Problem 85 - University Physics with Modern Physics

Question

A pump is required to lift 800 kg of water (about 210 gallons) per minute from a well 14.0 m deep and eject it with a speed of 18.0 m/s. (a) How much work is done per minute in lifting the water? (b) How much work is done in giving the water the kinetic energy it has when ejected? (c) What must be the power output of the pump?

Step-by-Step Solution

Answer

(a) 109,872 J, (b) 129,600 J, (c) 3,991.2 W.

Step 1: Determine the Work Done in Lifting the Water

The work done to lift the water can be calculated using the formula for gravitational work:\[ W_1 = mgh \]where \( W_1 \) is the work done lifting the water, \( m \) is the mass of the water, \( g \) is the acceleration due to gravity (approximately \(9.81 \ \text{m/s}^2\)), and \( h \) is the height lifted.Substitute the values:\[ W_1 = 800 \ \text{kg} \times 9.81 \ \text{m/s}^2 \times 14.0 \ \text{m} \]\[ W_1 = 109,872 \ \text{J} \] So, the work done per minute in lifting the water is \(109,872 \ \text{J}\).

Step 2: Calculate the Work Done in Giving the Water Kinetic Energy

The work done to give the water kinetic energy can be calculated using the kinetic energy formula:\[ W_2 = \frac{1}{2} mv^2 \]where \( W_2 \) is the work done, \( m \) is the mass of the water, and \( v \) is the velocity.Substitute the values:\[ W_2 = \frac{1}{2} \times 800 \ \text{kg} \times (18.0 \ \text{m/s})^2 \]\[ W_2 = \frac{1}{2} \times 800 \ \times 324 \]\[ W_2 = 129,600 \ \text{J} \]Hence, the work done in giving the water the kinetic energy is \(129,600 \ \text{J}\).

Step 3: Calculate the Total Work Done

The total work done by the pump per minute is the sum of the work done to lift the water and the work done to give it kinetic energy:\[ W_{\text{total}} = W_1 + W_2 \]\[ W_{\text{total}} = 109,872 \ \text{J} + 129,600 \ \text{J} \]\[ W_{\text{total}} = 239,472 \ \text{J} \] Thus, the total work done is \(239,472 \ \text{J}\).

Step 4: Determine the Power Output of the Pump

Power output of the pump is calculated by dividing the total work done by the time duration, which is 1 minute or 60 seconds:\[ P = \frac{W_{\text{total}}}{t} \]\[ P = \frac{239,472 \ \text{J}}{60 \ \text{s}} \]\[ P = 3,991.2 \ \text{W} \]Thus, the power output of the pump must be \(3,991.2 \ \text{W}\).

Key Concepts

Gravitational Work

When it comes to lifting objects against gravity, the concept of gravitational work is crucial. In simple terms, gravitational work refers to the energy required to lift a mass to a certain height against the gravitational force. The formula to calculate this work is given by: \[ W = mgh \]
Here, \( W \) represents the work done, \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and \( h \) is the height to which the object is lifted.

• Mass: The amount of matter in the object.
• Gravity: A constant that represents the force pulling objects toward the Earth's center.
• Height: The vertical distance the object is moved against gravity.

In the original problem, lifting 800 kg of water to a height of 14 meters requires a certain amount of energy. Using the formula, you can calculate this as approximately 109,872 Joules. This energy expenditure is what keeps the water elevated and ready for its next phase of movement.

Kinetic Energy

Once the water is lifted, it must be ejected with velocity. Here is where kinetic energy comes into play. Kinetic energy is the energy that an object possesses due to its motion. The formula for kinetic energy is: \[ KE = \frac{1}{2} mv^2 \]
Where \( KE \) is the kinetic energy, \( m \) is the mass of the object, and \( v \) is the velocity at which the object is moving.

• Mass: The weight of the water being ejected.
• Velocity: The speed at which the water exits the pump.

For the given problem, water is ejected at a speed of 18 m/s. Hence, the kinetic energy given to the water can be calculated to be about 129,600 Joules. This energy is what propels the water as it exits, allowing it to move freely away from the pump.

Power Output

Power is defined as the rate at which work is done or energy is transferred over time. It helps us understand how quickly energy is used or generated. The formula to determine power is: \[ P = \frac{W}{t} \]
Where \( P \) is power, \( W \) is the total work done, and \( t \) is the time over which the work is done. In this case, the total work includes both the gravitational work and the kinetic energy.

• Work: Total energy expended to lift and move the water.
• Time: Duration of 60 seconds, equivalent to one minute.

By calculating the power output for the pump lifting and ejecting water, we find it to be approximately 3,991.2 Watts. Knowing the power output is essential for assessing how effectively and efficiently the pump operates over the set period.