A pump is required to lift 800 liters of water per minute from a well 15.0 m deep and eject it with a speed of 17.0 m/s.
How much work per minute does the pump do?
use
E = mgh + 1/2 mv^2
= (800 L)(1 kg / 1 L) (9.8 m/s^2) (15m) + 1/2 (800 L)(1 kg / 1 L) (17 m/s)^2
= 233200 J
I agree with that.
To find the work done per minute by the pump, we need to calculate the amount of work done to lift the water and the amount of work done to eject the water.
The work done to lift the water can be calculated using the formula:
Work = force × distance
First, let's calculate the force required to lift the water. The force can be determined using the formula:
Force = mass × acceleration
In this case, the mass of the water being lifted can be calculated using the density formula:
Mass = volume × density
Given that the density of water is approximately 1000 kg/m³ and the volume of water being lifted is 800 liters, we can convert the volume to cubic meters:
Volume in cubic meters = 800 liters × (1 cubic meter / 1000 liters) = 0.8 cubic meters
Now, we can calculate the mass:
Mass = 0.8 cubic meters × 1000 kg/m³ = 800 kg
Next, let's calculate the force required to lift the water. The acceleration in this case is due to gravity and is approximately 9.8 m/s²:
Force = 800 kg × 9.8 m/s² = 7840 N (Newtons)
Now, let's calculate the distance over which the work is done. The water is being lifted from a well that is 15.0 m deep. Therefore, the distance is 15.0 m.
Using the formula:
Work = force × distance
Work = 7840 N × 15.0 m = 117,600 N·m
Now, let's calculate the work done to eject the water. To calculate this, we'll use the formula:
Work = force × distance
The force required to eject the water is equal to the mass flow rate of water multiplied by the velocity of ejection.
Given that the mass flow rate is 800 liters per minute, we can convert it to kg/s:
Mass flow rate in kg/s = 800 liters/minute × (1 cubic meter / 1000 liters) × (1 minute / 60 seconds) = 0.0133 kg/s
Now we can calculate the force:
Force = mass flow rate × velocity
Force = 0.0133 kg/s × 17.0 m/s = 0.2261 N
The distance over which the work is done is not given in the question. Let's assume it is 1 meter for simplicity.
Work = force × distance
Work = 0.2261 N × 1.0 m = 0.2261 N·m
Now, we can find the total work done per minute by adding the work done to lift the water and the work done to eject the water:
Total Work = Work to lift water + Work to eject water
Total Work = 117,600 N·m + 0.2261 N·m
Total Work = 117,600 N·m + 0.2261 N·m = 117,600 N·m
Therefore, the pump does 117,600 joules of work per minute.