A trough is 5 meters long, 1 meters wide, and 4 meters deep. The vertical cross-section of the trough parallel to an end is shaped like an isoceles triangle (with height 4 meters, and base, on top, of length 1 meters). The trough is full of water (density 1000 kg/m3 ). Find the amount of work in joules required to empty the trough by pumping the water out of an outlet that is located 3 meters above the top of the tank. See problem 21, page 464 of the text for a diagram of this trough configuration. (Note: Use g=9.8 m/s2 as the acceleration due to gravity.)
To find the amount of work required to empty the trough, we need to calculate the gravitational potential energy of the water in the trough. The formula for gravitational potential energy is given by:
Potential Energy = weight * height
First, we need to find the weight of the water in the trough. The weight is equal to the mass multiplied by the acceleration due to gravity (g).
Weight = mass * g
To find the mass, we need to calculate the volume of water in the trough. The volume can be calculated by multiplying the length, width, and depth of the trough.
Volume = length * width * depth
Next, we need to convert the volume to mass using the density of water (which is 1000 kg/m^3).
Mass = Volume * Density
Once we have the mass, we can calculate the weight using the formula mentioned earlier.
Weight = Mass * g
Now, we can calculate the potential energy by multiplying the weight by the height from which the water is lifted.
Potential Energy = Weight * Height
Finally, we obtain the amount of work required by subtracting the initial potential energy (which is zero since the water is already at the highest point) from the final potential energy. The final potential energy is the weight of the water at the height of 3 meters above the top of the tank.
Amount of Work = Final Potential Energy - Initial Potential Energy
Let's calculate the values step by step:
Volume = 5m * 1m * 4m = 20 m^3
Mass = Volume * Density = 20 m^3 * 1000 kg/m^3 = 20000 kg
Weight = Mass * g = 20000 kg * 9.8 m/s^2 = 196000 N
Potential Energy = Weight * Height = 196000 N * 3 m = 588000 J
Amount of Work = Final Potential Energy - Initial Potential Energy = 588000 J - 0 J = 588000 J
Therefore, the amount of work required to empty the trough is 588000 Joules.
To find the amount of work required to empty the trough, we need to calculate the potential energy of the water and convert it to work.
Step 1: Find the volume of water in the trough.
The volume of a triangular prism can be calculated using the formula:
Volume = (base area) x (height)
In this case, the base area of the triangular cross-section is (1/2) x (base length) x (height), which is (1/2) x 1 x 4 = 2 cubic meters.
Therefore, the volume of water in the trough is 2 cubic meters.
Step 2: Determine the mass of the water.
The mass of water can be calculated using the formula:
Mass = (Density) x (Volume)
Given that the density is 1000 kg/m³ and the volume of water is 2 cubic meters, the mass of water in the trough is:
Mass = 1000 kg/m³ x 2 m³ = 2000 kg
Step 3: Calculate the height difference of the water.
The height difference between the outlet and the top of the trough is 3 meters.
However, because the trough is already filled to the top, the water needs to be lifted an additional 4 meters to reach the outlet. So the total height difference is 3 meters + 4 meters = 7 meters.
Step 4: Determine the potential energy of the water.
Potential energy can be calculated using the formula:
Potential Energy = (Mass) x (Gravity) x (Height)
Given that the mass of water is 2000 kg, acceleration due to gravity is 9.8 m/s², and the height difference is 7 meters, the potential energy of the water is:
Potential Energy = 2000 kg x 9.8 m/s² x 7 m = 137,200 J
Therefore, the amount of work required to empty the trough is 137,200 joules.