Surface Tension

Volume remains constant after coalescing. Thus,

where is radius of bigger drop and is radius of each smaller drop

Now, surface energy per unit surface area is the surface tension. So, Surface energy,
or
Therefore, surface energy of bigger drop


Surface energy of smaller drop

Hence, required ratio

The surface tension of liquid decreases with rise of temperature. The surface tension of liquid is zero at its boiling point and it vanishes at critical temperature. At critical temperatureintermolecular forces for liquid and gases becomes equal and liquid can expand without any restriction. For small temperature differences, the variation in surface tension with temperature is linear and is given by relation

where are the surface tension at and respectively and is the temperature coefficient of surface tension.

On the surface of the liquid in equilibrium molecules of the liquid possess maximum potential energy.

Radius of one drop of mercury is .
The volume of one drop
; Total volume of the two drops,

Let the radius of the large drop formed be The volume of the large drop is also V.

Now the surface area of the two drops is

and the surface area of the resultant drop is

Let T be the surface tension of mercury. Therefore the surface energy of the two drops before coalescing is

and the surface energy after coalescing,

A molecule on the surface experiences a net force (Cohesive force) in downward direction due to net number of molecules pulling it in the downward direction (molecule A as shown in figure underneath). In contrast, a molecule well inside is equally pulled in all directions by molecules around at (molecule Bas shown in figure).

Step 1: Use volume conservation.
Let radius of each small drop be .
Total volume of 1000 drops
Step 2: Write surface energy relation.
Surface energy is proportional to surface area.
Initial surface energy:
Final surface energy:
Step 3: Find ratio.

Step 4: Conclusion.
The ratio of final surface energy to initial surface energy is .

Step 1: Understanding capillary rise.
The difference in height of the liquid in the two limbs of a U-tube containing a capillary rise is given by the formula: where: - is the surface tension, - is the angle of contact (which is zero here), - is the radius of the tube, - is the density of the liquid, - is the acceleration due to gravity. Since the angle of contact is zero, . The difference in the height of the liquid in the two limbs is: Step 2: Applying the formula for each tube.
For the tube with diameter , the radius is . For the tube with diameter , the radius is . Now calculate the difference in height for each tube using the surface tension , the density , and : For the larger tube: For the smaller tube: Step 3: Total difference in height.
The total difference in the height of the liquid in the two limbs is: Step 4: Conclusion.
Thus, the difference in the level in the two limbs is 8.4 mm, which corresponds to option (D).

Step 1: Volume of mercury drops.
The volume of a sphere is given by . Since the process is isothermal, the total volume is conserved when the drops combine. Therefore, the volume of the resulting large drop is the sum of the volumes of the individual drops: Simplifying: Step 2: Conclusion.
Thus, the radius of the resultant drop is , which is option (D).

Step 1: Expression for orbital magnetic moment.
Orbital magnetic moment of an electron is where is orbital angular momentum.
Step 2: Angular momentum in ground state.
For ground state,
Step 3: Substitution.

Step 4: Conclusion.
The orbital magnetic moment is .

Step 1: Understanding the rise of water.
In a capillary tube, the rise of liquid is given by: where is the surface tension, is the angle of contact, is the density of the liquid, is the acceleration due to gravity, and is the radius of the capillary tube.
Step 2: Effect of depression on contact angle.
When the tube is depressed, the apparent contact angle is affected by the length above the water surface. The new apparent angle of contact is given by , which corresponds to the change in the capillary rise due to the depression.
Step 3: Conclusion.
Thus, the new apparent angle of contact is , corresponding to option (C).

Step 1: Expression for force constant of the wire.
For a stretched wire, the effective force constant is:

Step 2: Time period of vertical SHM.
Time period for a mass–spring system is:

Step 3: Substitute the value of .

Step 4: Write expression for frequency.

Step 5: Conclusion.
The correct expression for frequency is option (B).

Step 1: Relation between time period and force constant.
For SHM,

Step 2: Determine individual force constants.

Step 3: Resultant force constant.
When both forces act together in the same direction,

Step 4: Express new time period.

Step 5: Conclusion.
The new periodic time is .

Step 1: Formula for Excess Pressure.
The excess pressure inside a soap bubble is given by: where is the surface tension and is the radius of the bubble. The radius is inversely proportional to the volume , so we have: Since the excess pressure in the first bubble is three times that in the second bubble, we can equate the pressures and solve for the volume ratio: Step 2: Final Answer.
Thus, the value of is 9:1.

Step 1: Energy Considerations.
The total energy released in the process of coalescence is the difference between the total energy of the small drops and the total energy of the big drop. This difference in energy is converted into the kinetic energy of the big drop. The energy of a single drop is related to its surface area and surface tension, and for the big drop, we have: The total energy released is proportional to the difference in the radii of the small and big drops, leading to the final velocity of the big drop.
Step 2: Final Answer.
Thus, the speed of the big drop is given by .

Step 1: Understanding the concept of volume conservation.
Since the process is isothermal, the volume of the two smaller drops is conserved when they combine into a larger drop. The volume of a spherical drop is given by . Therefore, the total volume before and after the merging of the drops is: Simplifying the equation, we get the relation for the radius of the resultant drop: Step 2: Conclusion.
Thus, the radius of the resultant drop is , corresponding to option (B).

Step 1: Effect of soap on water.
Soap molecules reduce the surface tension of water by weakening the cohesive forces between water molecules.
Step 2: Cleaning action.
Lower surface tension allows the soap solution to spread easily and penetrate the fabric pores, helping to remove dirt and grease.
Step 3: Conclusion.
Therefore, soap solution is effective in cleaning because it decreases the surface tension of water.

Step 1: Volume conservation.
Let radius of each small drop be .

Step 2: Surface energy relation.
Surface energy surface area .

Step 3: Total surface energy before coalescence.

Step 4: Surface energy after coalescence.

Step 5: Ratio of surface energies.

Step 6: Conclusion.
The required ratio is .

Step 1: Formula for excess pressure in liquid drop.

Step 2: Relation between excess pressures.
Given:

Step 3: Finding ratio of radii.

Step 4: Ratio of surface areas.

Step 5: Conclusion.
The ratio of their surface areas is .

Step 1: Formula for capillary rise.
The upward force due to surface tension in a capillary tube is given by: where is the radius of the capillary and is the surface tension. The downward force due to the weight of the water column is: where is the density of water, is the acceleration due to gravity, is the cross-sectional area, and is the height of the water column. Step 2: Equating forces.
At equilibrium, the forces balance, so: Simplifying: Using , , and , we can solve for . The given upward force is dyne, which is N, so we calculate the inner circumference of the capillary as . Step 3: Conclusion.
Thus, the inner circumference of the capillary is 1.5 cm, which corresponds to option (A).

Step 1: Formula for output current.
The output current is related to the input voltage and the current gain by the equation: Substituting the values:
Step 2: Conclusion.
Thus, the correct answer is (B) .

Step 1: Consider surface area conservation.
For soap bubbles under isothermal conditions, total surface area remains conserved.

Step 2: Write surface area expressions.
Total initial surface area:
Final surface area of big bubble:


Step 3: Equate areas.


Step 4: Solve for .


Step 5: Conclusion.
The radius of the big bubble is .

Step 1: Understanding stress and strain.
Stress is given by , where is the cross-sectional area of the wire. Since both wires have the same diameter, the stress in both will be the same.
Step 2: Strain relationship.
Strain is given by . The strain will be different for the two wires as their materials differ. The elongation produced by the same force will vary depending on the Young’s modulus of the material.
Step 3: Conclusion.
The stress will be the same because the diameter is the same, but the strain will be different due to the different materials. Thus, the correct answer is (D).

Step 1: Understanding resolving power.
The resolving power of an optical instrument is inversely proportional to the wavelength of the light used. This means that the instrument with a smaller wavelength will have a greater resolving power.
Step 2: Calculating the ratio.
Let and be the resolving powers of instruments A and B, respectively. We have: Substitute the values for and :
Step 3: Conclusion.
The ratio of the resolving power of A to B is , hence the correct answer is (D) 4 : 3.

Step 1: Relation for capillary rise.
Capillary rise is given by Thus,

Step 2: Relate radius with cross-sectional area.


Step 3: Find relation between height and area.


Step 4: Apply given change in area.
If , then

Step 1: Expression for capillary rise.
Capillary rise is given by

Step 2: Effect of acceleration on effective gravity.
In a lift moving downward with acceleration , effective gravity becomes

Step 3: Effect on height of capillary rise.
When decreases, increases.

Step 1: Understanding the capillary rise.
The height to which liquid rises in a capillary tube is given by the formula: where is the surface tension, is the density of the liquid, is the acceleration due to gravity, and is the radius of the capillary tube.
Step 2: Effect of the cross-sectional area.
Since the capillary rise depends on the radius of the tube, and the radius is related to the cross-sectional area by , reducing the area by a factor of 10 will increase the height by a factor of . Therefore, the new height is .
Step 3: Conclusion.
The correct answer is (A), 9 cm.

Step 1: Understanding the pressure in a drop.
The excess pressure inside a drop is given by the formula , where is the surface tension and is the radius of the drop. When drops combine, the volume remains constant, but the radius of the new drop increases.
Step 2: Applying the formula.
If 27 smaller drops combine, the radius of the new drop will be , because the volume of a sphere is proportional to . Since the excess pressure is inversely proportional to the radius, the new excess pressure will be: Step 3: Conclusion.
The excess pressure inside the bigger drop is 3 units, which corresponds to option (C).

Step 1: Apply conservation of volume.
When a mercury drop breaks into smaller droplets, total volume remains conserved.

Step 2: Write volume expressions.
Volume of original drop:
Volume of one small droplet:


Step 3: Equate total volumes.


Step 4: Solve for .


Step 5: Conclusion.
The radius of each droplet is .

Step 1: Using the formula for kinetic energy.
Kinetic energy is given by: Where: - , - and are the velocities of the masses. Since the kinetic energies are equal: Simplifying, we get: Step 2: Solving for velocities.
We can solve for the velocities in terms of the masses: Substituting and : Step 3: Calculating the momenta.
Momentum is given by . Thus, the momenta of the two masses are: The ratio of the momenta is: Thus, the ratio of the magnitudes of the momenta is , corresponding to option (B).

Step 1: Surface energy of a drop.
The surface energy of a drop is given by: where is the surface tension and is the surface area. For a spherical drop, the surface area is given by .
Step 2: Initial and final surface energy.
- For two small drops, the total surface area before coalescing is: - After coalescence, the new radius of the large drop is , so the surface area of the large drop is:
Step 3: Energy ratio.
The ratio of the surface energies before and after the change is:
Step 4: Conclusion.
Thus, the ratio of the surface energies is , which is option (B).

Step 1: Understanding the coefficient of restitution.
The coefficient of restitution is a measure of the elasticity of a collision. For a perfectly elastic collision, , meaning no kinetic energy is lost during the collision.
Step 2: Conclusion.
Thus, the correct answer is (B), 1.

Step 1: Write excess pressure formula for soap bubble.
For a soap bubble, excess pressure is where is surface tension and is radius.
Step 2: Use given condition.

Step 3: Solve for radii ratio.

Step 4: Use volume relation.
Volume of a bubble
Step 5: Find ratio of volumes.

Step 6: Conclusion.
The ratio of volumes is .

Step 1: Understanding molecular forces in a liquid.
In a liquid, molecules within the sphere of influence experience attractive forces from other molecules around them. The force exerted by other molecules within the sphere of influence is an attractive force. Thus, the correct answer is (A) attracted by other molecules in the sphere of influence.

The height of water in a capillary tube is given by the capillary rise formula: where is the surface tension, is the angle of contact, is the radius of the tube, is the density of the liquid, and is the acceleration due to gravity. The volume of water in the capillary is: Simplifying: The mass of water in the capillary is: For a capillary with radius , the new mass is: which shows that the mass of water is directly proportional to the radius of the capillary tube. Thus, the new mass is: Final Answer:

Using the conservation of volume, the volume of the big drop is equal to the sum of the volumes of the 1000 smaller drops: where is the radius of the large drop and is the radius of each smaller drop. Simplify: The surface energy of a sphere is proportional to its surface area. The initial energy (for 1000 small drops) is: The final energy (for the large drop) is: Thus, the ratio of the final to initial energy is: Final Answer:

The pressure inside a soap bubble is given by the formula: where is the surface tension and is the radius of the bubble. Step 1: For the two bubbles with radii and , the pressures are: Step 2: The pressure difference across the common surface is: Step 3: Substitute the expressions for and : Step 4: Simplify: Step 5: Solve for : Final Answer:

The pressure difference inside and outside a bubble is given by the formula: Where: - (surface tension), - (radius of the bubble). Substituting the values: Thus, the pressure inside the bubble is more than the outside pressure.

Concept:
Photoelectric equation: Step 1: Initial condition.
Step 2: Final condition.
Step 3: Compare equations.
Step 4: Conclusion. Since work function exists:

Step 1: Relations
Capillary rise .
Mass of liquid in capillary .
Step 2: Substitution
Since , then .
This implies .
Step 3: Calculation
New mass .
Final Answer: (A)

Concept: Capillary rise occurs due to surface tension. The upward force due to surface tension balances the weight of the liquid column. The formula for capillary rise is For water in a glass tube, the angle of contact .
Step 1: {Substitute .}
Step 2: {Final expression for height of capillary rise.}