Problem 64 - College Physics

Question

A girl consumes \(8.4 \times 10^{6} \mathrm{~J}\) ( 2000 food calories) of energy per day while maintaining a constant weight. What is the average power she produces in a day?

Step-by-Step Solution

Answer

The average power produced is approximately 97.22 Watts.

Step 1: Understanding Power

Power is defined as the rate at which energy is transferred or converted. It can be calculated using the formula: \[ \text{Power} = \frac{\text{Energy}}{\text{Time}} \] where energy is measured in Joules (J) and time is measured in seconds (s).

Step 2: Convert Time to Seconds

Since we need to find the average power over a day, we need to convert the time from days to seconds. One day has 24 hours, each hour has 60 minutes, and each minute has 60 seconds. Therefore, the number of seconds in a day is:\[ \text{Time} = 24 \times 60 \times 60 = 86,400 \text{ seconds} \]

Step 3: Calculate the Average Power

Using the energy consumed per day and the time in seconds, we can now calculate the average power:\[ \text{Power} = \frac{8.4 \times 10^{6} \text{ J}}{86,400 \text{ s}} \]Now, calculate the power:\[ \text{Power} \approx 97.22 \text{ Watts} \]

Step 4: Round the Result (if necessary)

It's often practical to round the final answer to a reasonable number of significant figures. Here, we will round it to two decimal places: \[ \text{Power} = 97.22 \text{ Watts} \]

Key Concepts

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In the context of this exercise, the girl's body is converting the chemical energy from food (measured in Joules) into various forms of energy such as kinetic energy, thermal energy, and others that keep the body functioning. Understanding how this conversion works in the human body can be crucial for multiple fields, including nutrition and medicine.

Energy in food is usually measured in calories, which is another form of energy unit. One food calorie (capital C in Calorie) is equivalent to 4,184 Joules. Therefore, when we know the caloric intake, we can convert it into Joules, making it easier to work with when calculating power.

• 1 Calorie (food Calorie) = 4,184 Joules
• Commercial food often displays energy content in Calories but scientific calculations like these often revert to Joules for precision.

In our exercise, the girl's daily food intake provides her with enough chemical energy to maintain her weight, which is then converted into other necessary bodily energy forms.

Joules to Watts

Converting energy in Joules to average power in Watts is a common calculation in physics, particularly when assessing energy usage over time. Power is essentially how fast energy is being used, and it is measured in Watts, where one Watt is equal to one Joule per second. This equation illustrates how energy is expended over time through the power expression:\[ \text{Power} = \frac{\text{Energy (Joules)}}{\text{Time (seconds)}} \]

In our exercise, the energy consumed by the girl per day, which is given as \(8.4 \times 10^6\) Joules, can be used along with the time span of a day (in seconds) to calculate her average power output. This process involves dividing the total energy in Joules by the time period in seconds. So, you get the average energy used per second, which is power expressed in Watts.

• 1 Watt = 1 Joule/second
• Total Energy (Joules) divided by Time (seconds) = Average Power (Watts)

Accurately performing this calculation allows physicists and engineers to quantify power usage or provide guidelines for energy efficiency.

Time Conversion in Physics

Time conversion in physics involves translating time from one unit into another to facilitate calculations. In our exercise, we start with a time span of one day and need to convert it to seconds, which is the SI unit for time. This conversion is crucial because it's common practice in physics to use base SI units to ensure consistency and reliability in results.

To convert from days to seconds, one must use the following steps: remember that

• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds

Multiplying these together provides the number of seconds in one day, which is 86,400 seconds. This consistent base provides a fundamental unit that allows for simpler calculations. Keeping time in seconds ensures accuracy when calculating average power or other rate-based quantities because it creates a direct relationship between energy and time. Such conversions are vital in achieving precise results and understanding larger concepts in physics.