Question

In order to keep a leaking ship from sinking, it is necessary to pump \(12 \mathrm{~kg}\) of water each second from below deck \(2.1 \mathrm{~m}\) upward and over the side. What is the minimum horsepower motor that can be used to save the ship \((1 \mathrm{hp}=746 \mathrm{~W})\) ?

Short answer

The minimum motor required is approximately 0.331 horsepower.

Step-by-step solution

Understanding the Problem

We need to calculate the work done to pump water and then convert this to horsepower. Each second, 12 kg of water needs to be lifted 2.1 m.

Calculating the Work Done Each Second

The work done to lift an object is given by the formula: \( W = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\)), and \( h \) is the height. Here, \( m = 12 \, \text{kg} \), \( g = 9.81 \, \text{m/s}^2 \), \( h = 2.1 \, \text{m} \). Hence, \( W = 12 \times 9.81 \times 2.1 \).

Performing the Calculation

Carrying out the calculation: \( W = 12 \times 9.81 \times 2.1 \approx 247.212 \, \text{J} \). This is the work done each second, hence this is the power required in watts.

Converting Watts to Horsepower

Since 1 horsepower (hp) is equal to 746 watts, to convert the power requirement to horsepower, divide by 746. \( \text{Horsepower} = \frac{247.212}{746} \approx 0.331 \, \text{hp} \).

Conclusion

The minimum horsepower motor required is approximately 0.331 horsepower to keep the ship from sinking.

Key concepts

Understanding Work and Energy

In physics, work and energy are fundamental concepts that help us understand how force is applied over a distance to move an object. When you do work, you transfer energy to an object, which in turn can cause movement. This relationship is often captured by the formula:

• \( W = mgh \)

Where \( W \) represents work, \( m \) is mass, \( g \) is the acceleration due to gravity (approximated as \( 9.81 \, \text{m/s}^2 \)), and \( h \) is the height to which the object is lifted.
The energy transferred is measured in joules. In our scenario, moving water vertically is an example of work done against gravitational force. By pumping the water, energy is continuously supplied to keep lifting it over the deck, hence preventing the ship from sinking.
Understanding this concept is critical to solve problems related to energy transfer and practical applications like machinery and motors.

Power and Horsepower Explained

Power is the rate at which work is done or energy is transferred over time. It's measured in watts (W) in the metric system, and one watt is equivalent to one joule per second. However, in some contexts such as engineering and automotive industries, we use horsepower (hp) as a power unit. One horsepower is defined as 746 watts.
For our exercise, calculating the power required to pump water helps us determine the necessary motor power in both watts and horsepower. First, calculate the power in watts using the work done per second formula. Then, convert that value to horsepower to find the equivalent motor strength.

• Conversion formula: \( \text{Horsepower} = \frac{\text{Power in Watts}}{746} \)

These concepts are vital for maritime operations and any task requiring a conversion of energy to motion, highlighting horsepower's historical and practical relevance.

Basics of Mechanics

Mechanics is the area of physics concerned with motion and the forces that produce motion. It encompasses both the detailed motion of objects and their overall energy characteristics.
Mechanics provides tools to solve problems involving forces and movement, as in our example where we calculate the power needed to pump water off a ship. Concepts such as force, work, and energy computations are rooted in mechanical principles. Understanding these foundations supports broader engineering and physics applications.
By linking these concepts to practical problems, students grasp how forces lead to energy consumption and planning for effective systems such as motor designs. Mechanics serves as the core interface between theoretical physics and applied engineering, providing solutions to real-world challenges.