Problem 71 - College Physics

Question

\(\bullet\) \(\bullet\) Horsepower. In the English system of units, power is ex- pressed as horsepower (hp) instead of watts, where 1.00 hp \(=\) 746 W. Horsepower is often used for motors and automo- biles. (a) What should be the power rating in horsepower of a 100 w lightbulb? (b) How many watts does a 75 hp motor produce? (c) Electrical resistors are rated in watts to indicate how much power they can tolerate without burning up. If a resistor is rated at \(2.00 \mathrm{W},\) what should be its rating in horse- power? (d) Our sun radiates \(3.92 \times 10^{26} \mathrm{J}\) of energy each second. What is its horsepower rating? (e) How many kilo- watt-hours of energy is produced when you run a 25 hp motor for 90 minutes?

Step-by-Step Solution

Answer

(a) 0.134 hp, (b) 55,950 W, (c) 0.00268 hp, (d) \(5.25 \times 10^{23}\) hp, (e) 27.975 kWh.

Step 1: Convert Power to Horsepower (a)

To convert power from watts to horsepower, use the formula: \[\text{hp} = \frac{\text{Power in watts}}{746} \]For a 100 W light bulb, the calculation is:\[\text{hp} = \frac{100}{746} \approx 0.134 \text{ hp} \]

Step 2: Convert Horsepower to Watts (b)

To convert from horsepower to watts, use the formula:\[ \text{Watts} = \text{hp} \times 746 \]For a 75 hp motor, the calculation is:\[ \text{Watts} = 75 \times 746 = 55,950\, \text{W} \]

Step 3: Resistor Rating in Horsepower (c)

Use the same conversion formula as in (a):\[ \text{hp} = \frac{\text{Power in watts}}{746} \]For a 2.00 W resistor, the calculation is:\[ \text{hp} = \frac{2.00}{746} \approx 0.00268 \text{ hp} \]

Step 4: Convert Sun's Power Output to Horsepower (d)

Since the sun's radiation is given in J/s, which is the same as watts, convert it to horsepower:\[ \text{hp} = \frac{3.92 \times 10^{26}}{746} \approx 5.25 \times 10^{23} \text{ hp} \]

Step 5: Energy Produced by a Motor in kWh (e)

First, convert the horsepower to watts:\[ \text{Power in Watts} = 25 \times 746 = 18,650 \text{ Watts} \]Now, calculate the energy produced, using the formula for energy:\[ \text{Energy} = \text{Power} \times \text{Time} \]Convert 90 minutes to hours: 90 minutes = 1.5 hours. Then,\[ \text{Energy in kWh} = \frac{18,650 \times 1.5}{1000} = 27.975 \text{ kWh} \]

Key Concepts

Horsepower

Horsepower is a unit of power traditionally used in the English system of measurement, especially to quantify the power output of engines like those in cars and electrical motors. This concept was originally developed to compare the power of steam engines to the work capacity of draught horses. One horsepower is defined as the power necessary to move 550 pounds over one foot in one second, or equivalently, it can be described as 746 watts.

When you see a car engine rated in horsepower, it tells you just how powerful the engine is. For example, a 75 hp motor is capable of producing 75 times 746 watts.

Watts

The watt is the standard unit of power in the metric system and is mostly used around the world, including in scientific contexts. Named after James Watt, the inventor of the steam engine, it quantifies how much energy is transferred over a set period of time.

Power measured in watts is a straightforward way to indicate the amount of electrical energy being used by an appliance or produced by a generator. For instance, a light bulb rated at 100 watts implies it uses 100 joules of energy per second.

To compare with other units of power such as horsepower, remember that 1 horsepower equals 746 watts. Thus, if you want to clarify power in terms that are easy to relate to physical machines, converting to horsepower might be beneficial.

Energy Calculation

Energy calculations help us understand the amount of work done by power sources over time. The main formula here is \[ \text{Energy} = \text{Power} \times \text{Time} \]where energy is typically measured in joules, power in watts, and time in seconds. However, for larger scale energy use, such as household consumption, the kilowatt-hour (kWh) is a common unit.

For example, using a 25 horsepower motor over a period gives us a chance to understand consumption in kWh by first converting the horsepower to watts (25 hp \( \times \) 746 watts/hp), and then multiplying by the time in hours of usage.

Electrical Resistors

Electrical resistors are components that resist the flow of current in a circuit. They are crucial in controlling the amount of energy that runs through electronics. Resistors are rated in watts, which indicates the amount of power they can handle before they risk overheating or failing, known as the resistor's power dissipating capability.

If you have a resistor rated for 2 watts, it means it can safely dissipate up to 2 watts of heat without damaging the component. To convert the power limit of the resistor from watts into horsepower, you can use the conversion formula mentioned earlier: divide the watts by 746. This gives insight into how the power management in electrical circuits translates across different measurement systems.