Problem 109 - Chemistry: The Central Science

Question

A 200-lb man decides to add to his exercise routine by walking up three flights of stairs \((45 \mathrm{ft}) 20\) 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

The man's assumption is incorrect. By walking up three flights of stairs 20 times per day, he only burns 58.280 Cal, which is less than the calories in an extra serving of French fries (245 Cal). Thus, he will likely add to his weight if he eats the extra French fries.

Step 1: Convert weight to mass

First, we need to convert the man's weight from pounds to kg. 1 pound = 0.453592 kg Mass (m) = (200 lb) * (0.453592 kg/lb) = 90.7184 kg

Step 2: Convert height to meters

Next, we'll convert the height of the stairs from feet to meters. 1 foot = 0.3048 meter Height (h) = (45 ft) * (0.3048 m/ft) = 13.716 m

Step 3: Calculate gravitational potential energy

Now we can calculate the gravitational potential energy for one flight of stairs. GPE = m * g * h = (90.7184 kg) * (9.8 \(m/s^2\)) * (13.716 m) = 12195.414 Joules

Step 4: Calculate work done for 20 flights of stairs

The man walks up 20 flights of stairs, so we'll multiply the GPE by 20 to find the total work done. Total work done = GPE * 20 = 12195.414 J * 20 = 243,908.28 Joules

Step 5: Convert work to calories

Now, we'll convert the work done in Joules to calories. 1 Cal = 4184 J Total work done (in Cal) = (243,908.28 J) / (4184 J/Cal) = 58.280 Cal

Step 6: Compare work done to calories from French fries

Finally, we'll compare the work done in calories to the calories in a serving of French fries. Total work done (58.280 Cal) < French fries (245 Cal) Since the work done by the man (58.280 Cal) is less than the calories in an extra serving of French fries (245 Cal), his assumption is incorrect, and he will likely add to his weight if he eats the extra French fries.

Key Concepts

Gravitational Potential Energy

When you lift an object, you're working against gravity, and this work is stored as gravitational potential energy (GPE). This energy is a result of the object's position relative to the ground.
The formula for gravitational potential energy is:

• \( \text{GPE} = m \cdot g \cdot h \)

where

• \(m\) is the mass of the object in kilograms (kg),
• \(g\) is the acceleration due to gravity, which is approximately \(9.8 \ \text{m/s}^2\),
• \(h\) is the height in meters (m) the object is raised to.

By climbing stairs, like in the original exercise, the man increases his GPE each time he ascends. This principle shows how energy is required to lift your body against the gravitational pull of the Earth. It's an essential concept in understanding energy expenditure in physical activities.

Calories to Joules Conversion

In physical science and nutrition, energy is measured in different units: calories and joules. A calorie is often used in dietary contexts to describe food energy. To understand activity energy expenditure, we often use joules. One dietary Calorie (often written as Cal with a capital 'C' to distinguish it from a calorie, which is 1/1000 of a Calorie) can be converted to joules for compatibility in scientific calculations.

• The conversion factor is: \(1 \ \text{Cal} = 4184 \ \text{J}\).

For the man in the exercise, converting the work he does in climbing stairs from joules to calories allows comparison to the caloric content of a treat such as French fries. This conversion helps demonstrate that the energy he expends doesn't quite match the energy he would gain from the additional food intake, highlighting a common misconception in balancing exercise and diet.

Exercise and Energy Expenditure

Exercise is a crucial part of maintaining a healthy weight. It involves different types of energy expenditure, including work against gravity, which increases our potential energy. The amount of energy burned during such activities can be calculated, as shown in the original exercise with climbing stairs.
To balance out energy intake from foods, one must ensure the energy expended equals or exceeds the energy consumed. Misjudging this balance can lead to unintentional weight gain.
Here are a few important points about exercise and energy:

• Regular exercise helps regulate weight by burning calories.
• Different activities have varying levels of energy conversion, requiring different amounts of calories from your diet to compensate.
• Understanding these concepts can help make informed decisions about diet and exercise regimens to maintain health and fitness goals.

Recognizing how activities translate into energy expenditure clarifies the realistic impact of exercise on our daily calorie balance.