A fresh water pond that is 14.7 m deep is contained on one side by a cliff. The water has eroded a nearly horizontal "tube" through a bed of limestone, which allows the water to emerge on the other side of the cliff. If the "tube" has a diameter of 4.07 cm, and is located 6.0 m below the surface of the pond, what is the frictional force between the "tube" wall and a rock that is blocking the exit?
b) When the rock is removed, what volume of water flows out of the "tube" in 2.90 hr?
for part a)
F=(Pressure)*(Area)
F=[(rho)(g)(d)]*(Area)
where d is the distance of the tube beneath the surface of the water, and area is the cross sectional area of the tube, and rho the density of water.
make sure to convert the units of the diameter into cm before subbing it into the equation!
A)2345N
To find the frictional force between the "tube" wall and the rock, we need to calculate the pressure exerted by the water column on the rock.
1) Calculate the pressure exerted by the water column:
The pressure at a given height in a fluid is given by the formula P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.
Given:
- Depth of the pond (h): 14.7 m
- Diameter of the tube (d): 4.07 cm = 0.0407 m
- Depth of the tube (h_tube): 6.0 m
First, let's calculate the pressure at the base of the tube:
P_base = ρgh
P_base = ρ_water * g * (h - h_tube)
where ρ_water is the density of water and g is the acceleration due to gravity.
The density of water is approximately 1000 kg/m^3, and the acceleration due to gravity is approximately 9.8 m/s^2.
Substituting the given values:
P_base = 1000 kg/m^3 * 9.8 m/s^2 * (14.7 m - 6.0 m)
2) Calculate the force exerted by the water column on the tube wall:
The force exerted by the water on the tube wall is equal to the pressure multiplied by the cross-sectional area of the tube.
The cross-sectional area of the tube can be calculated using the formula for the area of a circle:
A = πr^2
A = π * (d/2)^2
where r is the radius of the tube.
Substituting the given diameter:
A = π * (0.0407 m / 2)^2
Now, we can calculate the force:
F = P_base * A
This gives us the frictional force between the tube wall and the rock.
To answer part (b) of the question, we need to calculate the volume of water that flows out of the tube in 2.90 hours.
3) Calculate the flow rate of water:
The flow rate (Q) is given by the formula Q = A * v,
where A is the cross-sectional area of the tube and v is the velocity of the water.
We can assume that the velocity of the water is constant throughout the tube.
Q = A * v
To find the volume of water that flows out in 2.90 hours, we multiply the flow rate by the time:
Volume = Q * time
Now, let's substitute the calculated values to find the answers to part (b).