Problem 39 - College Physics

Question

\(\bullet\) \(\bullet\) Food calories. The food calorie, equal to \(4186 \mathrm{J},\) is a measure of how much energy is released when food is metabo- lized by the body. A certain brand of fruit-and-cereal bar con- tains 140 food calories per bar. (a) If a 65 \(\mathrm{kg}\) hiker eats one of these bars, how high a mountain must he climb to "work off" the calories, assuming that all the food energy goes only into increasing gravitational potential energy? (b) If, as is typical, only 20\(\%\) of the food calories go into mechanical energy, what would be the answer to part (a)? (Note: In this and all other problems, we are assuming that 100\(\%\) of the food calories that are eaten are absorbed and used by the body. This actually not true. A person's "metabolic efficiency" is the percentage of calories eaten that are actually used; the rest are eliminated by the body. Metabolic efficiency varies considerably from person to person.)

Step-by-Step Solution

Answer

(a) 920 meters; (b) 184 meters.

Step 1: Understanding Given Values

We know from the problem that:- 1 food calorie = 4186 Joules.- The energy of one bar = 140 food calories.Thus, the energy in joules for one bar is:\[ E = 140 \times 4186 \text{ J} \].The mass of the hiker is 65 kg. We need to find how high the hiker must climb to work off this energy purely through gravitational potential energy and also consider that only 20% of food energy is used in mechanical work.

Step 2: Calculate Total Energy from the Bar

First, convert the 140 food calories into joules. \[ 140 \text{ calories} \times 4186 \text{ J/calorie} = 586040 \text{ J} \].This is the total energy provided by the bar.

Step 3: Calculate Height for Total Energy Consumption

The potential energy gained by climbing a height \( h \) is given by \( E = mgh \), where \( m = 65 \text{ kg} \) and \( g = 9.8 \text{ m/s}^2 \). Rearrange to find the height: \[ h = \frac{E}{mg} = \frac{586040}{65 \times 9.8} \].Calculate:\[ h = \frac{586040}{637} \approx 920 \text{ meters} \].

Step 4: Consider Efficiency of 20%

In most realistic scenarios, only 20% of food energy converts to mechanical energy. So, the effective energy used for climbing is:\[ E_{\text{mech}} = 0.2 \times 586040 = 117208 \text{ J} \].

Step 5: Calculate Height Considering Efficiency

With the usable energy for mechanical work, recalculate height:\[ h_{\text{eff}} = \frac{E_{\text{mech}}}{mg} = \frac{117208}{65 \times 9.8} \].Calculate:\[ h_{\text{eff}} = \frac{117208}{637} \approx 184 \text{ meters} \].

Key Concepts

Calories

Calories are a measure of energy. When we talk about food calories, we are referring to the amount of energy stored in food that can be used by the body for various activities. A food calorie, often written with a capital 'C' (Cal), is equal to 4186 joules. For instance, a fruit-and-cereal bar containing 140 food calories actually contains an energy equivalent of 586,040 joules.

This energy from calories is essential for our bodies to perform physical activities, maintain metabolic processes, and maintain a stable temperature. In the context of our exercise, we are interested in how much energy is used to climb a mountain. By consuming calories, the body converts this energy into forms that can be used to do work, such as climbing.

Thus, calories are not just about weight gain or loss. They are a unit of energy that quantifies how much work the body can potentially perform. Knowing how to track these units helps in understanding energy intake and expenditure.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses because of its position in a gravitational field. In simpler terms, it is the energy gained by an object as it is lifted against gravity. The formula to calculate gravitational potential energy is given by: \[ E = mgh \] where:

• \( E \) - Gravitational potential energy (in joules)
• \( m \) - Mass of the object (in kilograms)
• \( g \) - Acceleration due to gravity (9.8 m/s² on Earth's surface)
• \( h \) - Height above the reference point (in meters)

In the exercise, the hiker converts the energy from the food bar into gravitational potential energy by climbing. The energy from the food bar allows the hiker to elevate his potential energy by moving to a higher altitude. Calculating the height helps us understand the practical use of energy calories in terms of the energy needed to elevate his body mass by a certain height.
The height calculated using the kinetic energy was 920 meters with 100% efficiency and 184 meters considering 20% efficiency of energy conversion.

Metabolic Efficiency

Metabolic efficiency refers to how effectively the human body converts ingested food energy into usable energy for activities. It is expressed as a percentage indicating the proportion of caloric intake used for physical work or metabolic processes. Not all the energy consumed from food is converted into mechanical energy or used entirely by the body.

In the given problem, only 20% of the food's energy is used in mechanical work to climb. This means that if the body could perfectly utilize all the caloric energy (100% efficient), the potential movement energy would be much greater. However, real-world biology limits this efficiency, usually due to

• energy losses in maintaining bodily functions,
• heat production, and
• incomplete digestion and absorption of food.

Metabolic efficiency can vary greatly among individuals due to genetic factors, level of physical fitness, and metabolic health. Understanding this concept helps individuals manage energy balance more effectively because it highlights the difference between caloric intake and physical performance output.

Mechanical Energy Conversion

Mechanical energy conversion is the process of transforming energy stored in food into mechanical work. In our daily activities, the body converts chemical energy from food into work through metabolic processes. This can include activities such as walking, running, or climbing.

In physics, mechanical energy comprises two main types: kinetic energy and potential energy. During a climb, a hiker's body mainly converts food energy into gravitational potential energy, a type of mechanical energy. For example, the available energy from a food bar is partially converted into elevating the hiker’s body mass against gravity.

The efficiency of this conversion is critical, as only a fraction of the consumed food energy is usually converted into actual work. In the given problem, 20% efficiency means that out of all the calories consumed, only a portion contributes to climbing the height of a mountain, while the rest is dissipated as heat or used for other bodily functions.

• This demonstrates the difference between intake energy and performing energy in physical activities.
• It also provides a tangible understanding of how we may feel energetic yet still have energy losses.

Efficient mechanical energy conversion maximizes performance while minimizing waste, which is important for activities like exercise and endurance sports.