Question 1

All people come very close to being able to float in water. What therefore is the volume (in cubic meters) of a 50-kg woman?

Accepted Answer

According to Archimedes' principle, the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. Since humans are nearly able to float in water, it means that the weight of the water displaced by a 50-kg woman is approximately equal to her weight. The density of water is 1000 kg/m³, so the volume of water displaced by a 50-kg woman is 50/1000 = 0.05 m³, which is the same as the volume of the woman herself. Therefore, the volume of the woman is 0.050 m³.

Question 2

Some species of whales can dive to depths of one kilometer. What is the total pressure they experience at this depth? (ρ_sea = 1 020 kg/m³ and 1.01 × 10⁵ N/m² = 1 ATM.)

Accepted Answer

C

Question 3

A supertanker filled with oil has a total mass of 6.1 × 10⁸ kg. If the dimensions of the ship are those of a rectangular box 300 meters long, 80 meters wide, and 40 meters high, determine how far the bottom of the ship is below sea level. (ρ_sea = 1 020 kg/m³.)

Accepted Answer

D

Question 4

What is the total mass of the Earth's atmosphere? The radius of the Earth is 6.4 × 10⁶ m, and 1 ATM = 1.01 × 10⁵ N/m².

Accepted Answer

The total mass of the Earth's atmosphere can be calculated using the formula:

Mass = Pressure x Volume / (Ideal Gas Constant x Temperature)

First, we need to calculate the total volume of the Earth's atmosphere. We know that the Earth's radius is 6.4 x 10^6 m, so the total volume of the atmosphere can be calculated as:

Volume = (4/3) x π x (6.4 x 10^6)^3 = 1.086 x 10^21 m^3

Next, we can calculate the total pressure of the atmosphere using the given information that 1 ATM = 1.01 x 10^5 N/m^2:

Pressure = 1 ATM = 1.01 x 10^5 N/m^2

Finally, we need to assume a temperature and gas constant. The temperature of the atmosphere varies widely depending on altitude and other factors, but we can assume an average temperature of around 15°C or 288 K. The gas constant for air is 287 J/(kg·K). Using these values, we can calculate the total mass of the Earth's atmosphere:

Mass = (1.01 x 10^5 N/m^2) x (1.086 x 10^21 m^3) / (287 J/(kg·K) x 288 K)

Mass = 5.3 x 10^18 kg

Therefore, the best choice is C, 5.3 × 10^18 kg.

Question 5

Find the pressure in atmospheres in the water at the base of Dworshak Dam if the water in the reservoir is 200 meters deep. (1.01 × 10⁵ N/m² = 1 ATM.)

Accepted Answer

The pressure at a certain depth in a fluid is given by the formula P = ρgh, where ρ is the density of the fluid (for water, ρ = 1000 kg/m³), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the depth of the fluid. Substituting the given values, P = 1000 kg/m³ * 9.8 m/s² * 200 m = 1.96 × 10^6 N/m². Since 1 ATM = 1.01 × 10^5 N/m², the pressure in atmospheres is 1.96 × 10^6 N/m² / 1.01 × 10^5 N/m² = 19.4, which rounds to approximately 20.4 ATM.

Question 6

Water is flowing at 4.0 m/s in a circular pipe. If the diameter of the pipe decreases to 1/2 its former value, what is the velocity of the water downstream?

Accepted Answer

According to the principle of continuity for incompressible fluids, the flow rate must remain constant. If the diameter is halved, the cross-sectional area is reduced to one-fourth. Therefore, the velocity must increase by a factor of four to maintain the same flow rate, resulting in a velocity of 16 m/s.

Question 7

A stonecutter's chisel has an edge area of 0.7 cm². If the chisel is struck with a force of 42 N, what is the pressure exerted on the stone?

Accepted Answer

The answer of A stonecutter's chisel has an edge area...

Question 8

A hydraulic lift raises a 2 000-kg automobile when a 500-N force is applied to the smaller piston. If the smaller piston has an area of 10 cm², what is the cross-sectional area of the larger piston?

Accepted Answer

We can use the formula: 
force on smaller piston/area of smaller piston = force on larger piston/area of larger piston

Substituting the given values: 
500N / (10 cm^2) = force on larger piston / area of larger piston

Solving for area of larger piston: 
Area of larger piston = (500N / 10 cm^2) x (the given mass of the car / acceleration due to gravity)
Area of larger piston = (50 N/cm^2) x (2000 kg / 9.8 m/s^2)
Area of larger piston = 1000 cm^2

Therefore, the cross-sectional area of the larger piston is 1000 cm^2, which is option D.

Question 9

Find the average density of a white dwarf star if it has a mass equal to that of the sun (2.0 × 10³⁰ kg) and a radius equal to that of the Earth (6.4 × 10⁶ m).

Accepted Answer

The formula for density is:

Density = mass/volume

For a white dwarf star, since it is roughly spherical in shape, the volume can be calculated using the formula for the volume of a sphere:

Volume = 4/3π(r^3)

Where r is the radius of the star.

Substituting the given values:

Volume = 4/3π(6.4 × 10^6)^3 = 1.09 × 10^20 m^3

Now, we can find the density:

Density = mass/volume

Density = (2.0 × 10^30)/(1.09 × 10^20) = 1.8 × 10^19 kg/m^3

Therefore, the closest answer choice is C, 1.8 × 10^19 kg/m^3.

Question 10

To the nearest percent, what fraction of an iceberg is submerged? (ρ_ice = 917 kg/m³, ρ_sea = 1.03 × 10³ kg/m³.)

Accepted Answer

The answer of To the nearest percent, what fraction of...

Question 11

A hole is punched in a full milk carton, 10 cm below the top. What is the initial velocity of outflow?

Accepted Answer

The initial velocity of outflow can be calculated using Torricelli's law, which states that the speed of efflux, v, from a hole at a depth h below the surface of a liquid is given by $v = \sqrt{2gh}$, where g is the acceleration due to gravity (approximately $9.8 m/s^2$). Plugging in the given depth of 10 cm (0.1 m) gives $v = \sqrt{2 \times 9.8 m/s^2 \times 0.1 m} = \sqrt{1.96 m^2/s^2} = 1.4 m/s$.

Question 12

A blimp is filled with 200 m³ of helium. How much mass can the balloon lift? The density of helium is one-seventh that of air, and the density of air is 1/800 that of water. Consider the mass of the balloon to be negligible.

Accepted Answer

First, we need to find the volume of the blimp using the given mass of helium and the density of helium: Density of helium = 1/7 * Density of air Density of air = 1/800 * Density of water Density of helium = 1/7 * (1/800 * Density of water) Density of helium = 1/5600 * Density of water Mass of helium = Density of helium * Volume of blimp Volume of blimp = Mass of helium / Density of helium Volume of blimp = 200 m³ / (1/5600 * Density of water) Volume of blimp = 200 m³ * 5600 / Density of water Volume of blimp = 1,120,000 m³ / Density of water Now, we need to find the mass of air that the blimp displaces to determine the lift: Mass of air = Density of air * Volume of blimp Mass of air = (1/800 * Density of water) * (1,120,000 m³ / Density of water) Mass of air = 1400 kg Finally, we can find the lift by subtracting the mass of air from the mass of helium: Lift = Mass of helium - Mass of air Lift = 200 m³ - 1400 kg Lift = 214 kg Therefore, the best choice is B, which is the closest to the calculated lift of 214 kg.

Question 13

A fountain sends water to a height of 100 meters. What must be the pressurization (above atmospheric) of the water system? 1 ATM = 1.01 × 10⁵ N/m².

Accepted Answer

The pressure at the base of the fountain must be sufficient to overcome the force of gravity and raise the water to a height of 100 meters. The pressure required is:P = ρgh = (1000 kg/m³)(9.8 m/s²)(100 m) = 9.8 × 10^5 N/m²Converting to atmospheres:P = (9.8 × 10^5 N/m²)/(1.01 × 10^5 N/m²) = 9.7 ATMTherefore, the pressurization (above atmospheric) of the water system must be 9.7 ATM.

Question 14

Water pressurized to 3.5 × 10⁵ Pa is flowing at 5.0 m/s in a horizontal pipe which contracts to 1/3 its former area. What are the pressure and velocity of the water after the contraction?

Accepted Answer

The conservation of mass in fluid dynamics, specifically the continuity equation, states that the product of cross-sectional area and velocity of flow is constant for an incompressible fluid. If the area contracts to 1/3 of its original size, the velocity must triple to maintain this product, resulting in a velocity of 15 m/s. The Bernoulli's equation, which conserves energy in a flowing fluid, implies that a change in velocity will result in a change in pressure. However, without specific calculations or further information on the initial conditions and the exact relationship between pressure and velocity in this context, determining the exact new pressure is complex. The provided answer assumes an understanding of these principles and the effects of the area contraction on velocity, but it simplifies the outcome regarding pressure.

Question 15

Find the average density of a red giant star with a mass of 20 × 10³⁰ kg (approximately 10 solar masses) and a radius of 150 × 10⁹ m (equal to the Earth's distance from the sun).

Accepted Answer

The average density is calculated using the formula: density = mass/volume. The volume of a sphere is (4/3)πr³. Substituting the given mass and radius, the density is approximately 1.41 × 10−3 kg/m³.

Question 16

When water freezes, it expands about nine percent. What would be the pressure increase inside your automobile engine block if the water in there froze? The bulk modulus of ice is 2.0 × 10⁹ N/m², and 1 ATM = 1.01 × 10⁵ N/m².

Accepted Answer

The pressure increase can be calculated using the equation:

ΔP = B(ΔV/V)

where ΔP is the pressure increase, B is the bulk modulus of ice, ΔV is the change in volume due to freezing (which can be calculated as 0.09 times the initial volume of the water), and V is the initial volume of the water.

Plugging in the values, we get:

ΔP = (2.0 × 10^9 N/m^2)(0.09V/V)
ΔP = (1.8 × 10^8 N/m^2)V

To convert this to atmospheres, we divide by the conversion factor:

ΔP = (1.8 × 10^8 N/m^2)(1 atm/1.01 × 10^5 N/m^2)
ΔP = 1800 atm

Therefore, the pressure increase inside the engine block would be 1800 atm, which corresponds to option D.

Question 17

The pressure inside a commercial airliner is maintained at 1 ATM (1.01 × 10⁵ N/m²). What is the outward force exerted on a 1.0 m × 2.0 m cabin door if the outside pressure (at 10 km height) is 0.3 ATM?

Accepted Answer

The answer of The pressure inside a commercial airliner is...

Question 18

What is the net force inward acting on a spherical bathysphere of diameter 2.00 m at an ocean depth of 1 000 m? (The pressure inside the bathysphere is, hopefully, 1 ATM.) ρ_{(sea water)} = 1.02 × 10³ kg/m³.

Accepted Answer

The net force is calculated using the pressure difference between the inside and outside of the bathysphere. The pressure at 1000 m depth is ρgh = (1.02 × 10³ kg/m³)(9.81 m/s²)(1000 m) = 1.00 × 10⁷ Pa. The area of the sphere is 4πr² = 4π(1 m)² = 12.57 m². The net force is pressure difference times area: (1.00 × 10⁷ Pa - 1.01 × 10⁵ Pa) × 12.57 m² ≈ 1.26 × 10⁸ N.

Question 19

The water level in a reservoir is maintained at a constant level. What is the exit velocity in an outlet pipe 3.0 m below the water surface?

Accepted Answer

The exit velocity can be found using Torricelli's theorem, which states that the speed of efflux of a fluid under gravity through an orifice is given by $ v = \sqrt{2gh} $. Here, $ g = 9.8 \, \text{m/s}^2 $ and $ h = 3.0 \, \text{m} $. Thus, $ v = \sqrt{2 \times 9.8 \times 3.0} = \sqrt{58.8} \approx 7.7 \, \text{m/s} $.

Question 20

Determine the minimum area of a flat ice floe 1.0 meter thick if it is to support a 2 000-kg car above seawater. (ρ_ice = 920 kg/m³, ρ_sea = 1 020 kg/m³.)

Accepted Answer

To support the car and the ice floe, the weight of the displaced seawater must equal the total weight of the car and the ice. The volume of seawater displaced is equal to the volume of the ice floe submerged. Let $ V $ be the volume of the ice floe, $ A $ its area, and $ h = 1.0 \, \text{m} $ its thickness. Thus, $ V = A \times h $.The mass of the car is $ 2000 \, \text{kg} $, and the density of ice ($ \rho_{\text{ice}} $) is $ 920 \, \text{kg/m}^3 $. The mass of the ice floe is $ \rho_{\text{ice}} \times V = 920 \times A \times 1.0 $.The total weight to be supported is the weight of the car plus the weight of the ice, which is $ 2000 \, \text{kg} + 920A \, \text{kg} $.The density of seawater ($ \rho_{\text{sea}} $) is $ 1020 \, \text{kg/m}^3 $. The weight of the displaced seawater must equal the total weight, so:$$ \rho_{\text{sea}} \times V_{\text{displaced}} = 2000 + 920A $$Since $ V_{\text{displaced}} = A \times h $ (the volume of the ice submerged equals the volume of the ice because it needs to be completely submerged to support the car and itself), and $ h = 1.0 \, \text{m} $, we have:$$ 1020 \times A = 2000 + 920A $$Solving for $ A $:$$ 100A = 2000 $$$$ A = 20 \, \text{m}^2 $$Therefore, the minimum area of the ice floe needed to support the car above seawater is $ 20 \, \text{m}^2 $.

Quiz 14: Fluid Mechanics

All people come very close to being able to float in water. What therefore is the volume (in cubic meters) of a 50-kg woman?

Some species of whales can dive to depths of one kilometer. What is the total pressure they experience at this depth? (ρ_sea = 1 020 kg/m³ and 1.01 × 10⁵ N/m² = 1 ATM.)

A supertanker filled with oil has a total mass of 6.1 × 10⁸ kg. If the dimensions of the ship are those of a rectangular box 300 meters long, 80 meters wide, and 40 meters high, determine how far the bottom of the ship is below sea level. (ρ_sea = 1 020 kg/m³.)

What is the total mass of the Earth's atmosphere? The radius of the Earth is 6.4 × 10⁶ m, and 1 ATM = 1.01 × 10⁵ N/m².

Find the pressure in atmospheres in the water at the base of Dworshak Dam if the water in the reservoir is 200 meters deep. (1.01 × 10⁵ N/m² = 1 ATM.)

Water is flowing at 4.0 m/s in a circular pipe. If the diameter of the pipe decreases to 1/2 its former value, what is the velocity of the water downstream?

A stonecutter's chisel has an edge area of 0.7 cm². If the chisel is struck with a force of 42 N, what is the pressure exerted on the stone?

A hydraulic lift raises a 2 000-kg automobile when a 500-N force is applied to the smaller piston. If the smaller piston has an area of 10 cm², what is the cross-sectional area of the larger piston?

Find the average density of a white dwarf star if it has a mass equal to that of the sun (2.0 × 10³⁰ kg) and a radius equal to that of the Earth (6.4 × 10⁶ m) .

To the nearest percent, what fraction of an iceberg is submerged? (ρ_ice = 917 kg/m³, ρ_sea = 1.03 × 10³ kg/m³.)

A hole is punched in a full milk carton, 10 cm below the top. What is the initial velocity of outflow?

A blimp is filled with 200 m³ of helium. How much mass can the balloon lift? The density of helium is one-seventh that of air, and the density of air is 1/800 that of water. Consider the mass of the balloon to be negligible.

A fountain sends water to a height of 100 meters. What must be the pressurization (above atmospheric) of the water system? 1 ATM = 1.01 × 10⁵ N/m².

Water pressurized to 3.5 × 10⁵ Pa is flowing at 5.0 m/s in a horizontal pipe which contracts to 1/3 its former area. What are the pressure and velocity of the water after the contraction?

Find the average density of a red giant star with a mass of 20 × 10³⁰ kg (approximately 10 solar masses) and a radius of 150 × 10⁹ m (equal to the Earth's distance from the sun) .

When water freezes, it expands about nine percent. What would be the pressure increase inside your automobile engine block if the water in there froze? The bulk modulus of ice is 2.0 × 10⁹ N/m², and 1 ATM = 1.01 × 10⁵ N/m².

The pressure inside a commercial airliner is maintained at 1 ATM (1.01 × 10⁵ N/m²) . What is the outward force exerted on a 1.0 m × 2.0 m cabin door if the outside pressure (at 10 km height) is 0.3 ATM?

What is the net force inward acting on a spherical bathysphere of diameter 2.00 m at an ocean depth of 1 000 m? (The pressure inside the bathysphere is, hopefully, 1 ATM.) ρ_{(sea water)} = 1.02 × 10³ kg/m³.

The water level in a reservoir is maintained at a constant level. What is the exit velocity in an outlet pipe 3.0 m below the water surface?

Determine the minimum area of a flat ice floe 1.0 meter thick if it is to support a 2 000-kg car above seawater. (ρ_ice = 920 kg/m³, ρ_sea = 1 020 kg/m³.)

Quiz 14: Fluid Mechanics

All people come very close to being able to float in water. What therefore is the volume (in cubic meters) of a 50-kg woman?

Some species of whales can dive to depths of one kilometer. What is the total pressure they experience at this depth? (ρsea = 1 020 kg/m3 and 1.01 × 105 N/m2 = 1 ATM.)

A supertanker filled with oil has a total mass of 6.1 × 108 kg. If the dimensions of the ship are those of a rectangular box 300 meters long, 80 meters wide, and 40 meters high, determine how far the bottom of the ship is below sea level. (ρsea = 1 020 kg/m3.)

What is the total mass of the Earth's atmosphere? The radius of the Earth is 6.4 × 106 m, and 1 ATM = 1.01 × 105 N/m2.

Find the pressure in atmospheres in the water at the base of Dworshak Dam if the water in the reservoir is 200 meters deep. (1.01 × 105 N/m2 = 1 ATM.)

Water is flowing at 4.0 m/s in a circular pipe. If the diameter of the pipe decreases to 1/2 its former value, what is the velocity of the water downstream?

A stonecutter's chisel has an edge area of 0.7 cm2. If the chisel is struck with a force of 42 N, what is the pressure exerted on the stone?

A hydraulic lift raises a 2 000-kg automobile when a 500-N force is applied to the smaller piston. If the smaller piston has an area of 10 cm2, what is the cross-sectional area of the larger piston?

Find the average density of a white dwarf star if it has a mass equal to that of the sun (2.0 × 1030 kg) and a radius equal to that of the Earth (6.4 × 106 m) .

To the nearest percent, what fraction of an iceberg is submerged? (ρice = 917 kg/m3, ρsea = 1.03 × 103 kg/m3.)

A hole is punched in a full milk carton, 10 cm below the top. What is the initial velocity of outflow?

A blimp is filled with 200 m3 of helium. How much mass can the balloon lift? The density of helium is one-seventh that of air, and the density of air is 1/800 that of water. Consider the mass of the balloon to be negligible.

A fountain sends water to a height of 100 meters. What must be the pressurization (above atmospheric) of the water system? 1 ATM = 1.01 × 105 N/m2.

Water pressurized to 3.5 × 105 Pa is flowing at 5.0 m/s in a horizontal pipe which contracts to 1/3 its former area. What are the pressure and velocity of the water after the contraction?

Find the average density of a red giant star with a mass of 20 × 1030 kg (approximately 10 solar masses) and a radius of 150 × 109 m (equal to the Earth's distance from the sun) .

When water freezes, it expands about nine percent. What would be the pressure increase inside your automobile engine block if the water in there froze? The bulk modulus of ice is 2.0 × 109 N/m2, and 1 ATM = 1.01 × 105 N/m2.

The pressure inside a commercial airliner is maintained at 1 ATM (1.01 × 105 N/m2) . What is the outward force exerted on a 1.0 m × 2.0 m cabin door if the outside pressure (at 10 km height) is 0.3 ATM?

What is the net force inward acting on a spherical bathysphere of diameter 2.00 m at an ocean depth of 1 000 m? (The pressure inside the bathysphere is, hopefully, 1 ATM.) ρ(sea water) = 1.02 × 103 kg/m3.

The water level in a reservoir is maintained at a constant level. What is the exit velocity in an outlet pipe 3.0 m below the water surface?

Determine the minimum area of a flat ice floe 1.0 meter thick if it is to support a 2 000-kg car above seawater. (ρice = 920 kg/m3, ρsea = 1 020 kg/m3.)

Some species of whales can dive to depths of one kilometer. What is the total pressure they experience at this depth? (ρ_sea = 1 020 kg/m³ and 1.01 × 10⁵ N/m² = 1 ATM.)

A supertanker filled with oil has a total mass of 6.1 × 10⁸ kg. If the dimensions of the ship are those of a rectangular box 300 meters long, 80 meters wide, and 40 meters high, determine how far the bottom of the ship is below sea level. (ρ_sea = 1 020 kg/m³.)

What is the total mass of the Earth's atmosphere? The radius of the Earth is 6.4 × 10⁶ m, and 1 ATM = 1.01 × 10⁵ N/m².

Find the pressure in atmospheres in the water at the base of Dworshak Dam if the water in the reservoir is 200 meters deep. (1.01 × 10⁵ N/m² = 1 ATM.)

A stonecutter's chisel has an edge area of 0.7 cm². If the chisel is struck with a force of 42 N, what is the pressure exerted on the stone?

A hydraulic lift raises a 2 000-kg automobile when a 500-N force is applied to the smaller piston. If the smaller piston has an area of 10 cm², what is the cross-sectional area of the larger piston?

Find the average density of a white dwarf star if it has a mass equal to that of the sun (2.0 × 10³⁰ kg) and a radius equal to that of the Earth (6.4 × 10⁶ m) .

To the nearest percent, what fraction of an iceberg is submerged? (ρ_ice = 917 kg/m³, ρ_sea = 1.03 × 10³ kg/m³.)

A blimp is filled with 200 m³ of helium. How much mass can the balloon lift? The density of helium is one-seventh that of air, and the density of air is 1/800 that of water. Consider the mass of the balloon to be negligible.

A fountain sends water to a height of 100 meters. What must be the pressurization (above atmospheric) of the water system? 1 ATM = 1.01 × 10⁵ N/m².

Water pressurized to 3.5 × 10⁵ Pa is flowing at 5.0 m/s in a horizontal pipe which contracts to 1/3 its former area. What are the pressure and velocity of the water after the contraction?

Find the average density of a red giant star with a mass of 20 × 10³⁰ kg (approximately 10 solar masses) and a radius of 150 × 10⁹ m (equal to the Earth's distance from the sun) .

When water freezes, it expands about nine percent. What would be the pressure increase inside your automobile engine block if the water in there froze? The bulk modulus of ice is 2.0 × 10⁹ N/m², and 1 ATM = 1.01 × 10⁵ N/m².

The pressure inside a commercial airliner is maintained at 1 ATM (1.01 × 10⁵ N/m²) . What is the outward force exerted on a 1.0 m × 2.0 m cabin door if the outside pressure (at 10 km height) is 0.3 ATM?

What is the net force inward acting on a spherical bathysphere of diameter 2.00 m at an ocean depth of 1 000 m? (The pressure inside the bathysphere is, hopefully, 1 ATM.) ρ_{(sea water)} = 1.02 × 10³ kg/m³.

Determine the minimum area of a flat ice floe 1.0 meter thick if it is to support a 2 000-kg car above seawater. (ρ_ice = 920 kg/m³, ρ_sea = 1 020 kg/m³.)