Problem 69 - College Physics

Question

You have no doubt noticed that you usually shiver when you get out of the shower. Shivering is the body's way of generating heat to restore its internal temperature to the normal \(37^{\circ} \mathrm{C}\). and it produces approximately \(290 \mathrm{~W}\) of heat power per square meter of body area. A \(68 \mathrm{~kg}(150 \mathrm{lb}), 1.78 \mathrm{~m}(5 \mathrm{ft}, 10\) in.) person has approximately \(1.8 \mathrm{~m}^{2}\) of surface area. How long would this person have to shiver to raise his or her body temperature by \(1.0 \mathrm{C}^{\circ},\) assuming that none of this heat is lost by the body? The specific heat of the body is about \(3500 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K}).\)

Step-by-Step Solution

Answer

The person has to shiver for about 7.6 minutes.

Step 1: Calculate the Total Heat Needed

To calculate the total heat required to raise the body temperature by 1.0°C, use the formula \( Q = mc\Delta T \). Where \( m \) is the mass (68 kg), \( c \) is the specific heat capacity (3500 J/kg·K), and \( \Delta T \) is the temperature change (1.0°C or 1.0 K, since the change in Celsius and Kelvin is the same).\[ Q = 68 \times 3500 \times 1.0 = 238,000 \text{ J} \]

Step 2: Calculate the Heat Power Produced

Now, calculate the total heat power produced by the person shivering. The heat power generated per square meter is 290 W. If the surface area is 1.8 m², the total power is:\[ P = 290 \times 1.8 = 522 \text{ W} \]

Step 3: Calculate Time Required to Generate Required Heat

To find out how long it takes to generate the required amount of heat, use the formula \( Q = Pt \), where \( P \) is power and \( t \) is time. Rearrange the formula to solve for time:\[ t = \frac{Q}{P} \]Substitute the known values:\[ t = \frac{238,000}{522} \approx 456 \text{ seconds} \]

Step 4: Convert Time to Minutes

Convert the time from seconds to minutes by dividing by 60:\[ t \approx \frac{456}{60} \approx 7.6 \text{ minutes} \]

Key Concepts

Specific Heat Capacity

Understanding specific heat capacity is crucial when studying thermodynamics. It's the measure of how much heat energy is needed to raise the temperature of one kilogram of a substance by one degree Celsius (or one Kelvin). For example, the specific heat capacity of the human body is about 3500 J/(kg·K).
This means that 3500 joules of energy is needed to raise the temperature of one kilogram of body mass by 1°C.
Different substances have different specific heat capacities, affecting how quickly they heat up or cool down.

• Water, for example, has a high specific heat capacity, meaning it can absorb a lot of heat without a significant change in temperature.
• This property plays a key role in energy transfer and temperature regulation in many biological and physical processes.

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a cooler one. In this context, when you're shivering, your body is transferring internal energy to keep warm.
There are three primary methods of heat transfer:

• Conduction is direct heat transfer between substances in contact, such as your feet on cold tiles.
• Convection involves the movement of heat through fluids, like the warmth of a hot drink spreading in your stomach.
• Radiation transfers heat via electromagnetic waves, such as sunlight warming your skin.

These processes explain how energy is distributed, affecting how quickly or slowly an item heats up. Your body uses these methods to maintain its internal temperature efficiently.

Energy Conservation

Energy conservation in thermodynamics deals with how energy is neither created nor destroyed, only transformed or transferred.
In the exercise, you generate heat through shivering, converting chemical energy from your body into thermal energy.

• This ensures energy remains in balance, following the first law of thermodynamics.
• The heat produced by shivering is used to raise your body temperature.
• Through energy conversion, your body's mechanisms work to maintain homeostasis, even in cooling conditions.

By understanding energy conservation, it becomes clear how dynamically energy flows through different systems, helping organisms adapt to various temperatures.

Body Temperature Regulation

Body temperature regulation is a vital process to ensure the body works correctly. The average human body temperature is around 37°C, and it is crucial to maintain it within a narrow range.
When you step out of a shower and shiver, your body responds by generating heat to counteract heat loss and maintain this stable internal environment.

• Shivering is an involuntary response that generates heat through muscle activity.
• Other body responses include altering blood flow, adjusting metabolic heat production, and employing clothing or behavior to conserve heat.
• These mechanisms ensure the body's core temperature remains stable, safeguarding vital functions.

Understanding how the body regulates temperature provides insight into how it maintains an energy balance through thermoregulation processes.