Fluid Mechanics

Capillary rise is given by:

If , the rise is non-zero, so Statement-I is incorrect. Statement-II is correct as the contact angle depends on the materials involved.

To solve this problem, we need to identify the correct form of Bernoulli's equation. Bernoulli's equation is a principle in fluid dynamics that describes the conservation of energy in a flowing fluid. It is applicable to incompressible, non-viscous fluids. The equation relates the pressure energy, kinetic energy per unit volume, and potential energy per unit volume of a fluid flowing along a streamline.

The general form of Bernoulli's equation is given as:

• is the pressure energy per unit volume.
• is the kinetic energy per unit volume, where is the velocity of the fluid and is the density of the fluid.
• is the potential energy per unit volume, where is the acceleration due to gravity and is the height above a reference point.

Now let's analyze the given options:

1. :
   • This option uses and P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}:
     • This option correctly reflects the known form of Bernoulli's equation, matching our derived equation above.
   • :
     • This option incorrectly represents the kinetic energy term; kinetic energy should have the form , rather than .
   • :
     • This option incorrectly modifies the potential energy term. Potential energy per unit volume should be .

Therefore, the correct form of Bernoulli's equation is represented by option 2: .

To determine the ratio for the sphere to just float on water, we need to apply the principle of buoyancy. The sphere will float when the weight of the sphere equals the weight of the water displaced by it.

The volume of the whole sphere (including the cavity) is given by:

The volume of the cavity is:

Thus, the volume of the material of the sphere is the total volume minus the cavity volume:

Let the density of the sphere's material be where is the density of water.

The weight of the sphere is:

For the sphere to just float, the buoyant force equals the weight:

Substituting known values, we get:

Dividing through by :

Simplifying, we get:

Rearranging gives:

Thus, the ratio is:

Taking the cube root on both sides, we find the ratio:

Therefore, the correct option is:

To solve this problem, we need to understand the floating condition of the ice cube in both water and kerosene oil. The principle behind this is Archimedes' principle, which states that the weight of the fluid displaced by a submerged object is equal to the weight of the object.

The specific gravity gives us the density relative to water. For our problem:

• Specific gravity of ice ( ) = 0.9
• Specific gravity of kerosene oil ( ) = 0.8
• Specific gravity of water = 1.0 (by definition)

The density of ice is 0.9 times that of water, and the density of kerosene oil is 0.8 times that of water.

The buoyant force on ice from both water and kerosene must balance the weight of the ice. Therefore, we have:

Let be the volume of ice immersed in water, and be the volume of ice immersed in kerosene. Then:

• Weight of ice =
• Buoyant force in water =
• Buoyant force in kerosene =

Equating the weight of the ice to the total buoyant force:

Substituting the specific gravities:

Assuming the entire volume is submerged, .

Let’s simplify:

This simplifies to:

Thus, the ratio is .

Therefore, the correct answer is 1 : 1.

Analyze Statement I:
Statement I states that when the speed of liquid is zero everywhere, the pressure difference at any two points depends on the equation:

This is correct and is based on the hydrostatic pressure difference, which applies when the fluid is at rest or moving uniformly without velocity gradients.

Analyze Statement II Using Bernoulli’s Equation:
In a venturi tube, where the fluid is in motion, we can apply Bernoulli’s equation:

Simplifying for the pressure difference, we get:

The statement given, , is not a general result of Bernoulli’s equation and is incorrect as presented.

Conclusion:
Therefore, Statement I is correct (it applies to a static fluid or uniform motion with no speed variations), but Statement II is incorrect in the context of the venturi tube.

To solve the problem of determining the new terminal velocity of the larger droplet formed by the coalescence of 8 smaller droplets, we can use the physics of terminal velocity and volume conservation. First, consider that the terminal velocity of a droplet can be expressed as , where is the radius of the droplet. When droplets coalesce, their combined volume equals the volume of the larger drop. Given the radius mm for each of the smaller droplets and their terminal velocity cm/s, we can calculate the new radius of the larger droplet. The volume of a sphere is given by . Therefore, the total initial volume of 8 small droplets is and the volume of the larger drop is . Equating these, we get:

Solving for , we have:

Substituting mm, we find:

Next, calculate the terminal velocity of the larger droplet, using . The new terminal velocity is given by:

Substitute the values:

Therefore,

The calculated terminal velocity of the larger droplet is indeed 40 cm/s, which fits perfectly within the given range of 40 to 40 cm/s.

To find the viscosity of the oil, we can use Stokes' Law, which relates the drag force experienced by a sphere moving through a viscous fluid to its radius, speed, and the fluid's viscosity. The formula for terminal velocity under the influence of gravity for a sphere in a viscous medium is given by:

where:

• is the terminal velocity
• is the radius of the sphere
• is the density of the sphere material (steel in this case)
• is the density of the fluid (oil in this case)
• is the acceleration due to gravity
• is the viscosity of the fluid

We are given:

• Diameter of the sphere
• Terminal velocity
• Density of steel
• Density of oil
• Acceleration due to gravity

First, calculate the radius of the sphere:

Substitute these values into the terminal velocity equation and solve for the viscosity ( ):

Rearrange to solve for :

Calculating the values inside the equation:

• Difference in density:
• Radius squared:

Substitute these into the equation:

After solving, the viscosity of the oil ( ) is approximately:

Therefore, the correct answer is 1.99.

Let's analyze each statement:

A. The terminal velocity is given by:
$ r \rho_s \rho_l g \eta V \propto r^2 V r R \eta (2)\ \text{A, C and D only} $.

To determine the work done by gravity when the two cylindrical vessels are connected, we start by understanding the physics involved:

1. Both cylindrical vessels have a cross-sectional area .
2. One vessel has water up to a height , and the other up to a height .
3. The density of water, .
4. Acceleration due to gravity, .

When the vessels are connected at the bottom, water will flow until the height of water in both containers is equal. Let's denote the final height as .

To find , use the conservation of volume:

• Initial total volume of water:
• Final total volume of water is spread equally:

The work done by gravity is equivalent to the change in potential energy. The water mass moves from its initial height positions to .

The change in potential energy (and thus work done) is given by:

• Calculating individually for each vessel:
• Water falling from 10 m to 8 m:
  • Mass:
  • Potential energy change:
• Water rising from 6 m to 8 m:
  • Mass:
  • Potential energy change:

The net work done is:

Thus, the work done by the force of gravity after the vessels are connected is , which matches the correct option.

The velocity vector is the derivative of the position vector with respect to time: At s, the velocity is: The magnitude of the velocity is: The direction of the velocity is given by the angle with the axis: Thus, .
Final Answer: .

- Assertion (A) is correct because the photoelectric effect can indeed be suppressed by applying a sufficiently negative potential to the material, as the energy required to remove electrons becomes greater than the energy provided by the incident radiation.
- Reason (R) is also correct, as a negative electric potential does indeed stop the emission of electrons, but it does not depend on the frequency of the incident radiation, but rather on the energy required to overcome the binding energy of the electrons. Thus, (R) is correct but does not explain (A) fully. The correct answer is option (1).

Analyze each statement.
A: Incorrect, as the graph is not a parabola but rather a more complex function of radius and viscosity.
B: Incorrect, as terminal velocity varies with ball size and density.
C: Correct, as viscosity and terminal velocity are temperature-dependent.
D: Correct, as variations in terminal velocity can reflect differences in liquid density.
E: Correct, as the initial speed affects the drag force and settling time, influencing the measured viscosity.

Let's analyze each statement:

A. Surface tension arises due to extra energy of the molecules at the surface as compared to the molecules at the interior of a liquid.
The molecules at the surface experience a net inward force due to cohesive forces from the liquid below. This gives the surface molecules extra energy, leading to surface tension. Thus, statement A is incorrect.

B. As the temperature of a liquid rises, the coefficient of viscosity decreases.
This is because the cohesive forces between liquid molecules decrease with increasing temperature, making it easier for the liquid to flow. Thus, statement B is incorrect.

C. As the temperature of a gas increases, the coefficient of viscosity increases.
In gases, viscosity arises due to the transfer of momentum between gas molecules. At higher temperatures, the gas molecules move faster, leading to more frequent and effective collisions and thus increased momentum transfer. Thus, statement C is correct.

D. The onset of turbulence is determined by the Reynolds number.
The Reynolds number is a dimensionless quantity that characterizes the flow regime: low Reynolds numbers indicate laminar flow, while high Reynolds numbers indicate turbulent flow. Thus, statement D is correct.

E. In a steady flow, two streamlines never intersect.
Streamlines represent the direction of fluid flow at a given instant. If streamlines intersected, it would imply that a fluid particle at that point would have two different velocities simultaneously, which is not possible in steady flow. Thus, statement E is correct.

Final Answer:
The correct statements are C, D and E.

The final answer is .

• is the total energy of the particle,
• is the momentum of the particle,
• is the speed of light in a vacuum,
• is the rest mass of the particle.

Final Answer: .

To find the velocity of the water coming out of the hole, we can use Torricelli's theorem, which is an application of Bernoulli's principle for fluid dynamics. The velocity of efflux, , from a hole at the side of a tank can be given by:

where:

• is the acceleration due to gravity (10 ).
• is the effective height of the water column above the hole.

In addition to the gravitational force of the water column's height, we have an external load of 50 kg exerting additional pressure. The pressure due to this load can be calculated and converted to an equivalent height of water using the formula:

Here, the additional effective pressure head can be given by:

• (density of water)

Now calculate the height:

The initial height of the water column above the hole is .

So the total effective height above the hole .

Using Torricelli's theorem with the total height,

Thus, the velocity of the water coming out of the hole is .

To find the fractional compression of water at a depth of 2.5 km below sea level, we use the formula for fractional compression given by:

Where is the pressure applied and is the bulk modulus of the water.

The pressure at a depth is given by:

Given:

Substitute these values into the pressure formula:

Now, substitute and into the fractional compression formula:

To express this as a percentage, multiply by 100:

Thus, the fractional compression of water at the given depth is 1.25%.

Let be the energy of both the photon and the proton.

Photon:
Let be the wavelength of the photon. The energy of the photon is given by:
$ p p \frac{\lambda_p}{\lambda} \frac{1}{c} \sqrt{\frac{E}{2m_p}} $.

The bead moves under the influence of gravity and the restoring force from the spring.
Step 1: Apply energy conservation. At the top of the hoop, the bead has potential energy due to gravity and the spring. At the bottom, the bead will have kinetic energy.
Step 2: At the top, the potential energy is: where is the height of the bead from the bottom.
Step 3: The spring potential energy at the top is zero since the spring is at its equilibrium length.
Step 4: At the bottom, the kinetic energy is: The spring at the bottom has a compression of , so the spring potential energy is: Step 5: Apply conservation of mechanical energy: Step 6: Solve for : Final Conclusion: The velocity of the bead when the spring becomes is given by , which is Option (3).

To solve the problem of finding the relationship between the speed of water flowing in a pipe and the pressure difference, we can apply Bernoulli's principle. Bernoulli's principle states that in a streamline flow of an ideal fluid, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is a constant.

The general form of Bernoulli's equation for a horizontal flow (neglecting height-related potential energy difference) is:

Where:

• is the pressure exerted by the fluid.
• is the density of the fluid.
• is the velocity of the fluid.

Now, when the valve is closed, the pressure is and the velocity is zero because the fluid is at rest. Thus, the Bernoulli equation becomes:

When the valve is opened, the pressure drops to and the water starts to flow with velocity . Applying Bernoulli's equation, we get:

Equating the two equations, we have:

Simplifying for the velocity , we get:

This shows that the velocity is proportional to . Therefore, the correct answer is .

The photoelectric equation is given by: where: - is the stopping potential, - is the frequency of incident light, - is Planck's constant, - is the charge of the electron, - is the work function. From this equation, we can see that is linear with respect to , with a slope of , and the intercept gives the value of .

• (A) is true because the relation between and is linear.
• (B) is also true because the slope of the line gives , and rearranging gives .
• (C) is true because the slope of the vs line is related to .
• (D) is false because the value of the electric charge is required to find from the slope.
• (E) is false because to determine the work function, knowing is essential, and the value of cannot be determined from the vs curve alone without knowing .

Final Answer: (3) (A), (B) and (C) only.

When two identical conducting spheres are in contact, their charges are shared equally. The total charge on both spheres is: Thus, the charge on each sphere after they are in contact will be: Using Coulomb's law for the force of repulsion between the two spheres: Substitute the known values for the force and charge and solve for the distance : Solving for , we find .
Final Answer: .

To solve this question, we need to evaluate each statement individually in the context of standard physics principles.

Analysis of Statement-I: "The hot water flows faster than cold water."

This statement deals with the viscosity of fluids. Viscosity is the measure of a fluid's resistance to flow. As the temperature of water increases, its viscosity decreases. Therefore, hot water has a lower viscosity than cold water, allowing it to flow more easily. Hence, Statement-I is true.

Analysis of Statement-II: "Soap water has higher surface tension as compared to fresh water."

Surface tension is the tendency of fluid surfaces to shrink into the minimum surface area possible and is caused by the attraction between the liquid molecules. Adding soap to water decreases its surface tension because soap molecules disrupt the cohesive forces between water molecules. Soap acts as a surfactant, which means it reduces the surface tension of water. Therefore, Statement-II is false.

Now, let's evaluate the options:

• Option 1: Statement-I is false but Statement-II is true. (Incorrect, as Statement-I is true and Statement-II is false.)
• Option 2: Statement-I is true but Statement-II is false. (Correct, as analysis showed this statement arrangement to be true.)
• Option 3: Both Statement-I and Statement-II are true. (Incorrect, as Statement-II is false.)
• Option 4: Both Statement-I and Statement-II are false. (Incorrect, as Statement-I is true.)

Based on the analysis, the correct answer is: Statement-I is true but Statement-II is false.

Step 1: Evaluate the truth of and . Sound travels faster in solids than in gases due to the higher density and elastic properties of solids, not gases.
Step 2: Determine the correctness of . In fact, solids generally have a higher Bulk modulus than gases, contradicting the statement of .
Conclusion: Assertion is true, stating that sound waves travel faster in solids is correct, but Reason is false as gases typically have a lower Bulk modulus than solids.

Step 1: Statement A is true. Surface tension occurs due to excess molecular energy at the surface.
Step 2: Statement B is incorrect. As the temperature of a liquid increases, the coefficient of viscosity decreases.
Step 3: Statement C is true. As the temperature of a gas increases, the coefficient of viscosity increases.
Step 4: Statement D is true. Reynolds number determines whether the flow is laminar or turbulent.
Step 5: Statement E is true. In a steady flow, streamlines do not intersect. Final Conclusion: The correct statements are A, D, and E, which corresponds to Option (4).