Q84P - 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 for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

Step-by-Step Solution

Answer

1. The work done are 0.07 J/beat to 0.175 J/beat.
2. No, it is not possible for a human-powered aircraft to fly for extended periods by flapping its wings.

Step 1: Identify the given data

The mass of the bird is $m_b=0.07 kg$.

The minimum power output is $p_{min}=10 \mathrm{watt}/\mathrm{kg}$.

The minimum power output is $p_{max}=25 \mathrm{watt}/\mathrm{kg}$.

The number of flaps per second is N = 10 beat/s.

Step 1: The relationship between the power and work

The relationship between the power and work is given by,

$\mathit{P}\mathbf{=}\frac{\mathbf{W}}{\mathbf{t}}$ 

Here, P, is power, W is work done, and t is time.

Step 3: Determine the amount of work done by such a hummingbird in each wingbeat (a)

Find the minimum power as follows.

                                        $$\begin{gathered} P_{min}=P_{min}\times m_b \\ =\left(10 watt/kg\right)\left(0.07 kg\right) \\ =0.7 watt \end{gathered}$$      

 

Find the maximum power as follows.

                                        $$\begin{gathered} P_{max}=P_{max}\times m_b \\ =\left(25 watt/kg\right)\left(0.07 kg\right) \\ =1.75 watt \end{gathered}$$        

 

Find the work done corresponding to the minimum power as follows.

                                         $$\begin{gathered} W_{min}=\frac{0.7 J/s}{10 beat/s} \\ =0.07 J/beat \end{gathered}$$   

 

Find the work done corresponding to the maximum power as follows.

                                          $$\begin{gathered} W_{max}=\frac{1.75 J/s}{10 beat/s} \\ =0.175 J/beat \end{gathered}$$

 

Therefore, the required work done are 0.07 J/beat, and 0.175 J/beat.

Step 4: Determine the whether it possible for a human-powered aircraft to fly for extended periods by flapping its wings (b)

Power output of the athlete is P = 1.4 kW or P = 1400 W.

 

The Mass of athlete is m = 70 kg.

 

Steady power output of the athlete is P = 500 W.

                                                

 Power output per kilogram of the body mass is given by,

                                               

                                                       $$\begin{gathered} \frac{\mathrm{P}}{\mathrm{M}}=\frac{500 \mathrm{W}}{70 \mathrm{kg}} \\ \approx 7 \mathrm{W}/\mathrm{g} \end{gathered}$$

Since, the power output in order to fly ranges from 10 to 25 Watts, and the value of is P/m less when compared to the range of flying.

Now, the athlete can expend $\frac{1400 \mathrm{W}}{70 \mathrm{kg}}=20 \mathrm{W}/\mathrm{kg}$ for short interval of time. Hence, it is not possible for a human powered aircraft to fly for extended periods by flapping its wings.