Water is lifted out of a well 25.0 m deep by a motor rated at 1.50 hp.
Assuming efficiency 90%, how many kilograms of water can be lifted in 1 min?
To solve this problem, we need to calculate the work done by the motor to lift the water out of the well, taking into account its rated power and efficiency.
First, let's convert the power from horsepower (hp) to watts (W). Since 1 hp is about equal to 745.7 W, we'll have:
1.50 hp = 1.50 * 745.7 W = 1118.55 W
Next, let's calculate the work done by the motor in lifting the water. The work done is equal to the force applied (in this case, the weight of the water) multiplied by the distance it is lifted. The weight of the water can be calculated using its mass and the acceleration due to gravity.
The mass can be found using the density of water, which is approximately 1000 kg/m³. Since the volume of the water lifted is equal to the cross-sectional area of the well multiplied by its depth, we'll have:
Volume = Area * Depth
Volume = π * (radius of well)² * Depth
Since the radius of the well is not given, let's assume it to be 1 meter for simplicity. Therefore:
Volume = π * (1)² * 25 = 78.54 m³
The mass of the water can be calculated by multiplying the volume by its density:
Mass = Volume * Density
Mass = 78.54 m³ * 1000 kg/m³
Mass = 78540 kg
Now, we need to calculate the work done by the motor. The work done is given by the formula:
Work = Force * Distance
Since the force is equal to the weight of the water and the distance is the depth of the well, we'll have:
Work = Weight of water * Depth
Work = Mass * Gravity * Depth
Let's assume the acceleration due to gravity to be 9.8 m/s². Therefore:
Work = 78540 kg * 9.8 m/s² * 25.0 m
Work = 192,261,000 J (joules)
Since the efficiency of the motor is given as 90%, we need to account for the energy loss due to inefficiency. To find the actual work done by the motor, we divide the calculated work by the efficiency:
Actual Work = Work / Efficiency
Actual Work = 192,261,000 J / 0.90
Actual Work = 213,623,333 J (joules)
Finally, we need to calculate the amount of work done per minute, as requested. Since 1 minute is equal to 60 seconds, we'll have:
Work per Minute = Actual Work / Time
Work per Minute = 213,623,333 J / 60 s
Work per Minute ≈ 3,560,389 J/min
Therefore, the motor can lift approximately 3,560,389 joules of water in 1 minute.
To solve this problem, we need to determine the amount of work done by the motor in 1 minute and then convert it to the amount of water lifted.
First, let's convert the power rating of the motor to watts:
1 horsepower (hp) = 746 watts
So, the power of the motor is:
1.50 hp × 746 W/hp = 1119 W
Next, we can calculate the work done by the motor in 1 minute:
Work = Power × Time
= 1119 W × (1 min × 60 s/min)
= 1119 W × 60 s
= 67140 J
Now, let's calculate the mass of water that can be lifted using the work-energy principle:
Work = Change in Potential Energy
The potential energy of an object is given by the equation:
Potential Energy = Mass × Gravity × Height
The mass of water can be calculated as follows:
Mass = Work / (Gravity × Height)
where:
- Work is the calculated work done by the motor (67140 J)
- Gravity is the acceleration due to gravity (9.8 m/s^2)
- Height is the depth of the well (25.0 m)
Substituting the given values into the equation:
Mass = 67140 J / (9.8 m/s^2 × 25.0 m)
= 271.29 kg
Therefore, the motor can lift approximately 271.29 kilograms of water in 1 minute.