Problem 84 - University Physics with Modern Physics

Question

All birds, independent of their size, must maintain a power output of 10\(-\)25 watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (\(Patagona gigas\)) has mass 70 g and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A 70-kg athlete can maintain a power output of 1.4 kW for no more than a few seconds; the \(steady\) power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

Step-by-Step Solution

Answer

The hummingbird does 0.1225 J of work per wingbeat. It is impractical for humans to fly by flapping wings.

Step 1: Convert Mass to Kilograms

The mass of the hummingbird is given as 70 grams. To convert this to kilograms, divide by 1000: \(70 \text{ g} = 0.07 \text{ kg}\).

Step 2: Calculate Power Output

Using the given range, let's assume a middle value of 17.5 watts per kilogram for the power output. Therefore, the power output \(P\) for the hummingbird is \(P = 0.07 \text{ kg} \times 17.5 \text{ W/kg} = 1.225 \text{ W}\).

Step 3: Find Work Done in One Second

The power output refers to energy per unit time. Since the hummingbird flaps its wings 10 times per second, the energy expended each second is 1.225 Joules, which is the work done in one second.

Step 4: Calculate Work Done per Wingbeat

Since the hummingbird flaps its wings 10 times per second, divide the total work done in one second by the number of wingbeats to find the work per wingbeat: \(\text{Work per wingbeat} = \frac{1.225 \text{ J}}{10} = 0.1225 \text{ J}\).

Step 5: Calculate Human Power-to-Weight Ratio

The athlete has a mass of 70 kg and can maintain a power output of 1.4 kW for short periods. Convert this to watts per kg by dividing the total power by the mass: \(\frac{1400 \text{ W}}{70 \text{ kg}} = 20 \text{ W/kg} \). The steady power output of 500 W gives \(\frac{500 \text{ W}}{70 \text{ kg}} \approx 7.14 \text{ W/kg} \).

Step 6: Evaluate Human Flight Feasibility

Since the sustained power output per kilogram for humans (7.14 W/kg) is significantly lower than what is required for continuous flapping flight (10-25 W/kg), it is impractical for a human-powered aircraft to fly for extended periods by flapping wings.

Key Concepts

Power Output

In the context of flight, power output is fundamental. This is the rate at which work is done or energy is transferred. For flying creatures, such as birds, power output per kilogram of body mass must fall within a range to sustain flight. For example, the Andean giant hummingbird must maintain a power range between 10 and 25 watts per kilogram.

To understand power output, let's break it down:

• Power (\( P \)) is expressed as watts (W), where 1 watt equals 1 joule per second.
• Formula: Power is calculated as work done over time, \( P = \frac{W}{t} \). For birds, maintaining this energy helps them stay aloft with vigorous, constant wingbeats.

Considering a bird's mass, such as the 70-gram hummingbird, helps determine the total power output needed for flight. By examining this power output, we indirectly understand how much energy a bird expends to remain airborne with each flap of its wings.

Wingbeat Energy Calculation

To understand how much energy is used in a single wingbeat, we focus on the concept of work.

In physics, work is the energy transferred to or from an object via the application of force along a displacement. For the hummingbird, we need to know how much work gets done in each wingbeat when it hovers:

• Given: The bird flaps its wings 10 times per second and expends 1.225 joules of energy per second.
• Formula used: Work per wingbeat can be calculated by dividing total energy expended in one second by the number of wingbeats: \( \text{Work per wingbeat} = \frac{\text{Total work per second}}{\text{Wingbeats per second}} \).

This approach reveals that the hummingbird does \( 0.1225 \) joules of work per wingbeat. Understanding this helps illustrate the demands placed on a small creature during flight, highlighting the efficiency and power of their wing muscle movements.

Human-Powered Flight Feasibility

The possibility of human-powered flight using wing flapping is an intriguing concept, yet faced with practical challenges. The energy required in terms of power output for sustained flight is economically demanding for humans.

Key points to consider include:

• Typical athletes can only maintain a continuous power output of about \( 7.14 \) watts per kilogram, much lower than needed for flight. Birds need around 10-25 watts per kilogram.
• For brief moments, a human can generate a power output similar to birds (around 20 watts per kilogram), but this cannot be sustained for long.

Given this mismatch in energy requirements, it’s clear that, despite our engineering achievements, replicating bird flight patterns using direct human power remains unrealistic. Instead, human-powered aircraft designs focus on steady, power-conserving methods like pedaling propellers, aligning more with our natural endurance capabilities.