Problem 66 - Physics for Scientists and Engineers with Modern Physics

Question

(II) How much work can a 3.0 -hp motor do in \(1.0 \mathrm{~h} ?\)

Step-by-Step Solution

Answer

The 3.0-hp motor can do 8,056,800 joules of work in 1 hour.

Step 1: Understanding Horsepower

The term "horsepower," abbreviated as 'hp,' is a unit of power that is equal to 746 watts. Therefore, a 3.0-hp motor can perform work at a rate of \(3.0 \times 746\) watts.

Step 2: Converting Power to Work

To find out how much work the motor can do, we need to multiply the power (in watts) by the time (in seconds). The power is 2238 watts (from \(3.0 \times 746\)), and the time is 1 hour, which equals \(3600\) seconds.

Step 3: Calculating the Work Done

Calculate the work done by multiplying the power in watts with the time in seconds. \[\text{Work} = 2238 \text{ watts} \times 3600 \text{ seconds}\]

Step 4: Evaluating the Expression

Now, calculate the expression to find the total work done:\[\text{Work} = 2238 \times 3600 = 8,056,800 \text{ joules}\]

Key Concepts

Horsepower

Horsepower is a measurement of power, which is the rate at which work is done. Originally, it was created to quantify how much work a horse could do. Though horses don't have the main role in today's industries, this unit is still widely used.

• 1 horsepower (hp) equals 746 watts.
• It is commonly used for measuring the power output of engines and motors.

When you have a motor rated at 3.0 horsepower, this means the motor can output power at a rate of 3.0 times 746 watts. For the example given, that totals to 2238 watts. This power measurement is essential in determining how much work the motor can do over a certain period of time.

Power Conversion

To understand how much work a motor can perform, we must convert power into work. Power is the rate of doing work, while work refers to the total effort exerted.

• Power (measured in watts) describes how quickly work is being done.
• Work (measured in joules) indicates the overall effect accomplished by the power over time.

This conversion is straightforward through the formula:\[\text{Work (in joules)} = \text{Power (in watts)} \times \text{Time (in seconds)}\]Using the given example, the motor runs at 2238 watts for an entire hour. We need to convert the hour into seconds because watts are power per second. Hence, 1 hour equals 3600 seconds. Multiplying power (2238 watts) by time (3600 seconds) gives us the total work.

Joules

Joules are a fundamental unit of work in the International System of Units (SI). It tells us how much energy is expended when a force acts over a distance.

• 1 joule equals the energy transferred when 1 watt of power works for 1 second.
• Joules are used broadly in science to quantify energy, work, and heat.

In the context of our problem, once we perform the calculation \[\text{Work} = 2238 \text{ watts} \times 3600 \text{ seconds}\]the result is 8,056,800 joules. This number represents the total amount of work a 3.0-horsepower motor can do in one hour. Understanding joules is crucial for grasping how various forms of energy are interrelated and interchangeable.