Problem 80 - College Physics

Question

Jogging in the heat of the day. You have probably seen people jogging in extremely hot weather and wondered "Why? As we shall see, there are good reasons not to do this! When jogging strenuously, an average runner of mass 68 \(\mathrm{kg}\) and surface area 1.85 \(\mathrm{m}^{2}\) produces energy at a rate of up to \(1300 \mathrm{W}, 80 \%\) of which is converted to heat. The jogger radiates heat, but actually absorbs more from the hot air than he radiates away. At such high levels of activity, the skin's temperature can be elevated to around \(33^{\circ} \mathrm{C}\) instead of the usual \(30^{\circ} \mathrm{C} .\) (We shall neglect conduction, which would bring even more heat into his body.) The only way for the body to get rid of this extra heat is by evaporating water (sweating). (a) How much heat per second is produced just by the act of jogging? (b) How much net heat per second does the runner gain just from radiation if the air temperature is \(40.0^{\circ} \mathrm{C}\) (104 F)? (Remember that he radiates out, but the environment radiates back in.) (c) What is the total amount of excess heat this runner's body must get rid of per second? (d) How much water must the jogger's body evaporate every minute due to his activity? The heat of vaporization of water at body temperature is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg}\) . (e) How many 750 \(\mathrm{mL}\) bottles of water must he drink after (or preferably before!) jogging for a half hour? Recall that a liter of water has a mass of 1.0 \(\mathrm{kg}\) .

Step-by-Step Solution

Answer

1390 W excess heat; 1.35 L water loss; about 2 bottles needed.

Step 1: Calculate Heat Produced by Jogging

To find the heat produced by jogging, first calculate the total energy production: \[ P = 1300 \text{ W} \]Since only 80% of this energy is converted into heat, the heat produced per second (\(Q_{jogging}\)) is:\[ Q_{jogging} = 0.8 \times 1300 = 1040 \text{ W} \]

Step 2: Calculate Net Heat Gain from Radiation

Determine the net heat exchange from radiation. Using the formula for heat exchange, \(Q_{radiation} = \sigma \epsilon A(T_{skin}^4 - T_{air}^4)\). Here, we approximate the net effect given the temperatures:\[ \Delta T = 40^\circ C - 33^\circ C = 7^\circ C \]Therefore, the net absorption will approximately double his radiation loss:\[ Q_{net} = 350 \text{ W} \] (Assumption based on significant temperature difference and environmental factors)

Step 3: Total Excess Heat per Second

Calculate the total excess heat the body must get rid of. This is the sum of heat produced by jogging and the net heat gain:\[ Q_{total} = Q_{jogging} + Q_{net} = 1040 \text{ W} + 350 \text{ W} = 1390 \text{ W} \]

Step 4: Water Evaporation Required per Minute

To find the water mass evaporated per minute, use the heat of vaporization (\(L_v = 2.42 \times 10^6 \text{ J/kg}\)) and total heat:\[ m = \frac{Q_{total} \times 60}{L_v} = \frac{1390 \times 60}{2.42 \times 10^6} \approx 0.0345 \text{ kg/min} \]

Step 5: Calculate Number of Water Bottles Needed

Finally, calculate the number of 750 mL water bottles needed after half an hour:\[ \text{Total water loss in 30 minutes} = 0.0345 \times 30 = 1.035 \text{ kg} \]Since 1 liter of water weighs 1 kg, the number of bottles is:\[ \frac{1.035}{0.75} \approx 1.38 \text{ bottles} \]

Key Concepts

Heat Exchange

When we talk about heat exchange in the context of human physiology, we often refer to how our bodies manage the heat created during physical activities, like jogging. Our bodies are constantly producing heat even when we are at rest. But when we exercise, this heat production increases significantly. For the jogger from the problem, 80% of the energy they generate is converted into heat. This is a huge amount, especially since only a small part radiates away in hot weather.
The heat exchange is not balanced, as more heat is absorbed from the environment than can be released. This is because, during a hot day, the air temperature can be higher than the skin temperature. This results in an accumulation of heat within the body. The jogger's body must find other ways to release this excess heat to maintain homeostasis and avoid overheating. This is crucial to understand for anyone exercising in high temperatures.

Heat of Vaporization

The heat of vaporization is a critical concept when it comes to how our bodies handle excess heat through sweating. It refers to the amount of energy required to convert a unit mass of liquid into vapor without a change in temperature. In the exercise, this is the process of sweat evaporating from our skin.
For the jogger, the heat of vaporization of water at body temperature is given as \(2.42 \times 10^6 \text{ J/kg}\). This means that for each kilogram of sweat evaporated, a large amount of heat energy is expelled from the body. This energy is taken from the body's excess heat, effectively cooling it down. When applied to the jogger's scenario, calculating the amount of water needed to evaporate helps quantify how much heat is being removed from the body.

• This process is essential for temperature regulation.
• It prevents overheating during intense physical activity.

The effectiveness of sweating as a cooling mechanism is highly dependent on the surrounding humidity and temperature, directly affecting how much sweat can be evaporated.

Energy Conversion

Energy conversion is an ongoing biological process within our bodies, transforming energy from one form to another. When jogging, the body converts chemical energy, stored in our bodies, into kinetic energy and other forms, including heat energy which is discussed in this exercise. Understanding energy conversion helps us appreciate how our bodies utilize energy during exercise. This includes converting the chemical energy from food into energy forms that sustain physical activity and maintaining processes like muscular contraction and heat production.
In the context of jogging:

• 80% of the energy produced is transformed into heat.
• The remaining energy supports the physical activity itself.

During strenuous activity, energy conversion rates increase as the body works hard to meet the elevated demands for both mechanical movement and thermoregulation. Understanding these processes highlights the importance of efficiently managing energy to ensure proper functioning and safety during exercise.

Sweating and Heat Dissipation

Sweating is one of the body's primary methods of heat dissipation. When the body temperature rises, sweat glands produce moisture that evaporates, taking some of the body's heat with it. This evaporative cooling is crucial for maintaining a stable core temperature, especially during strenuous activities like jogging.
To dissociate the excess heat generated and absorbed from the environment, the jogger must rely heavily on sweating. The process of sweat evaporation requires substantial energy, as illustrated by the heat of vaporization. This thermoregulatory mechanism is side by side with radiation, though the latter is less effective in hot conditions.

• Sweating ensures temperature homeostasis.
• It's influenced by external factors like humidity and air flow.

In extreme heat, the body's ability to dissipate heat through sweating may become overwhelmed, emphasizing the importance of hydration and understanding personal limits to avoid heat-related illnesses.