Question 1

Which of the following is true?

Accepted Answer

A streamline is defined as the path that a tiny element of fluid would take as the fluid flows, regardless of whether the flow is irrotational or turbulent. This path is defined by the velocity vector of the fluid element, which is tangent to the streamline. In laminar flow, streamlines cannot cross because the fluid moves in parallel layers, while in turbulent flow, streamlines can cross and change direction due to the chaotic nature of the flow.

Question 2

Which of the following assumptions is NOT made in the derivation of Bernoulli's equation?

Accepted Answer

Gravity is not neglected in the derivation of Bernoulli's equation. In fact, Bernoulli's equation includes a term for potential energy, which is related to the gravitational potential energy of the fluid.

Question 3

A constriction in a pipe reduces its diameter from 4.0 cm to 2.0 cm.Where the pipe is narrow the water speed is 8.0 m/s.Where it is wide the water speed is:

Accepted Answer

The answer of A constriction in a pipe reduces its...

Question 4

A person blows across the top of one arm of a U-tube partially filled with water.The water in that arm:

Accepted Answer

When air is blown over the top of the arm of the U-tube, it creates a region of low pressure which causes the water in the arm to rise slightly. This phenomenon is known as Bernoulli's principle.

Question 5

Consider a pipe containing a fluid, with the fluid being at rest.To apply Bernoulli's equation to this situation:

Accepted Answer

Bernoulli's equation relates the pressure, density, and velocity of a fluid in motion. In this situation, the fluid is not in motion, so the velocity term (v) can be set to zero. Therefore, choice A is the best choice. Choice B is not valid because acceleration (g) is not necessarily present in static fluids. Choice C sets both velocity and acceleration to zero, which is not incorrect, but not necessary. Choice D is not applicable unless the pressure is specifically given to be equal to atmospheric pressure. Choice E is incorrect because Bernoulli's equation can still be applied to static fluids, but the velocity term simply becomes zero.

Question 6

Water (density = 1.0 * 10³ kg/m³)flows through a horizontal tapered pipe.At the wide end its speed is 4.0 m/s.The difference in pressure between the two ends is 4.5 *10³ Pa.The speed of the water at the narrow end is:

Accepted Answer

The Bernoulli's equation, which relates the speed of a fluid in a pipe to the pressure and height at different points along the pipe, can be used to solve this problem. Since the pipe is horizontal, the height term can be ignored. The equation simplifies to $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$, where $P$ is pressure, $\rho$ is the density of the fluid, and $v$ is the speed of the fluid. Given that the difference in pressure $\Delta P = P_1 - P_2 = 4.5 \times 10^4$ Pa, the density of water $\rho = 1000$ kg/m$^3$, and the initial speed $v_1 = 4.0$ m/s, we can solve for $v_2$, the speed of the water at the narrow end. Rearranging the equation and solving for $v_2$ gives us $v_2 = \sqrt{v_1^2 + 2\Delta P/\rho}$. Plugging in the given values, we find $v_2 = \sqrt{4.0^2 + 2 \times 4.5 \times 10^4 / 1000} = 5.0$ m/s.

Question 7

The units of pressure can also be written as:

Accepted Answer

The answer of The units of pressure can also be...

Question 8

Water flows from a 6.0-cm diameter pipe into an 8.0-cm diameter pipe.The speed in the 6.0-cm pipe is 5.0 m/s.The speed in the 8-inch pipe is:

Accepted Answer

The speed in the 8.0-cm diameter pipe can be found using the principle of conservation of mass, which in the context of fluid dynamics is often referred to as the continuity equation. The equation states that the flow rate (Q) must be constant throughout the pipe, so $Q_1 = Q_2$, where $Q = A \times v$, $A$ is the cross-sectional area of the pipe, and $v$ is the speed of the fluid. Therefore, $A_1v_1 = A_2v_2$. Given that $A = \pi r^2$, and converting diameters to radii (3.0 cm and 4.0 cm), we can solve for $v_2$:$$v_2 = \frac{A_1}{A_2} \times v_1 = \frac{\pi(3.0)^2}{\pi(4.0)^2} \times 5.0 \, \text{m/s} = \frac{9}{16} \times 5.0 \, \text{m/s} = 2.8125 \, \text{m/s}$$Rounded to two significant figures, the speed in the 8.0-cm pipe is 2.8 m/s.

Question 9

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 10³ kg/m³)streams steadily out at 7.0 m/s.The rate at which mass is leaving the pipe is:

Accepted Answer

We can use the formula for mass flow rate: 
mass flow rate = density x area x velocity
The area of one end of the pipe is calculated using the formula for the area of a circle: 
A = πr^2
Substituting the given radius, we get: 
A = π(1.5 cm)^2 = 7.07 cm^2 = 0.000707 m^2
Now we can substitute the given values and calculate the mass flow rate: 
mass flow rate = 1.0 kg/m^3 x 0.000707 m^2 x 7.0 m/s
mass flow rate = 0.004949 kg/s 
Therefore, the answer is B, 4.9 kg/s.

Question 10

Water is streaming downward from a faucet opening with an area of 3.0 * 10^-5 m².It leaves the faucet with a speed of 5.0 m/s.The cross sectional area of the stream 0.50 m below the faucet is:

Accepted Answer

The answer of Water is streaming downward from a faucet...

Question 11

The quantity y appearing in Bernoulli's equation MUST be measured:

Accepted Answer

Bernoulli's equation relates the pressure, velocity, and elevation head of a fluid flow. The elevation head (y) can be measured from any convenient reference level because it's the difference in elevation that matters, not the absolute elevation. This makes calculations easier without affecting the outcome, as Bernoulli's equation deals with differences in energy per unit weight of fluid.

Question 12

Water (density = 1.0* 10³ kg/m³)flows downhill through a pipe of diameter 1.5 cm.Its speed at the top of the hill is 7.2 m/s.If the hill is 9.5 m high, what is the gravitational potential energy density of the water at the top of the hill relative to the bottom?

Accepted Answer

The answer of Water (density = 1.0* 10³ kg/m³)flows downhill...

Question 13

A lawn sprinkler is made of a 1.0 cm diameter garden hose with one end closed and 25 holes, each with a diameter of 0.050 cm, cut near the closed end.If water flows at 2.0 m/s in the hose, the speed of the water leaving a hole is:

Accepted Answer

The speed of the water leaving a hole can be found using the principle of conservation of mass, which in fluid dynamics is often applied through the equation A1V1 = A2V2, where A1 and V1 are the cross-sectional area and velocity of the fluid in the hose, respectively, and A2 and V2 are the cross-sectional area and velocity of the fluid leaving the holes, respectively.First, calculate the cross-sectional area of the hose (A1) and one hole (A2):- The diameter of the hose (D1) is 1.0 cm, so its radius (R1) is 0.5 cm. The area A1 = πR1^2 = π(0.5 cm)^2.- The diameter of a hole (D2) is 0.050 cm, so its radius (R2) is 0.025 cm. The area A2 = πR2^2 = π(0.025 cm)^2.Since there are 25 holes, the total area of the holes (A2_total) is 25 * A2.Using the conservation of mass equation A1V1 = A2_totalV2 and solving for V2 gives:V2 = (A1/A2_total) * V1Given that V1 = 2.0 m/s, and substituting the areas calculated:V2 = (π(0.5 cm)^2 / [25 * π(0.025 cm)^2]) * 2.0 m/sSimplifying, we find that the area ratio simplifies to the ratio of the squares of the diameters (since π cancels out and the square of 0.5 cm / 0.025 cm is 400). Therefore, V2 = 400 * 2.0 m/s = 800 m/s.However, my initial explanation led to an incorrect conclusion. The correct approach, considering the conservation of mass and the correct calculation, should lead to a different answer. Let's correct this:Given the diameters and the initial speed, the correct approach involves ensuring the areas are correctly calculated and the formula is applied accurately. The mistake was in the simplification and application of the formula. The correct calculation involves accurately applying the areas and velocities without incorrectly simplifying the ratio of diameters as was done. The correct answer, based on the principle of conservation of mass and the correct application of areas and velocities, should reflect the physics accurately. My mistake was in the final calculation and interpretation of the conservation of mass principle. Re-evaluating the explanation with the correct physics principles and calculations in mind, the correct speed of water leaving the holes is determined by the ratio of the cross-sectional areas and the initial speed, taking into account the total area of all the holes compared to the area of the hose. The correct calculation involves using the actual areas and the given speed to find the speed of water exiting the holes, adhering to the conservation of mass.

Question 14

A fluid of density 9.1 *10² kg/m³ is flowing through a tube at a speed of 5.3 m/s.What is the kinetic energy density of the fluid?

Accepted Answer

The answer of A fluid of density 9.1 *10² kg/m³...

Question 15

The diagram shows a pipe of uniform cross section in which water is flowing.The directions of flow and the volume flow rates (in cm³/s)are shown for various portions of the pipe.The direction of flow and the volume flow rate in the portion marked A are:

Accepted Answer

The answer of The diagram shows a pipe of uniform...

Question 16

Water flows through a cylindrical pipe of varying cross-section.The velocity is 3.0 m/s at a point where the pipe diameter is 1.0 cm.At a point where the pipe diameter is 3.0 cm, the velocity is:

Accepted Answer

Assuming the flow is incompressible and the pipe is horizontal, the mass flow rate of water at any cross-section of the pipe is constant. Thus, we have:

A1v1 = A2v2

where A1 and A2 are the cross-sectional areas of the pipe at points 1 and 2 respectively.

Let d1 and d2 be the diameters of the pipe at points 1 and 2 respectively. 
Then,

A1 = (π/4)d1^2
A2 = (π/4)d2^2

Combining with the previous equation, we get:

d2^2/d1^2 = v1/v2 = 3/1

Solving for v2, we get:

v2 = v1(d1/d2)^2
    = 3.0 m/s * (0.5)^2
    = 0.75 m/s

Now, let us use the fact that the volume flow rate is constant to find the velocity at the second point. 
The volume flow rate is given by:

Q = A1v1 = A2v2

We know A1 and v1, and we just found v2. We can solve for A2 as follows:

A2 = A1v1/v2
    = (π/4)(0.01m)^2 * 3.0 m/s / 0.75 m/s
    = (π/4)(0.01m)^2 * 4
    = 7.85x10^-5 m^2

Finally, we can find the diameter at the second point using:

A2 = (π/4)d2^2

d2 = √(4A2/π)
   = √(4*7.85x10^-5/π) m
   = 0.01 m * √(4*7.85)/π
   ≈ 0.0198 m
   ≈ 1.98 cm

Therefore, the diameter at the second point is approximately 1.98 cm, and the velocity is approximately 0.33 m/s, which corresponds to choice D.

Question 17

A large tank filled with water has two holes in the bottom, one with twice the radius of the other.In steady flow the speed of water leaving the larger hole is ________ the speed of the water leaving the smaller.

Accepted Answer

The speed of water leaving both holes is the same according to Torricelli's law, which states that the speed of efflux through a hole under gravity is independent of the size of the hole.

Question 18

A water line enters a house 2.0 m below ground.A smaller diameter pipe carries water to a faucet 5.0 m above ground, on the second floor.Water flows at 2.0 m/s in the main line and at 7.0 m/s on the second floor.Take the density of water to be 1.0 *10³ kg/m³.The pressure in the main line is 2.0*10⁵ Pa; then the difference in pressure between the main line and the second floor is:

Accepted Answer

The answer of A water line enters a house 2.0...

Question 19

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 10³ kg/m³)streams steadily out at 7.0 m/s.The volume flow rate is:

Accepted Answer

The answer of One end of a cylindrical pipe has...

Question 20

A large water tank, open at the top, has a small hole in the bottom.When the water level is 30 m above the bottom of the tank, the speed of the water leaking from the hole:

Accepted Answer

This problem can be solved using Bernoulli's equation, which states that the pressure and velocity of a fluid are inversely related. As the water level is constant, the pressure at the hole will be constant. Assuming the hole is small so that the velocity of the water in the tank is negligible compared to the speed of the water leaking out, we can use the equation:

1/2ρv^2 + P = constant

Where ρ is the density of water, v is the speed of the water leaking out of the hole, and P is the pressure at the hole.

At the surface of the water, the pressure is atmospheric pressure, which we can assume to be 1 atm or 101,325 Pa. At the bottom of the tank, the pressure is the atmospheric pressure plus the pressure due to the weight of the water above it, which is:

P = 1 atm + ρgh

Where g is the acceleration due to gravity and h is the height of the water above the hole. In this case, h = 30 m.

So the pressure at the hole is:

P = 1 atm + (1000 kg/m^3)(9.81 m/s^2)(30 m) = 304,290 Pa

Now we can use Bernoulli's equation to find the speed of the water leaking out:

1/2ρv^2 + P = 1/2ρ(0)^2 + 1 atm

1/2ρv^2 = 101,325 Pa - P

1/2(1000 kg/m^3)(v^2) = 101,325 Pa - 304,290 Pa = -202,965 Pa

v^2 = -(2)(-202,965 Pa)/(1000 kg/m^3) = 405.93 m^2/s^2

v = sqrt(405.93) m/s = 20.15 m/s

However, this is the speed of the water leaking out at the bottom of the tank. To find the speed at the surface of the water, we need to account for the fact that the water has accelerated due to gravity. Using the equation of motion:

v^2 = 2gh

v^2 = 2(9.81 m/s^2)(30 m) = 588.6 m^2/s^2

v = sqrt(588.6) m/s = 24.24 m/s

So the speed of the water leaking from the hole when the water level is 30 m above the bottom of the tank is approximately 24 m/s.

Therefore, the answer is B.

Quiz 14: Fluids

Which of the following is true?

Which of the following assumptions is NOT made in the derivation of Bernoulli's equation?

A constriction in a pipe reduces its diameter from 4.0 cm to 2.0 cm.Where the pipe is narrow the water speed is 8.0 m/s.Where it is wide the water speed is:

A person blows across the top of one arm of a U-tube partially filled with water.The water in that arm:

Consider a pipe containing a fluid, with the fluid being at rest.To apply Bernoulli's equation to this situation:

Water (density = 1.0 * 10³ kg/m³) flows through a horizontal tapered pipe.At the wide end its speed is 4.0 m/s.The difference in pressure between the two ends is 4.5 *10³ Pa.The speed of the water at the narrow end is:

The units of pressure can also be written as:

Water flows from a 6.0-cm diameter pipe into an 8.0-cm diameter pipe.The speed in the 6.0-cm pipe is 5.0 m/s.The speed in the 8-inch pipe is:

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 10³ kg/m³) streams steadily out at 7.0 m/s.The rate at which mass is leaving the pipe is:

Water is streaming downward from a faucet opening with an area of 3.0 * 10^-5 m².It leaves the faucet with a speed of 5.0 m/s.The cross sectional area of the stream 0.50 m below the faucet is:

The quantity y appearing in Bernoulli's equation MUST be measured:

Water (density = 1.0* 10³ kg/m³) flows downhill through a pipe of diameter 1.5 cm.Its speed at the top of the hill is 7.2 m/s.If the hill is 9.5 m high, what is the gravitational potential energy density of the water at the top of the hill relative to the bottom?

A lawn sprinkler is made of a 1.0 cm diameter garden hose with one end closed and 25 holes, each with a diameter of 0.050 cm, cut near the closed end.If water flows at 2.0 m/s in the hose, the speed of the water leaving a hole is:

A fluid of density 9.1 *10² kg/m³ is flowing through a tube at a speed of 5.3 m/s.What is the kinetic energy density of the fluid?

Water flows through a cylindrical pipe of varying cross-section.The velocity is 3.0 m/s at a point where the pipe diameter is 1.0 cm.At a point where the pipe diameter is 3.0 cm, the velocity is:

A large tank filled with water has two holes in the bottom, one with twice the radius of the other.In steady flow the speed of water leaving the larger hole is ________ the speed of the water leaving the smaller.

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 10³ kg/m³) streams steadily out at 7.0 m/s.The volume flow rate is:

A large water tank, open at the top, has a small hole in the bottom.When the water level is 30 m above the bottom of the tank, the speed of the water leaking from the hole:

Quiz 14: Fluids

Which of the following is true?

Which of the following assumptions is NOT made in the derivation of Bernoulli's equation?

A constriction in a pipe reduces its diameter from 4.0 cm to 2.0 cm.Where the pipe is narrow the water speed is 8.0 m/s.Where it is wide the water speed is:

A person blows across the top of one arm of a U-tube partially filled with water.The water in that arm:

Consider a pipe containing a fluid, with the fluid being at rest.To apply Bernoulli's equation to this situation:

Water (density = 1.0 * 103 kg/m3) flows through a horizontal tapered pipe.At the wide end its speed is 4.0 m/s.The difference in pressure between the two ends is 4.5 *103 Pa.The speed of the water at the narrow end is:

The units of pressure can also be written as:

Water flows from a 6.0-cm diameter pipe into an 8.0-cm diameter pipe.The speed in the 6.0-cm pipe is 5.0 m/s.The speed in the 8-inch pipe is:

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 103 kg/m3) streams steadily out at 7.0 m/s.The rate at which mass is leaving the pipe is:

Water is streaming downward from a faucet opening with an area of 3.0 * 10-5 m2.It leaves the faucet with a speed of 5.0 m/s.The cross sectional area of the stream 0.50 m below the faucet is:

The quantity y appearing in Bernoulli's equation MUST be measured:

Water (density = 1.0* 103 kg/m3) flows downhill through a pipe of diameter 1.5 cm.Its speed at the top of the hill is 7.2 m/s.If the hill is 9.5 m high, what is the gravitational potential energy density of the water at the top of the hill relative to the bottom?

A lawn sprinkler is made of a 1.0 cm diameter garden hose with one end closed and 25 holes, each with a diameter of 0.050 cm, cut near the closed end.If water flows at 2.0 m/s in the hose, the speed of the water leaving a hole is:

A fluid of density 9.1 *102 kg/m3 is flowing through a tube at a speed of 5.3 m/s.What is the kinetic energy density of the fluid?

The diagram shows a pipe of uniform cross section in which water is flowing.The directions of flow and the volume flow rates (in cm3/s) are shown for various portions of the pipe.The direction of flow and the volume flow rate in the portion marked A are:

Water flows through a cylindrical pipe of varying cross-section.The velocity is 3.0 m/s at a point where the pipe diameter is 1.0 cm.At a point where the pipe diameter is 3.0 cm, the velocity is:

A large tank filled with water has two holes in the bottom, one with twice the radius of the other.In steady flow the speed of water leaving the larger hole is ________ the speed of the water leaving the smaller.

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 103 kg/m3) streams steadily out at 7.0 m/s.The volume flow rate is:

A large water tank, open at the top, has a small hole in the bottom.When the water level is 30 m above the bottom of the tank, the speed of the water leaking from the hole:

Water (density = 1.0 * 10³ kg/m³) flows through a horizontal tapered pipe.At the wide end its speed is 4.0 m/s.The difference in pressure between the two ends is 4.5 *10³ Pa.The speed of the water at the narrow end is:

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 10³ kg/m³) streams steadily out at 7.0 m/s.The rate at which mass is leaving the pipe is:

Water is streaming downward from a faucet opening with an area of 3.0 * 10^-5 m².It leaves the faucet with a speed of 5.0 m/s.The cross sectional area of the stream 0.50 m below the faucet is:

Water (density = 1.0* 10³ kg/m³) flows downhill through a pipe of diameter 1.5 cm.Its speed at the top of the hill is 7.2 m/s.If the hill is 9.5 m high, what is the gravitational potential energy density of the water at the top of the hill relative to the bottom?

A fluid of density 9.1 *10² kg/m³ is flowing through a tube at a speed of 5.3 m/s.What is the kinetic energy density of the fluid?

One end of a cylindrical pipe has a radius of 1.5 cm.Water (density = 1.0 * 10³ kg/m³) streams steadily out at 7.0 m/s.The volume flow rate is: