Mechanical Properties Of Fluid

To solve this problem, we need to use the concept of bulk modulus (K), which is defined by the formula:

where:

• is the change in pressure.
• is the change in volume.
• is the original volume.
• The negative sign indicates that pressure increases when volume decreases.

We are given:

• Bulk modulus,
• Original volume,
• Volume change percentage,

Calculate the change in volume ( ):

Now, substitute the values back into the bulk modulus formula:

Solve for :

Substitute into the equation:

Thus, the increase in pressure required is .

To solve the problem, we need to use the principle of conservation of mass, specifically the equation of continuity for incompressible fluids, which states that the product of the cross-sectional area of the tube and the velocity of the fluid flow through that area is a constant. Mathematically, it is represented as:

where:

• and are the cross-sectional areas of the first and second tubes, respectively.
• and are the flow velocities in the first and second tubes, respectively.

Given the radii:

• The radius of the first tube is , thus its cross-sectional area .
• The radius of the second tube is , thus its cross-sectional area .

Substituting the areas into the continuity equation:

The terms cancel out, leaving:

Rearranging to solve for , we find:

Therefore, the speed of water in the second tube is .

The problem involves the relationship between pressure and volume for an air bubble as it rises to the surface of a lake. We can use Boyle's Law, which states that for a constant temperature, the product of pressure and volume remains constant, i.e., . Given:

Let be the pressure at the bottom of the lake and be the pressure at the surface.

Atmospheric pressure cm Hg.

Density of mercury density of water .

The pressure at the bottom is the sum of atmospheric pressure and the pressure due to water column:

Using Boyle’s Law:

Therefore:

Substitute in the pressure equation:

Convert from cm of Hg to pascals:

cm Hg cm Pascal/cm (since 1 cm Hg = 1333 Pa)

Pascal

Using the conversion of mercury to water density:

kg/m / (40/3)

Calculate pressure in water as :

Pascal

Finally, solve for :

Pascal

meters

Thus, the depth of the lake is 20 meters.

Water Escape Calculation

We need to calculate the amount of water that escapes from a hole in a pipe near the lower end of a large water storage tank.

The hole has an area of , and the height of the water in the tank is above the point of the leak.

Step 1: Calculate the Speed of Water

Using Torricelli's Law, the speed of the water is given by the equation:

where:

• is the acceleration due to gravity,
• is the height of the water above the hole.

Substituting the values:

Step 2: Calculate the Volume of Water Escaping Per Second

The flow rate is given by:

where:

• is the area of the hole,
• is the velocity of the water.

Substituting the values:

Thus, the amount of water escaping in 1 second is: