Problem 66 - College Physics

Question

A pump lifts \(200 \mathrm{~kg}\) of water per hour a height of \(5.0 \mathrm{~m} .\) What is the minimum necessary power output rating of the water pump in watts and horsepower?

Step-by-Step Solution

Answer

The pump needs at least 2.725 watts or 0.00365 horsepower.

Step 1: Calculate the Work Done

First, calculate the work done to lift the water. The work done is given by the formula: \[ W = m imes g imes h \]where:- \( m = 200 \) kg is the mass,- \( g = 9.81 \) m/s² is the acceleration due to gravity,- \( h = 5.0 \) m is the height.Substitute the values:\[ W = 200 imes 9.81 imes 5.0 = 9810 \text{ Joules} \]

Step 2: Calculate the Power in Watts

Power is the rate of doing work. The formula to find power when work and time are given is:\[ P = \frac{W}{t} \]where:- \( W = 9810 \) Joules is the work done,- \( t = 3600 \) seconds (1 hour = 3600 seconds).Substitute the values:\[ P = \frac{9810}{3600} \approx 2.725 \text{ Watts} \]

Step 3: Convert Power to Horsepower

To convert from watts to horsepower, use the conversion factor \( 1 \text{ horsepower} = 745.7 \text{ Watts} \).\( HP = \frac{P}{745.7} \)Substitute the value:\[ HP = \frac{2.725}{745.7} \approx 0.00365 \text{ horsepower} \]

Key Concepts

Work and Energy

When we talk about work and energy in physics, we are discussing how forces interact with objects to either move them or change their state of energy. Work is defined as the energy transferred to or from an object via the application of force along a displacement. In simple terms, work is done when you move something. The formula for work is \( W = F \times d \), where \( F \) is the force applied and \( d \) is the displacement in the direction of the force.
In the context of lifting an object, the work done can be expressed using gravitational force. So, we use \( W = m \times g \times h \), where \( m \) is mass, \( g \) is acceleration due to gravity, and \( h \) is height. This formula takes into account the gravitational pull and the distance the object is moved.
Understanding work and gravitational potential energy is important when determining how much energy is stored or needed, for instance, raising 200 kg of water a height of 5 m requires 9,810 Joules of energy.

Power and Energy Conversion

Power is the rate at which work is performed or energy is transferred. It is how quickly something can accomplish work. Power is measured in watts (W) where 1 watt is equal to 1 joule per second. The equation for power is \( P = \frac{W}{t} \), which shows it depends on both the amount of work done \( W \) and the time \( t \) it takes.
In the given problem, the water pump performs 9,810 Joules of work over 3,600 seconds (or 1 hour). That means the pump needs a minimum power output rating of approximately 2.725 watts to lift the water to the desired height. The task can be seen as converting mechanical energy to potential energy as the water is lifted.
When we need to express power in different units, such as horsepower, we use a conversion factor. Power can be converted from watts to horsepower with the factor 1 horsepower = 745.7 watts. Thus, for the pump, 2.725 watts converts to approximately 0.00365 horsepower.

Unit Conversion in Physics

Unit conversion is a fundamental aspect in solving physics problems. It allows us to translate measurements into different units, ensuring consistency and making calculations easier to understand or compare.
For instance, when calculating power, it's crucial to know that work (measured in joules) and time (in seconds) combine to give power in watts. Changing units systematically, like from watts to horsepower, enables engineers and scientists to communicate findings in universally understood terms.
Always remember to use appropriate conversion factors for accuracy. A common factor is that 1 hour equals 3,600 seconds, essential when calculating power over time. For the conversion between watts and horsepower, use the factor 1 horsepower = 745.7 watts. By mastering these conversions, you ensure precision in interpreting and expressing energy and power in various contexts.