Surface Tension

The correct answer is 2



Therefore , the value of r is 2cm

The correct answer is (A) : 2.8 × 10^–4 J

⇒ ΔE = T(ΔS)
= T × 4π(nr′² –r²), n = 64
= T × 4π × (4 – 1)r²
⇒ ΔE = 0.075 × 4 × 3.142 (3) × 10^–4 J
= 2.8 × 10^–4 J

To calculate the gain in potential energy when an object is taken to a height above the surface of the Earth, where this height is equal to three times the radius of the Earth, we use the formula for gravitational potential energy in the gravitational field:

The gravitational potential energy at a height above the Earth's surface is given by:

where:

• is the gravitational constant.
• is the mass of Earth.
• is the mass of the object.
• is the radius of the Earth.
• is the height above the Earth's surface.

Instead of for the surface, the change in potential energy is:

At the surface of the Earth, the gravitational potential energy is:

At a height above the surface, it is:

The gain in potential energy, therefore, is:

Now, using the simplified gravitational potential energy change considering , we have:

Substitute , , and :

Therefore, the gain in potential energy is 48 MJ.

The correct answer is 1.
Surface Tension =T
R : Radius of bigger drop
r : Radius of smaller drop
Volume will remain same

Step 1: Analyze the terms in the binding energy equation.

.

- Volume term:

- Surface energy term:

- Coulomb term:

- Asymmetry term:

- Pairing term:

Step 2: Match the options with the equation.

- C and D are directly supported by the equation.

Final Answer: The most appropriate choice is option (3).

Given:

• Surface tension ( ) =
• Initial radius ( ) =
• Final radius ( ) =

Step 1: Formula for Work Done

The work done ( ) in increasing the surface area of a soap bubble is given by:

where:

• : Surface tension
• : Change in surface area.

Step 2: Calculate the Change in Surface Area

A soap bubble has two surfaces (inner and outer). The initial surface area ( ) and final surface area ( ) are given by:

The change in surface area ( ) is:

Substitute and :

Simplify:

Step 3: Calculate the Work Done

Substitute and into the formula for work done:

Using :

Simplify step by step:

Final Answer:

The work done is .

When the disc rolls without slipping, the topmost point moves along a cycloidal path. The total displacement is calculated as follows:

Step 1: Distance moved by the center of the disc.
The center of the disc moves a horizontal distance equal to the arc length subtended by half the circumference:

Step 2: Vertical displacement of point .
Point starts at the top of the disc and ends at the bottom after half a rotation, so the vertical displacement is:

Step 3: Resultant displacement.
The resultant displacement is given by:

Substitute the values:

Factor :

To solve this problem, we need to understand the relationship between the surface energy of droplets when small droplets coalesce to form a larger droplet.

Consider the following:

1. Let the radius of one small droplet be . Then the volume of one small droplet is given by:
2. The total number of small droplets is 1000. Therefore, the total volume of the small droplets is:
3. When these 1000 small droplets combine to form one big droplet, the volume of the big droplet is the same as the total volume of all small droplets. Let the radius of the big droplet be . Therefore, the volume of the big droplet is:
4. Equating the volumes of the big droplet and the total small droplets:
5. Solving for , we have:
6. Therefore, since the cube root of 1000 is 10.
7. The surface energy is directly proportional to the surface area of the droplet. The total surface area of all small droplets is:
8. The surface area of the big droplet is:
9. The ratio of the new surface area to the original surface area is:
10. Hence, the surface energy becomes times the original total surface energy of all small droplets.

Thus, the correct answer is .

Work Done in Expanding a Soap Bubble:
The work done in increasing the surface area of a soap bubble is given by:
where is the surface tension and is the increase in surface area.

Calculate Initial and Final Surface Areas:
Initial radius .
Initial surface area .
Suppose the new radius is . Then the final surface area is:

Calculate the Change in Surface Area :

Use the Work Done to Solve for :
Given and , we have:

Simplifying,

Using :

Conclusion:
The new radius of the soap bubble is .

The excess pressure in a soap bubble is given by the formula , where is the surface tension, and is the radius of the soap bubble.
Given that this pressure is balanced by the pressure due to a liquid column, we have:
, where , , and .
Thus, we equate the pressures: Substituting the known values: Simplifying, we find: Solving for :
The diameter is twice the radius:
Hence, the diameter of the soap bubble is , which is within the provided range of 7,7.

The correct answer is :
By conservation of volume
Surface energy of 1000 droplets =

Surface energy will decrease in the process of formation of bigger drop, hence energy is released and temperature increases

To find the work done in dividing a liquid drop into smaller drops, we need to consider the change in surface energy due to the change in total surface area.

1. Calculate the initial surface area of the original drop:
   • The original drop is a sphere with radius .
   • The surface area of a sphere is given by the formula .
2. Calculate the total surface area of the smaller drops:
   • The original drop is divided into 27 identical smaller drops.
   • If the radius of each small drop is , then using the volume conservation:
   • This implies:
   • The surface area of each small drop is .
   • The total surface area of 27 smaller drops is .
3. Calculate the change in surface area and work done:
   • The change in surface area is .
   • Work done (which is the change in surface energy) is given by .

Therefore, the work done in the process is , which matches the correct answer given in the options.

To find the length of water that rises in the capillary tube, we use the capillary rise formula, which is given by:

where:

• is the surface tension of water:
• is the contact angle, which is for water and glass, thus
• is the density of water:
• is the acceleration due to gravity:
• is the radius of the capillary tube:

Substituting these values into the formula, we get:

Calculating the above expression yields:

The tube is inclined at to the vertical, so the length of water in the tube will be affected by this inclination. To find the actual length of water risen in the tube, we use:

Thus, the actual length is:

Given the options, the answer closest to is multiplied by . Therefore, the correct answer is:

The problem is about finding the surface energy released when two identical spherical water drops of radius coalesce into a single larger drop. The surface tension of the liquid is .

Concept Used:

The surface energy of a liquid drop is directly proportional to its surface area:

When two drops combine, the total volume remains constant. Since the new radius changes, the total surface area decreases, releasing energy equal to the difference in initial and final surface energies.

Step-by-Step Solution:

Step 1: Apply volume conservation:

Step 2: Calculate the total initial surface energy of two small drops:

Step 3: Calculate the final surface energy of the single large drop:

Step 4: Energy released is the difference between the initial and final energies:

Final Computation & Result:

The surface energy released when two identical drops of radius coalesce is:

Correct Option: (A)

To solve the problem of finding the pressure difference in the bubble, we first need to understand the factors influencing the pressure. An air bubble submerged in a liquid experiences two main pressures: the hydrostatic pressure due to the liquid column above it and the pressure due to surface tension.

1. Calculate Hydrostatic Pressure:
The hydrostatic pressure at depth is given by:

where:
(density of the liquid),
(acceleration due to gravity),
(depth in meters).

Thus:

.

2. Calculate Pressure Due to Surface Tension:
The pressure inside a bubble due to surface tension is given by the formula:

where:
(surface tension),
(radius of the bubble).

Thus:

.

3. Calculate the Total Pressure Inside the Bubble:
The total pressure inside the bubble is:

Since we are finding the difference between the pressure inside the bubble and atmospheric pressure, we subtract from both sides:

The pressure difference is:

.

We conclude that the difference between the pressure inside the bubble and atmospheric pressure is 2190 Pa. This precise calculation validates the solution correctly, considering the provided values.

We are given two liquids A and B having contact angles and in a capillary tube. The ratio is defined as:

We need to determine which statement about the nature of their meniscus (concave or convex) is correct when is negative or zero.

Concept Used:

The shape of a meniscus depends on the contact angle :

• If , then → concave meniscus (liquid wets the surface).
• If , then → convex meniscus (liquid does not wet the surface).

Step-by-Step Solution:

Step 1: Analyze the expression for .

The sign of depends on the signs of and .

Step 2: Consider the case when is negative.

If is negative, then one cosine term must be positive and the other negative.

• So, one liquid has (concave meniscus), and
• the other liquid has (convex meniscus).

Step 3: Determine which one corresponds to which case.

For to be negative, and have opposite signs. Hence:

• If → Liquid A has concave meniscus.
• If → Liquid B has convex meniscus.

Step 4: Now, consider the case when .

This means . At this angle, the liquid does not rise or fall, and the meniscus is flat. Thus, Liquid A neither wets nor repels the surface, while Liquid B’s nature depends on its own contact angle .

Final Computation & Result:

Hence, the correct interpretation is:

Final Answer: The correct statement is — If is negative, then liquid A has concave meniscus and liquid B has convex meniscus.

The surface energy of a single drop is given by , where is the radius and is the surface tension. - Initially, there are two drops, so the total surface energy is . - After the two drops coalesce, the radius of the new drop becomes , so the surface energy of the new drop is . The surface energy released is the difference between the initial and final surface energy, which is . Thus, the correct answer is .

To solve this problem, we need to use the concepts of pressure difference and surface tension. The pressure difference caused by surface tension for an air bubble is given by the formula:

Here, is the surface tension of the liquid, and is the radius of the air bubble.

Given that the pressure inside the bubble is 2100 N/m greater than the atmospheric pressure, we can write:

Also, given that the radius (converted to meters), we can substitute these values into the equation:

Simplifying for gives:

Therefore, the surface tension of the liquid is .

The correct answer is 0.05.