Pressure

Percentage change

So, The Correct Option is (B) : 1.67

By Pascal’s law, pressure is equally transmitted to enclosed water in all directions. Statement II directly reflects this. Statement I is also true because at a given depth (level) in a static fluid, the pressure is the same in all directions. This is due to the hydrostatic pressure which depends only on the density of the fluid and depth.

Step 1: Write the formula for pressure. The pressure exerted by the smaller piston is given by: where: - is the force exerted due to the weight of the vehicle, - is the area of cross-section of the cylinder.
Step 2: Calculate the force . The force is the weight of the vehicle: where: - , - .
Step 3: Convert the area to SI units. The given area is: Convert it to square meters:
Step 4: Calculate the pressure. Substitute and into the formula for pressure: Simplify:
Final Answer: The maximum pressure the smaller piston would have to bear is:

To solve the problem, we need to find the rate of flow of glycerin through a conical pipe given the density, cross-sectional areas, and pressure drop. Let's break this down step by step.
Step 1: Understand the given data
- Density of glycerin, ρ = 1.25 × 10³ kg/m³
- Area at one end of the pipe, A₁ = 10 cm² = 10 × 10^-4 m² = 1 × 10^-3 m²
- Area at the other end of the pipe, A₂ = 5 cm² = 5 × 10^-4 m²
- Pressure drop, ΔP = 3 N/m²
Step 2: Apply the continuity equation
Using the principle of conservation of mass, we can apply the continuity equation:
A₁v₁ = A₂v₂
From this, we can express v₂ in terms of v₁:
v₂ = (A₁/A₂)v₁ = (10 × 10^-4 / 5 × 10^-4)v₁ = 2v₁
Step 3: Apply Bernoulli's equation
According to Bernoulli's equation, we have:
P₁ + (1/2)ρv₁² = P₂ + (1/2)ρv₂²
Rearranging gives us:
P₁ - P₂ = (1/2)ρ(v₂² - v₁²)
Substituting ΔP = P₁ - P₂ = 3 N/m² and v₂ = 2v₁:
3 = (1/2)ρ((2v₁)² - v₁²)
3 = (1/2)ρ(4v₁² - v₁²) = (1/2)ρ(3v₁²)
3 = (3/2)ρv₁²
Step 4: Solve for v₁
Substituting the value of ρ:
3 = (3/2)(1.25 × 10³)v₁²
3 = (3 × 1.25 × 10³ / 2)v₁²
v₁² = (3 × 2) / (3 × 1.25 × 10³) = 2 / (1.25 × 10³)
v₁² = 2 / 1250 = 1 / 625 (m/s)²
v₁ = √(1/625) = 1/25 = 0.04 m/s
Step 5: Calculate the rate of flow
The rate of flow Q is given by:
Q = A₁v₁
Substituting A₁ = 1 × 10^-3 m² and v₁ = 0.04 m/s:
Q = (1 × 10^-3)(0.04) = 4 × 10^-5 m³/s
Step 6: Express in terms of x
The rate of flow is given as x × 10^-5 m³/s:
Q = 4 × 10^-5 ⇒ x = 4
Final Answer
The value of x is 4.

1. Write the Expression for the Pressure Inside the Bubble:

The pressure inside the bubble is given by the formula:

Where:

• is the atmospheric pressure.
• is the density of the liquid.
• is the acceleration due to gravity.
• is the depth of the bubble.
• is the surface tension of the liquid.
• is the radius of the bubble.

2. Substitute the Given Values into the Formula:

Given:

, , , , , .

Substituting the values into the formula:

Final Answer

Thus, the pressure inside the bubble is greater than the atmospheric pressure by 1150 Pa.

To determine the pressure difference between the inside and outside of a soap bubble, we use the concept of surface tension and the geometrical properties of the bubble.

When a bubble forms, it consists of a thin film of liquid. The bubble has two surfaces in contact with air: one on the inside and one on the outside.

The excess pressure inside a bubble is due to the surface tension acting on these two surfaces. For a soap bubble, this pressure difference can be expressed using the formula for excess pressure due to surface tension:

• is the surface tension of the bubble.
• is the radius of the bubble.

The formula arises because a soap bubble has two surfaces (inside and outside). Normally, for a single surface, the formula is , but since a soap bubble has two surfaces, the surface tension formula must account for both, doubling the typical expression.\

Now, let's analyze the options given:

• : This is the correct formula for the excess pressure inside a soap bubble over the pressure outside.
• : This option is incorrect because the dimensions and dependency are reversed.
• : This does not account for twice the surface tension from two surfaces and thus is incorrect.
• : Correct for a single spherical surface, but not for a bubble with two surfaces.

Therefore, the correct answer is: .

Step 1: Use the ideal gas law.

For an ideal gas: Since are the same for both gases, the ratio of pressures depends only on the ratio of moles:

Step 2: Calculate moles of each gas.

For hydrogen ( ):

For oxygen ( ):

Step 3: Find the ratio of pressures.

Final Answer:

The bulk modulus is defined as: Rearranging: The volume of the cube is: Converting to : Substituting values: Converting to mm :