Mechanical Properties Of Solids

Elastic potential energy per unit volume

From definition of Young's modulus of wire

Stress strain
Given, strain=X
=

Hooke in 1679 , showed experimentally that if strain is small, then the stress is proportional to strain. The ratio of stress to strain is constant for material of the given body and is called modulus of elasticity . Thus,
constant
Hence, if stress is increased, then the ratio of stress to strain remains constant.

We know that

Bulk modulus

Answer (b)

Step 1: Escape velocity formula.
Escape velocity is given by:

Step 2: Given data.
For earth:
For planet:

Step 3: Escape velocity of the planet.

Step 4: Ratio of escape velocities.

Step 5: Conclusion.
The required ratio is .

Step 1: Use formula for elongation of a wire.
where is force, is length, is cross-sectional area, and is Young’s modulus.

Step 2: Write elongation expressions.
For thick wire: For thin wire:

Step 3: Find ratio.

Step 1: Expression for kinetic energy.
Kinetic energy is given by
Step 2: Applying error propagation.
For a quantity of the form , the percentage error is
Step 3: Substituting given values.

Step 4: Conclusion.
The percentage error in kinetic energy is 8%.

Step 1: Write the expression for angular momentum.
For circular motion, angular momentum is given by:

Step 2: Express velocity in terms of angular momentum.

Step 3: Use centripetal force formula.
Centripetal force is:

Step 4: Substitute the value of velocity.

Step 5: Conclusion.
The centripetal force is given by , hence option (C) is correct.

Step 1: Understanding the relationship.
The load versus elongation graph reflects the stiffness of the material. A thicker wire would have a steeper slope because it can withstand a greater load for the same elongation. The thicker the wire, the higher the slope.
Step 2: Analyzing the options.
(A) line OQ: This line represents a less steep slope, indicating a thinner wire.
(B) line OP: This line has the steepest slope, indicating the thickest wire.
(C) line OR: This line represents a moderately steep slope, indicating a medium thickness.
(D) line OS: This line represents a slope less steep than OP but steeper than OQ.
Step 3: Conclusion.
Thus, the thickest wire is represented by line OP.

Step 1: Understanding the energy stored in a stretched wire.
The energy stored per unit volume in a stretched wire is given by: where stress is proportional to the force applied and strain is proportional to the elongation of the wire. The energy is directly proportional to the square of the elongation.
Step 2: Finding the increase in energy.
The energy for the new elongation of is calculated by comparing the energy with the previous elongation. The increase in potential energy is: Step 3: Conclusion.
Thus, the increase in energy stored per unit volume is , corresponding to option (C).

Step 1: Using Poisson's ratio.
Poisson's ratio is defined as the ratio of lateral strain to longitudinal strain. The lateral strain (change in diameter) is given by: The longitudinal strain is given as , and Poisson’s ratio is 0.4, so: Step 2: Calculating the change in diameter.
The change in diameter is given by: However, the change in diameter per unit length results in a factor of , so the final answer is .
Step 3: Conclusion.
Thus, the change in diameter is , corresponding to option (C).

Step 1: Relation of shear deformation.
Shear strain is given by For the first cube,
Step 2: Effect of doubling the side.
For the second cube, height becomes . Since the force and material are the same, shear stress and hence shear strain remain the same.
Step 3: Calculating new displacement.

Step 4: Conclusion.
The displacement of the top layer becomes .