Elasticity

With rise in temperature, the Young's modulus of elasticity generally decreases for most materials.

Concept Used:

Young's modulus (E) is defined as the ratio of stress to strain in the elastic region of a material. It is a measure of the stiffness of a material. The temperature dependence of Young's modulus is related to the interatomic bonding forces and thermal expansion of the material.

Step-by-Step Explanation:

Step 1: Understand the relationship between interatomic forces and Young's modulus.

Young's modulus is given by . At the atomic level, it depends on the curvature of the potential energy curve between atoms: , where U is the interatomic potential and r is the interatomic distance.

Step 2: Consider the effect of temperature on interatomic spacing.

As temperature increases, materials expand due to increased atomic vibrations. This increases the average interatomic distance (r). For most bonding potentials, the second derivative decreases as r increases beyond the equilibrium position.

Step 3: Analyze the effect on bond stiffness.

The force constant (k) of the atomic bond, which determines Young's modulus, decreases with increasing interatomic separation. Since thermal expansion causes increased interatomic spacing, the bond becomes less stiff, resulting in a decrease in Young's modulus.

Step 4: Consider exceptions and general behavior.

While most materials show a decrease in Young's modulus with increasing temperature, some materials like invar (a nickel-iron alloy) show minimal change due to their near-zero thermal expansion coefficient.

Therefore, with rise in temperature, the Young's modulus of elasticity decreases for most materials.

The problem asks for the cross-sectional area of a wire that sags under the weight of a suspended mass. We are given the wire's initial length, its Young's modulus, the suspended mass, and the angle the wire makes with the horizontal at the supports.

Concept Used:

The solution involves combining two main physics principles:

1. Mechanical Equilibrium: The system is in static equilibrium. The vertical components of the tension in the two segments of the wire must balance the gravitational force (weight) of the suspended mass.
2. Elasticity (Young's Modulus): The tension in the wire causes it to stretch. The relationship between stress, strain, and Young's modulus ( ) is given by: where is the tension, is the cross-sectional area, is the elongation, and is the original length.
3. Small Angle Approximation: For a small angle (in radians), . This approximation is used to relate the geometry of the sag to the strain.

Step-by-Step Solution:

Step 1: Analyze the forces and find the tension ( ).

The suspended mass is in equilibrium. Its weight acts downwards and is balanced by the vertical components of the tension from the two halves of the wire. Let be the angle each half of the wire makes with the horizontal.

The equilibrium equation in the vertical direction is:

Given that radian is a small angle, we can use the approximation .

We can now solve for the tension :

Step 2: Analyze the geometry to find the strain.

The initial length of the wire is . Let's denote the half-length as . When the mass is suspended, the wire sags by a distance and each half has a new length . From the diagram, we have a right-angled triangle with sides , , and hypotenuse .

For a small angle , we have .

Using the binomial approximation for small , we can find :

The total elongation of the wire is . The strain is .

Since , the strain can be expressed as:

Step 3: Use Young's Modulus to calculate the area ( ).

The formula for Young's Modulus is .

Rearranging for the area :

We have:

Substituting these values:

Step 4: Express the answer in the required format.

The problem asks for the value of A in the form _____ .

Our calculated value is .

Therefore, the value to be filled in the blank is 1.

The question requires evaluating the statements related to the concept of elasticity in physics. Let’s analyze both the assertion and the reason carefully.

1. Understanding the Assertion (A):
   • The assertion states: "The property of a body, by virtue of which it tends to regain its original shape when the external force is removed, is Elasticity."
   • This statement is true. Elasticity is the physical property of materials where they return to their original shape after the deforming forces are removed. This is a fundamental characteristic of elastic materials.
2. Analyzing the Reason (R):
   • The reason states: "The restoring force depends upon the bonded inter atomic and inter molecular force of solid."
   • This statement is also true. The restoring force in a material, which leads to elasticity, indeed depends on the interatomic and intermolecular forces. These internal forces try to restore the structure back to its original shape once the external forces are withdrawn.
3. Evaluating the Relationship:
   • Both statements are true, and the reason (R) provides the correct explanation for the assertion (A).
   • The elasticity of a material is directly related to the internal forces (inter atomic and inter molecular forces) that act to return the material to its original shape after the deforming force is removed.

Based on the analysis, the correct option is: Both (A) and (R) are true and (R) is the correct explanation (A).

To solve this problem, we need to understand the relationship between excess pressure inside a soap bubble and its radius, and subsequently its volume.

The formula for excess pressure ( ) inside a soap bubble is given by:

where is the surface tension of the liquid and is the radius of the soap bubble.

According to the question, the excess pressure inside the first bubble is thrice that inside the second bubble, so:

By substituting the formula for excess pressure, we get:

After simplifying, the above equation becomes:

This implies:

The volume of a sphere (or bubble) is given by:

Thus, the volumes of the bubbles can be represented as:

• Volume of the first bubble:
• Volume of the second bubble:

After substituting into the equation for , we find:

Therefore, the ratio of the volumes of the first and second bubble is:

Based on this calculation, the correct answer is 1 : 27.

Given:

Initial pressure of the liquid

Final pressure of the liquid

The change in pressure ( ) is:

The change in volume ( ) is:

The bulk modulus is given as:

Now, using the formula for bulk modulus:

We can calculate the volume :

Substituting the given values:

Thus, the volume is:

To determine the angular displacement of the rod axis from its original position, we need to use the formula for angular displacement due to shear force in a cylindrical body:

Where:

• is the shear force applied at the top of the rod.
• is the length of the rod.
• is the shear modulus of the material.
• is the cross-sectional area of the rod.

Given values are:

• N
• m
• N/m²
• The radius of the rod is 4 cm, which is m

First, calculate the cross-sectional area of the cylindrical rod:

m²

Substitute these values into the formula:

Now, simplify:

Thus, the angular displacement is . Hence, the correct answer is:

We are given:

• When tension , length .
• When tension , length .

We must find the original (unstretched) length of the string.

Concept Used:

For an elastic string, extension is proportional to the applied tension :

where is the force constant of the string.

Step-by-Step Solution:

Step 1: Write the two equations:

Step 2: Divide or eliminate :

Step 3: Substitute known values:

Step 4: Cross multiply:

Final Computation & Result: