Problem 39 - Fundamentals of Physics

Question

A certain brand of hot-dog cooker works by applying a potential difference of \(120 \mathrm{~V}\) across opposite ends of a hot dog and allowing it to cook by means of the thermal energy produced. The current is \(10.0 \mathrm{~A},\) and the energy required to cook one hot dog is \(60.0 \mathrm{~kJ}\). If the rate at which energy is supplied is unchanged, how long will it take to cook three hot dogs simultaneously?

Step-by-Step Solution

Answer

It takes 2.5 minutes to cook three hot dogs.

Step 1: Understand the Power Formula

Power is the rate at which energy is transferred or the rate at which work is done. The formula for electrical power (in watts) is given by the product of the current (I) and the voltage (V): \[ P = IV \]where \( I = 10.0 \, \text{A} \) and \( V = 120 \, \text{V} \). We can use this to calculate the rate of energy transfer.

Step 2: Calculate the Electrical Power

Using the formula for power:\[ P = IV = 10.0 \, \text{A} \times 120 \, \text{V} = 1200 \, \text{W} \]This means the hot dog cooker supplies energy at a rate of 1200 watts, or 1200 joules per second.

Step 3: Calculate Total Energy Required for Three Hot Dogs

Each hot dog requires an energy of \(60.0 \, \text{kJ} = 60,000 \, \text{J}\). For three hot dogs, the total energy required is:\[ 3 \times 60,000 \, \text{J} = 180,000 \, \text{J} \]

Step 4: Determine the Time to Cook Three Hot Dogs

Since power \(P\) is the rate at which energy is transferred, we can find the time \(t\) needed to transfer a given amount of energy \(E\) by rearranging the power formula \(P = \frac{E}{t}\):\[ t = \frac{E}{P} \]Substitute the values for \(E\) and \(P\):\[ t = \frac{180,000 \, \text{J}}{1200 \, \text{W}} = 150 \, \text{s} \]

Step 5: Convert Time into Minutes

To convert the time from seconds to minutes, divide by 60:\[ t = \frac{150}{60} = 2.5 \, \text{minutes} \]Thus, it will take 2.5 minutes to cook three hot dogs simultaneously.

Key Concepts

Thermal Energy

Thermal energy is the energy that comes from heat. When you apply electricity to an object, like a hot dog in this case, that energy can be transformed into heat. The hot dog cooker uses electrical energy to generate the needed thermal energy to cook the hot dogs.

As heat is produced, the molecules in the hot dog begin to move more rapidly, causing the temperature to rise until it's cooked. In terms of energy, each hot dog requires a total of 60,000 joules (60.0 kJ) to become ready. Understanding thermal energy helps explain how the process of cooking occurs in electrical appliances such as this one.

Voltage

Voltage is essential since it determines how much electric potential energy is available to move electrons from one point to another.

In our hot dog cooker example, a potential difference, which is also known as voltage, of 120 volts (V) is applied across the hot dogs. This means there is enough force to drive electrical current through the hot dogs, which in turn generates heat.

Voltage is like the pressure that pushes charges through a conductor. It is a vital part of any electrical circuit, and higher voltages can mean more energy available for use, as long as the current remains constant.

Current

Current refers to the flow of electric charge through a conductor. It's measured in amperes (A). In this hot dog cooker, the electric current is 10.0 A, which means that at any time, 10 ampere worth of electric charges are flowing through the hot dogs.

This electric current is crucial for transferring the energy that will be converted into heat. When you have a stable and consistent current, it ensures that the energy is supplied at a constant rate, which is important for consistent cooking results. Current, combined with voltage, affects how much power the electric circuit can deliver, which is calculated with the formula: \[ P = IV \] This ensures that the energy is turned efficiently into thermal energy to cook the hot dogs.

Energy Transfer Rate

The energy transfer rate is how fast energy is provided or consumed in a given timeframe. It is measured in watts (W), which is synonymous with joules per second. The rate at which the hot dog cooker transfers energy is 1200 W, meaning it delivers 1200 joules of energy every second.

For these hot dogs, the total energy needed is 180,000 J. Dividing the required energy by the energy transfer rate gives us the cooking time in seconds, which is then converted into minutes. This rate is crucial because it directly influences how quickly the hot dogs cook. The higher the energy transfer rate, the shorter the time to reach cooking completion, given the same amount of required energy. Maintaining a steady energy transfer rate ensures reliability and efficiency in electrical appliances.