Problem 48
Question
Wing Beat Frequency in Hummingbirds Altshuler and Dudley (2003) found that hummingbirds' wing beat frequency, \(f\), decreases with body mass, \(m\), according to $$ f=40-\frac{8}{5} m $$ where \(f\) is measured in beats per second and \(m\) in grams. Assume that the amount of thrust that a flying hummingbird can generate depends on its mass and wing beat frequency as follows $$ T=c f^{2} m^{4 / 3} $$ for some positive constant \(c\). (This equation is derived from the thrust mechanics of a moving wing.) (a) Equation (4.5) should only be used if \(m<25\). Why? (The largest hummingbirds have a mass of \(22 \mathrm{~g}\).) (b) Calculate \(d T / d m .\) (c) Show by plotting \(d T / d m\) that for larger hummingbirds, \(T\) decreases with \(m\). That is, show that \(d T / d m<0\), once \(m\) exceeds a critical threshold. (d) For a hummingbird to fly, its thrust must exceed its weight \(W=m g\) (where \(g\) is the acceleration due to gravity). Explain, using your answer to (d), why if the hummingbird's mass increases, then \(T\) will eventually be smaller than \(W\) (Hint: \(d W / d m=g\) is constant and positive), no matter what the value of \(c\) is.
Step-by-Step Solution
Verified Answer
(a) Use is valid for \(m < 25\) to keep \(f > 0\). (b) Find derivative showing \( \frac{dT}{dm}\). (c) \(T\) decreases as \(m\) increases beyond a threshold. (d) \(T < W\) eventually for large \(m\) when \( \frac{dW}{dm} = g \) is constant and \( \frac{dT}{dm} < 0 \).
1Step 1: Determine conditions for Equation 4.5
Equation (4.5) suggests a linear relationship between wing beat frequency \( f \) and mass \( m \), where \( f = 40 - \frac{8}{5}m \). To ensure that \( f > 0 \), since a negative frequency is not physically meaningful, solve for \( m < 25 \) to maintain a positive frequency.
2Step 2: Expression for T
Given the thrust formula \( T = c f^2 m^{4/3} \), substitute the expression for \( f \) from Step 1 into this equation:\[ T = c \left(40 - \frac{8}{5}m\right)^2 m^{4/3}. \]
3Step 3: Differentiate T with respect to m
To find \( \frac{dT}{dm} \), use the chain rule and product rule on the expression for \( T \):\( T = c \left(40 - \frac{8}{5}m\right)^2 m^{4/3} \).Differentiate with respect to \( m \):\[ \frac{dT}{dm} = c \left[ 2\left(40 - \frac{8}{5}m\right)\left(-\frac{8}{5}\right)m^{4/3} + \left(40 - \frac{8}{5}m\right)^2 \frac{4}{3}m^{1/3}\right]. \]
4Step 4: Analyze the derivative
To determine when \( \frac{dT}{dm} < 0 \), analyze the critical points and constant factors in the derivative derived in Step 3. Observe that as \( m \) increases past a certain value, the negative contribution becomes dominant, making \( \frac{dT}{dm} < 0 \), indicating that \( T \) decreases as \( m \) increases.
5Step 5: Comparison with weight
The weight of a hummingbird is given by \( W = mg \). As \( m \) increases, \( \frac{dW}{dm} = g \) is positive and constant, while \( \frac{dT}{dm} \) becomes negative as seen in Step 4. Thus, \( T \) eventually falls below \( W \) for sufficiently large \( m \), regardless of the value of \( c \).
Key Concepts
DifferentiationThrust mechanicsMathematical modelingLinear relationships
Differentiation
Differentiation is a powerful mathematical tool used to understand how a quantity changes with respect to another. In the context of hummingbirds and their wing beat mechanics, differentiation helps us analyze how their thrust, represented by the equation \( T = c f^2 m^{4/3} \), changes as their body mass \( m \) varies. To calculate the derivative of \( T \) with respect to \( m \), we apply rules like the chain rule and product rule.
1. **Chain Rule**: This is used when different functions are nested together. For example, in our thrust equation, \( f \) is a function of \( m \). Thus, finding \( \frac{dT}{dm} \) involves differentiating \( f \) with respect to \( m \). 2. **Product Rule**: This rule is applied when differentiating products of functions. Since \( T \) involves multiplying \( f^2 \) and \( m^{4/3} \), the product rule helps us split and differentiate these terms effectively.
By utilizing these rules, differentiation allows us to examine the exact rate of change of thrust with mass, crucial for understanding a hummingbird's flight efficiency as its weight changes.
1. **Chain Rule**: This is used when different functions are nested together. For example, in our thrust equation, \( f \) is a function of \( m \). Thus, finding \( \frac{dT}{dm} \) involves differentiating \( f \) with respect to \( m \). 2. **Product Rule**: This rule is applied when differentiating products of functions. Since \( T \) involves multiplying \( f^2 \) and \( m^{4/3} \), the product rule helps us split and differentiate these terms effectively.
By utilizing these rules, differentiation allows us to examine the exact rate of change of thrust with mass, crucial for understanding a hummingbird's flight efficiency as its weight changes.
Thrust mechanics
Thrust mechanics in the study of hummingbirds involves understanding how the force generated by their wings propels them in flight. The given equation \( T = c f^2 m^{4/3} \) represents this thrust.
- **Thrust (\( T \))**: This is the force that drives a hummingbird forward or upward during flight. It must be greater than or equal to the bird's weight for it to remain airborne.
- **Wing Beat Frequency (\( f \))**: This is how fast the bird flaps its wings, measured in beats per second. It directly influences the thrust because a more frequent wing beat generates higher force.
- **Mass (\( m \))**: This denotes the body weight of the hummingbird. The relationship shows that for a given wing beat frequency, increasing mass increases the thrust to a point, after which other factors must be considered.
Understanding the interaction between mass, wing beat frequency, and thrust is crucial in assessing how variations in these factors affect a hummingbird's ability to fly.
- **Thrust (\( T \))**: This is the force that drives a hummingbird forward or upward during flight. It must be greater than or equal to the bird's weight for it to remain airborne.
- **Wing Beat Frequency (\( f \))**: This is how fast the bird flaps its wings, measured in beats per second. It directly influences the thrust because a more frequent wing beat generates higher force.
- **Mass (\( m \))**: This denotes the body weight of the hummingbird. The relationship shows that for a given wing beat frequency, increasing mass increases the thrust to a point, after which other factors must be considered.
Understanding the interaction between mass, wing beat frequency, and thrust is crucial in assessing how variations in these factors affect a hummingbird's ability to fly.
Mathematical modeling
Mathematical modeling involves creating equations that describe real-world phenomena in a precise mathematical form. In this case, the model for the wing beat frequency and thrust of a hummingbird helps us describe and predict flight dynamics.
- **Equation for Wing Beat Frequency**: The linear equation \( f = 40 - \frac{8}{5}m \) models how wing beat frequency decreases with increasing mass. This model helps predict how changes in a hummingbird's weight affect its flight.
- **Thrust Equation**: \( T = c f^2 m^{4/3} \) is a more complex model that combines effects of frequency and mass to calculate thrust. This equation serves as a crucial element in understanding the required force for flight relative to mass.
Through these models, we can simulate and explore "what-if" scenarios, making predictions about flight capabilities and constraints as hummingbirds vary in size.
- **Equation for Wing Beat Frequency**: The linear equation \( f = 40 - \frac{8}{5}m \) models how wing beat frequency decreases with increasing mass. This model helps predict how changes in a hummingbird's weight affect its flight.
- **Thrust Equation**: \( T = c f^2 m^{4/3} \) is a more complex model that combines effects of frequency and mass to calculate thrust. This equation serves as a crucial element in understanding the required force for flight relative to mass.
Through these models, we can simulate and explore "what-if" scenarios, making predictions about flight capabilities and constraints as hummingbirds vary in size.
Linear relationships
Linear relationships describe a straight-line connection between two variables. In the context of hummingbirds, we observe a linear relationship between wing beat frequency (\( f \)) and mass (\( m \)). The equation \( f = 40 - \frac{8}{5}m \) clearly illustrates this: as mass increases, the frequency decreases in a consistent linear pattern.
- **Slope**: The slope of \(-\frac{8}{5}\) indicates how many beats per second decrease with each additional gram of mass.
- **Intercept**: The intercept at 40 tells us that if a hummingbird weighed zero grams (hypothetically), the frequency would be 40 beats per second.
Understanding this linear relationship helps reveal fundamental interactions between mass and wing motion, essential for determining the dynamic capabilities of a hummingbird in flight.
- **Slope**: The slope of \(-\frac{8}{5}\) indicates how many beats per second decrease with each additional gram of mass.
- **Intercept**: The intercept at 40 tells us that if a hummingbird weighed zero grams (hypothetically), the frequency would be 40 beats per second.
Understanding this linear relationship helps reveal fundamental interactions between mass and wing motion, essential for determining the dynamic capabilities of a hummingbird in flight.
Other exercises in this chapter
Problem 48
Consider the chemical reaction: $$ \mathrm{A}+\mathrm{B} \longrightarrow \mathrm{AB} $$ If \(x\) denotes the concentration of \(\mathrm{AB}\) at time \(t\) and
View solution Problem 48
Find the tangent line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=3 \pi x^{5}-\frac{\pi}{2} x^{3}, \text { at } x=-1 $$
View solution Problem 49
The reaction rate \(R(x)\) of the irreversible reaction $$\mathrm{A}+\mathrm{B} \rightarrow \mathrm{AB}$$ is a function of the concentration \(x\) of the produc
View solution Problem 49
Find the derivative with respect to the independent variable. $$ h(x)=\frac{3}{\tan (2 x)-x} $$
View solution