Problem 72 - College Physics

Question

\(\bullet\) \(\bullet\) Maximum sustainable human power. The maximum sustainable mechanical power a human can produce is about \(\frac{1}{3}\) hp. How many food calories can a human burn up in an hour by exercising at this rate? (Remember that only 20\(\%\) of the food energy used goes into mechanical energy.)

Step-by-Step Solution

Answer

A person can burn about 1069 food calories per hour at this exercise rate.

Step 1: Convert Horsepower to Watts

The first step is to convert horsepower (hp) to watts, as we need the power in standard units to perform calculations. Given that 1 hp = 746 watts, we can find that \(\frac{1}{3}\) hp equals:\[\frac{1}{3} \times 746 = 248.67 \text{ watts}\] This means the maximum sustainable power is 248.67 watts.

Step 2: Calculate Total Energy Used in Watts

Only 20% of the energy from food goes into mechanical energy. To find the total energy used, we set up the equation where mechanical energy is 20% of total energy: \[ 248.67 = 0.2 \times \text{Total Energy} \]Thus, solving for Total Energy gives:\[ \text{Total Energy} = \frac{248.67}{0.2} = 1243.35 \text{ watts}\] This is the total energy consumed per second.

Step 3: Convert Energy to Calories per Hour

Now we need to convert the energy from watts (joules/second) to food calories for an hour of exercise. First, convert watts to joules per second: \[ 1243.35 \text{ watts} = 1243.35 \text{ joules/second} \]Then, calculate for one hour (3600 seconds):\[ 1243.35 \times 3600 = 4,476,060 \text{ joules} \]Finally, convert joules to calories, noting that 1 food calorie = 4184 joules:\[ \frac{4,476,060}{4184} \approx 1069 \text{ food calories} \] Therefore, approximately 1069 food calories can be burned per hour at this rate.

Key Concepts

Unit conversion

Converting units is an essential part of solving many physics problems. When dealing with mechanical power, it is often measured in horsepower (hp), especially for machinery or engines. However, for scientific calculations, it's crucial to convert this power into watts because watts are the standard unit of power.

• 1 horsepower is equivalent to 746 watts.

To convert horsepower to watts, you multiply the value by 746. In our example, \[ \frac{1}{3} \text{ hp} = \frac{1}{3} \times 746 = 248.67 \text{ watts}. \] Understanding unit conversions ensures precision in calculations and helps maintain consistency across different scientific measurements.

Mechanical power

Mechanical power refers to the rate at which work is done or energy is transferred. Humans can generate mechanical power through activities such as cycling, running, or lifting objects.
In our exercise, the maximum sustainable mechanical power a person can exert is given as \(\frac{1}{3}\) hp (horsepower), which we calculated as 248.67 watts.

• Mechanical power is calculated as work done over time.
• The efficiency of energy conversion influences how much power can be sustained over time.

Knowing the mechanical power helps us understand how much work can be continuously performed by a person and what energy input is required.

Sustainable power

Sustainable power is the maximum power output that a person can maintain over a prolonged period without exhaustion. In everyday terms, it’s what you can manage while going for a long run or bike ride.
Sustainable power is only a fraction of the peak power a person can achieve for brief moments. For humans, about \(\frac{1}{3}\) hp is a manageable level for continuous exertion.
This concept is linked to endurance and energy efficiency.

• Understanding sustainable power is vital for athletes aiming to optimize performance.
• It's crucial for designing exercise programs tailored to individual energy outputs.

Recognizing this balance helps in planning sustained physical activities without overexertion.

Energy consumption

Energy consumption refers to how much energy is used by the body to perform physical activities. When calculating energy consumption, it is important to consider the mechanical efficiency of the body.
In our problem, only 20% of the food energy consumed is converted to mechanical energy. This means to find the total energy utilized, we solve:\[ 248.67 \text{ watts} = 0.2 \times \text{Total Energy} \]This equation is rearranged to get the total energy expenditure:\[ \text{Total Energy} = \frac{248.67}{0.2} = 1243.35 \text{ watts per second}.\]Efficiency plays a significant role in energy consumption. Not all energy from food is used effectively, and a lot is lost as heat.

Food calories conversion

Converting energy from food calories to joules helps quantify how much energy is consumed and used during physical activities. This conversion is crucial since dietary energy is measured in food calories while scientific energy is measured in joules.

• 1 food calorie equals 4184 joules.

Once we calculate the energy in joules of an activity lasting an hour, we convert it to food calories to understand the nutritional needs:\[ 4,476,060 \text{ joules} \div 4184 \approx 1069 \text{ food calories}.\]By understanding food calorie conversion, individuals can balance their energy intake with their physical output, aiding in effective nutrition and exercise planning.