Construct Hydrometer: Length, Average Density, Mark Spacing Answered

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- A hydrometer is to be constructed with a cylindrical floating rod. Nine fiduciary marks are to be placed along the rod to indicate densities of 0.98 g/cm3, 1.00 g/cm3 ,1.02 g/cm3, 1.04 g/cm3, . . . 1.14 g/cm3 . the row of marks is to start 0.200 cm from the top end of the rod and end 1.80 cm. from the top end .
a). What is the required length of the rod?
b). What must be its average density ?
c). Should the marks be equally spaced ?
Explain your answer.

-Water is forced out of a fire extinguisher by air pressure. How much gauge air pressure in the tank (above atmosphere) is required for the water jet to have speed of 30.0 m/s when the water level in the tank is 0.500 m. below
the nozzle ?

 

[anathema]

Hello,

I would like to provide some insights and answers to the questions posed in this forum post.

a) The required length of the rod can be calculated using the given information. We know that the marks start 0.200 cm from the top end and end 1.80 cm from the top end. This means that the distance between the first and last mark is (1.80-0.200) cm = 1.60 cm. Since there are nine marks, the distance between each mark is 1.60/9 = 0.1778 cm. Therefore, the required length of the rod is 1.80 cm + 0.1778 cm = 1.9778 cm.

b) The average density of the rod can be calculated by taking the average of the given densities. This will give us (0.98 + 1.00 + 1.02 + 1.04 + ... + 1.14)/9 = 1.04 g/cm3. Therefore, the average density of the rod should be 1.04 g/cm3.

c) No, the marks should not be equally spaced. This is because the given densities are not equally spaced. In order to accurately measure the density, the marks should be placed at equal intervals corresponding to the density values. This will ensure that the readings obtained from the hydrometer are accurate and precise.

Moving on to the second question about the water jet from a fire extinguisher, we can use the Bernoulli's principle to calculate the required gauge air pressure. According to this principle, the sum of the pressure, kinetic energy, and potential energy per unit volume of a fluid remains constant. In this case, the water is the fluid and the air pressure is acting as the external force.

Using the equation for the Bernoulli's principle: P + 1/2ρv^2 + ρgh = constant, where P is pressure, ρ is density, v is velocity, g is acceleration due to gravity, and h is height, we can rearrange the equation to solve for the pressure.

P = constant - 1/2ρv^2 - ρgh

Substituting the given values, we get:

P = constant - 1/2(1000 kg/m^3)(30.0 m/s)^2 - (1000 kg/m^3)(9.8

 

[m2003]

a). The required length of the rod can be calculated by subtracting the starting point (0.200 cm) from the end point (1.80 cm), giving a length of 1.60 cm. This length will allow for nine fiduciary marks to be placed along the rod at equal intervals.

b). The average density of the rod can be calculated by taking the midpoint of the density range, which would be 1.06 g/cm3. This average density will ensure that the rod will float at a consistent level in water, allowing for accurate readings to be taken.

c). The marks should be equally spaced in order to accurately measure the density of a liquid. If the marks were not equally spaced, it could lead to inaccurate readings and hinder the functionality of the hydrometer. Additionally, equally spaced marks will make it easier to interpolate between the marks for more precise measurements.

-For the fire extinguisher question, the gauge air pressure required can be calculated using the Bernoulli's equation, which states that the sum of the pressure, kinetic energy, and potential energy per unit volume of a fluid is constant. We can set the initial pressure (atmospheric pressure) and velocity (30.0 m/s) to zero, and solve for the gauge pressure at the nozzle. This would give us a gauge pressure of approximately 29.4 kPa. However, this calculation does not take into account factors such as friction and air resistance, so the actual required pressure may be slightly higher.