclusion.
Thus, the correct answer is (C) 81 : 1.

Step 1: Write definition of bulk modulus.
Bulk modulus is defined as
Step 2: Relate density with volume.
Density is For constant mass,
Step 3: Substitute in bulk modulus relation.

Step 4: Consider effective pressure.
Since pressure acts from all sides, effective bulk modulus becomes .
Step 5: Final expression.

Step 6: Conclusion.
The increase in density is .

Step 1: Write the formula for elongation of a wire.
The elongation of a wire is given by where is force, is original length, is cross-sectional area and is Young’s modulus.
Step 2: Identify constant quantities.
Since both wires are of the same material and same length, and are stretched by the same force, the quantities , , and remain constant.
Thus,
Step 3: Relate area with mass.
Mass of a wire is given by Since and are constant,
Step 4: Calculate the ratio of elongations.

Step 5: Conclusion.
The ratio of elongations of the two wires is .

Step 1: Find the restoring force due to extension of wire.
When the mass is displaced vertically, the wire is stretched. The restoring force due to elasticity is where is the extension.
Step 2: Identify the effective spring constant.
Comparing with Hooke’s law , we get
Step 3: Write frequency formula for SHM.
For a mass–spring system,
Step 4: Substitute the value of .

Step 5: Conclusion.
The frequency of oscillation is

Step 1: Understanding the force due to thermal expansion.
When a metal rod is heated, it expands. The force required to prevent this expansion is related to the Young's modulus and the change in length due to temperature. The formula for the force required to prevent the expansion is: where is the Young's modulus, is the cross-sectional area, is the coefficient of linear expansion, and is the temperature change.
Step 2: Conclusion.
Thus, the correct answer is (C) .

Concept:
Angular acceleration is given by: where is torque and is moment of inertia. Step 1: Position of centre of mass. For a uniform rod pivoted at one end:
Step 2: Torque due to gravity. Force acts downward at centre of mass. Perpendicular distance:
Step 3: Moment of inertia. For rod about one end:
Step 4: Angular acceleration.
Step 5: Conclusion.

Step 1: Condition
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Step 2: Solve for phase
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Step 3: Time Calculation
Substitute :
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Final Answer: (C)

Step 1: Lorentz Force Formula
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Step 2: Cross Product
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Using unit vector rules: .
Step 3: Magnitude
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Final Answer: (A)