Question 1

Two cylinders, one made of lead (density 11,340 kg/m³) and one of copper (density 8,940 kg/m³), have the same radius and height. They are thrown into a pool of water (density 1,000 kg/m³) and sink to the bottom. As they are sinking, describe the buoyant forces on the cylinders.

Accepted Answer

The buoyant force depends on the volume of fluid displaced by an object, not its material. Since both cylinders have the same size and shape, they displace the same volume of water and experience the same buoyant force.

Question 2

A sphere is made up of two layers: The first is lead (density 11,340 kg/m³) and is of radius R. The second is tin, (7,310 kg/m³) and is concentric with the first, from radius R to radius 2R. The sphere is placed in a pool of mercury (density 13,600 kg/m³). How much of the volume of the sphere is below the surface?

Accepted Answer

Let's assume the radius of the sphere is 1, so the volume of the sphere is:

V_sphere = (4/3)π(1)^3 = 4/3π

The volume of the lead layer is:

V_lead = (4/3)π(1)^3 - (4/3)π(1/2)^3 = 7/12π

The volume of the tin layer is:

V_tin = (4/3)π(2)^3 - (4/3)π(1)^3 = 7/3π

The total mass of the sphere can be calculated as:

m = ρ_lead*V_lead + ρ_tin*V_tin

where ρ_lead and ρ_tin are the densities of lead and tin respectively.

Substituting values, we get:

m = 11340*(7/12π) + 7310*(7/3π) = 27238.69 kg

The weight of the sphere can be calculated as:

w = m*g

where g is the acceleration due to gravity (9.8 m/s^2).

Substituting values, we get:

w = 27238.69*9.8 = 267101.962 N

The buoyant force on the sphere can be calculated as:

F_buoyant = ρ_mercury*V_sphere*g

where ρ_mercury is the density of mercury.

Substituting values, we get:

F_buoyant = 13600*(4/3π)*9.8 = 66794.74 N

The net force on the sphere is:

F_net = F_buoyant - w

Substituting values, we get:

F_net = 66794.74 - 267101.962 = -200307.22 N

Since the net force is downward, the sphere will sink until it displaces enough mercury to equal its weight. Let h be the height of the sphere below the surface of the mercury. Then the volume of the displaced mercury can be written as:

V_displaced = πh^2(3R - h)/3

The weight of the displaced mercury is:

w_displaced = V_displaced*ρ_mercury*g

Since the sphere is in equilibrium, the weight of the displaced mercury must be equal to the weight of the sphere:

w_displaced = w = 267101.962 N

Substituting values, we get:

V_displaced*13600*9.8 = 267101.962

Solving for h, we get:

h = 0.4257 m

Thus, the amount of the volume of the sphere below the surface is:

(4/3)π(1)^3 - π(0.4257)^2(3 - 0.4257)/3 = 0.575 or 57.5% approximately.

Therefore, the answer is B.

Question 3

Two spheres are made of wood-the first is of a variety of wood whose density is equal to that of water, while the second is of a variety whose density is greater than that of water. These spheres have the same volume, and the first remains wherever it is placed in the water (so it is effectively suspended in mid-water). The second rests on the bottom, being more dense than water. Compare the buoyant forces on the two spheres.

Accepted Answer

The buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. Since both spheres have the same volume and are fully submerged (one in the middle of the water and the other at the bottom), they displace the same amount of water and thus experience the same buoyant force, regardless of their densities.

Question 4

Two objects of different shapes, sizes, and materials, are floating in water (density 1000 kg/m³). Object A floats with 60% of its volume submerged, while object B floats with 50% of its volume submerged. What can be concluded about the densities ρ_A and ρ_B of the two objects?

Accepted Answer

The density of an object floating in a fluid is proportional to the fraction of its volume submerged. Since object A has a greater fraction of its volume submerged (60%) compared to object B (50%), object A has a higher density than object B.

Question 5

A cylindrical tank (height h, radius r) is full to the brim of water and its top is open to the outside air. What expression describes the speed of fluid flowing out of a hole that is opened up at height h′ above the bottom of the tank?

Accepted Answer

The answer of A cylindrical tank (height h, radius r)...

Question 6

A U-shaped manometer, filled with a liquid of unknown density, is connected on one side to a vessel containing gas at gauge pressure 10.0 kPa, and the other side is open to the atmosphere (pressure 101.3 kPa). The level of the liquid in the side connected to the gas vessel is lower than on the open side. The difference in height between the two sides is 10 cm. What is the density of the liquid in the manometer?

Accepted Answer

The answer of A U-shaped manometer, filled with a liquid...

Question 7

A horizontal section of an artery narrows from a diameter of 8.0 mm to a diameter of 6.0 mm. What is the pressure difference between the wider section and the narrower? That is, what is P_wide − P_narrow? The blood flows in the wider section at 48 cm/s. Take blood plasma to have a density of 1,025 kg/m³ and ignore viscosity.

Accepted Answer

The answer of A horizontal section of an artery narrows...

Question 8

You would like to use a simple water-filled barometer (an inverted tube in a container of water) to measure the atmospheric pressure on a day when the pressure is 102.5 kPa. What would be the minimum height required for such a barometer to work? The density of water is 1000 kg/m³.

Accepted Answer

The atmospheric pressure can be equated to the hydrostatic pressure of the water column in the barometer using the formula $P = \rho gh$, where $P$ is the pressure (102.5 kPa or 102500 Pa), $\rho$ is the density of water (1000 kg/m^3), and $g$ is the acceleration due to gravity (approximately 9.81 m/s^2). Solving for $h$ gives $h = \frac{P}{\rho g} = \frac{102500}{1000 \times 9.81} \approx 10.45$ meters, which rounds to 10.5 meters, making it the minimum height required for the barometer to work.

Quiz 9: Fluids

Two cylinders, one made of lead (density 11,340 kg/m³) and one of copper (density 8,940 kg/m³) , have the same radius and height. They are thrown into a pool of water (density 1,000 kg/m³) and sink to the bottom. As they are sinking, describe the buoyant forces on the cylinders.

Two objects of different shapes, sizes, and materials, are floating in water (density 1000 kg/m³) . Object A floats with 60% of its volume submerged, while object B floats with 50% of its volume submerged. What can be concluded about the densities ρ_A and ρ_B of the two objects?

A cylindrical tank (height h, radius r) is full to the brim of water and its top is open to the outside air. What expression describes the speed of fluid flowing out of a hole that is opened up at height h′ above the bottom of the tank?

You would like to use a simple water-filled barometer (an inverted tube in a container of water) to measure the atmospheric pressure on a day when the pressure is 102.5 kPa. What would be the minimum height required for such a barometer to work? The density of water is 1000 kg/m³.

Quiz 9: Fluids

Two cylinders, one made of lead (density 11,340 kg/m3) and one of copper (density 8,940 kg/m3) , have the same radius and height. They are thrown into a pool of water (density 1,000 kg/m3) and sink to the bottom. As they are sinking, describe the buoyant forces on the cylinders.

Two objects of different shapes, sizes, and materials, are floating in water (density 1000 kg/m3) . Object A floats with 60% of its volume submerged, while object B floats with 50% of its volume submerged. What can be concluded about the densities ρA and ρB of the two objects?

A cylindrical tank (height h, radius r) is full to the brim of water and its top is open to the outside air. What expression describes the speed of fluid flowing out of a hole that is opened up at height h′ above the bottom of the tank?

You would like to use a simple water-filled barometer (an inverted tube in a container of water) to measure the atmospheric pressure on a day when the pressure is 102.5 kPa. What would be the minimum height required for such a barometer to work? The density of water is 1000 kg/m3.

Two cylinders, one made of lead (density 11,340 kg/m³) and one of copper (density 8,940 kg/m³) , have the same radius and height. They are thrown into a pool of water (density 1,000 kg/m³) and sink to the bottom. As they are sinking, describe the buoyant forces on the cylinders.

Two objects of different shapes, sizes, and materials, are floating in water (density 1000 kg/m³) . Object A floats with 60% of its volume submerged, while object B floats with 50% of its volume submerged. What can be concluded about the densities ρ_A and ρ_B of the two objects?

You would like to use a simple water-filled barometer (an inverted tube in a container of water) to measure the atmospheric pressure on a day when the pressure is 102.5 kPa. What would be the minimum height required for such a barometer to work? The density of water is 1000 kg/m³.