Problem 115 - Chemistry: The Central Science in SI Units, Global Edition

Question

A \(100-\mathrm{kg}\) man decides to add to his exercise routine by walking up six flights of stairs \((30 \mathrm{~m}) 10\) times per day. He figures that the work required to increase his potential energy in this way will permit him to eat an extra order of French fries, at 245 Cal, without adding to his weight. Is he correct in this assumption?

Step-by-Step Solution

Answer

By calculating the potential energy gained from walking up six flights of stairs 10 times a day, we find the total potential energy in calories. Comparing this value to the energy provided by an extra order of French fries (245 calories), we can determine whether the man's assumption is correct. \(PE = 100\,\text{kg} \times 9.81\,\text{m/s²} \times 30\,\text{m}\) and \(Total\: PE\: (\text{calories}) = \frac{Total\: PE\, (\text{Joules})}{4.184\, \text{J/cal}}\).

Step 1: Calculate the potential energy per flight of stairs

To start, we will find the potential energy (PE) gained by the man after walking up a single flight of stairs. Potential energy is given by the formula: \(PE = mgh\) where m is the mass of the man (100 kg), g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height of one flight of stairs (30 m). Therefore, we can calculate the potential energy as follows: \(PE = 100\,\text{kg} \times 9.81\,\text{m/s²} \times 30\,\text{m}\)

Step 2: Calculate the total potential energy gained per day

Now that we have the potential energy gained per flight of stairs, we need to determine the total potential energy gained in a day. As the man walks up the stairs 10 times a day, we can multiply the potential energy gained per flight by 10: \(Total\: PE = 10 \times PE\)

Step 3: Convert potential energy to calories

Now we need to convert the total potential energy from joules (the SI unit for energy) to calories because the energy content of French fries is given in calories. We know that: \(1\,\text{calorie} = 4.184\, \text{J}\) So we can convert the total potential energy to calories: \(Total\: PE\, (\text{calories}) = \frac{Total\: PE\, (\text{Joules})}{4.184\, \text{J/cal}}\)

Step 4: Compare the energy from the potential energy gained to the energy provided by French fries

Now that we have the total potential energy in calories, we can compare it to the energy provided by an extra order of French fries (245 calories). If the total potential energy gained is greater than or equal to the energy from French fries, then the man's assumption is correct. You can evaluate whether the man's assumption is correct based on the comparison of energy gained and energy consumed.

Key Concepts

Caloric Energy Conversion

The process of converting energy from food into usable energy within the human body is essential for bodily functions and physical activity. One of the units of measurement for energy is the calorie. In exercise scenarios, the energy expended by the body is often measured in calories to better relate to the energy obtained from food.
In physics, energy calculations typically use the unit "joules." To relate this to calories, the conversion is needed:

• 1 calorie is equivalent to 4.184 joules.

Understanding this conversion is crucial when comparing potential energy gains from an activity, such as climbing stairs, to the caloric intake from food items like French fries. This helps determine if physical activities balance added caloric intake, maintaining weight stability.

Exercise and Energy Balance

Achieving and maintaining a stable weight involves balancing the calories consumed with those expended through daily activities and exercise. When engaging in exercises like walking up stairs, the energy used increases the body's caloric expenditure.
Here's how this method works:

• The activity increases potential energy, which requires the body to expend calories equivalent to the energy gained.
• If energy from exercise matches the caloric intake from an additional food item, weight remains stable.

In our example, if climbing the stairs offsets the calories from eating French fries, the man's weight should remain unchanged. This illustrates energy balance principles, where energy input equals energy output.

Gravitational Potential Energy

Gravitational potential energy relates to the energy stored in an object due to its elevated position relative to a lower position. It depends on three main factors: mass, gravity, and height.
The potential energy gained while ascending can be calculated using the formula:

• \(PE = mgh\), where:
• \(m\) is mass in kilograms,
• \(g\) is the acceleration due to gravity (\(9.81 \, \text{m/s}^2\)), and
• \(h\) is the height in meters.

For the man climbing stairs, this calculation helps determine the total energy used per ascent and throughout his daily routine. The man’s frequent trips up the stairs aggregate his total energy expenditure.

Physics of Motion

The physics of motion deals with the principles governing movement, including forces, energy, and the laws of motion set by Newton. When we climb stairs, several physical forces come into play:

• The force of gravity opposes the upward motion, necessitating extra energy to overcome it.
• The body must exert a force greater than gravitational pull to ascend, utilizing energy stored as potential energy.

This scenario illustrates important physics concepts, such as work and energy transformation, where work done against gravity transforms kinetic energy into potential energy. Understanding these can greatly enhance comprehension of everyday movements and their energy requirements.