Chapter 7: Problem 41
What is the de Broglie wavelength, in \(\mathrm{cm},\) of a \(12.4-\mathrm{g}\) hummingbird flying at \(1.20 \times 10^{2} \mathrm{mph} ?(1\) mile \(=\) \(1.61 \mathrm{~km} .)\)
Short Answer
Expert verified
The de Broglie wavelength of the hummingbird, in cm, is \( \lambda \times 100\).
Step by step solution
01
Convert weight to kilograms
Firstly, the weight of the hummingbird, given in grams, should be converted to kilograms since the SI unit of weight is kilogram. We transform it into kilograms by multiplying by a factor of \(10^{-3}\). So, the weight in kilograms is \(12.4 * 10^{-3} = 0.0124 \: kg\).
02
Convert speed to m/s
Similarly, the speed must be converted to meters per second. We do that by the following transformations: \[1.20 \times 10^{2}\: mph \times \frac{1.61\: km}{mile} \times \frac{1000\: m}{km} \times \frac{1\: hr}{3600\: s} = 53.64444\: m/s\]. Thus, the speed in m/s is 53.64444.
03
Compute Momentum
The momentum, denoted by 'p', of an object is given by the equation \(p = mass * velocity\). We substitute the mass and velocity calculated in the previous steps, so \(p = 0.0124\: kg * 53.64444\: m/s = 0.66555136\: kg.m/s\).
04
Compute de Broglie wavelength
The de Broglie wavelength is given by the formula \(\lambda = h/p\), where \(h\) is Planck's constant. The accepted value for Planck's constant is \(6.62607015 \times 10^{-34}\: m^2 kg/s\). Thus, we substitute \(h\) and \(p\) into the formula, so \(\lambda = 6.62607015 \times 10^{-34} / 0.66555136\).
05
Convert wavelength to cm
The de Broglie wavelength is given in meters and the problem asked for it in cm. Convert it by multiplying by 100. Thus, the de Broglie wavelength in cm is \( \lambda \times 100\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Momentum Calculation
The concept of momentum is fundamental in physics, especially when studying motion and its properties. Momentum, usually denoted by the letter 'p', is the product of the mass 'm' of an object and its velocity 'v'. The formula is beautifully succinct:
\[\begin{equation}P = m \times v\end{equation}\]
Understanding how to calculate momentum is essential, as it allows us to predict the subsequent motion of objects after they interact. In classical mechanics, when a hummingbird flies, its momentum can be calculated using this equation.
Imagine our hummingbird from the exercise. It has a mass and a speed, which, when multiplied together, give us the momentum of the bird in flight. For instance, with a mass of 0.0124 kg and a velocity of 53.64444 m/s, the momentum is computed simply as the multiplication of these two numbers, providing us with a snapshot of the bird's state of motion at that instance. Thus, for many problems in physics, knowing how to calculate this momentum paves the way to understand more complex phenomena, such as collisions or the properties of waves associated with particles, as described by the de Broglie hypothesis.
\[\begin{equation}P = m \times v\end{equation}\]
Understanding how to calculate momentum is essential, as it allows us to predict the subsequent motion of objects after they interact. In classical mechanics, when a hummingbird flies, its momentum can be calculated using this equation.
Imagine our hummingbird from the exercise. It has a mass and a speed, which, when multiplied together, give us the momentum of the bird in flight. For instance, with a mass of 0.0124 kg and a velocity of 53.64444 m/s, the momentum is computed simply as the multiplication of these two numbers, providing us with a snapshot of the bird's state of motion at that instance. Thus, for many problems in physics, knowing how to calculate this momentum paves the way to understand more complex phenomena, such as collisions or the properties of waves associated with particles, as described by the de Broglie hypothesis.
Unit Conversion
In physics, precise calculations often involve converting units from one system to another. This can be crucial when calculating physical quantities because formulas are typically defined using specific units. When units aren't correctly converted, it can lead to errors, which can significantly affect the outcome of scientific experiments and problems, like the de Broglie wavelength calculation.
To facilitate conversion between units, one should have a basic understanding of the relationships between different units of measurement. For example, knowing that 1 mile equals 1.61 kilometers, or that 1 hour consists of 3600 seconds, is vital for converting miles per hour to meters per second. It's a matter of multiplying or dividing by conversion factors: the numerical values that relate one unit to another. These factors are used sequentially to transform the original unit into the desired one, like a chain of stepping stones across a stream. In our hummingbird example, we converted the bird's speed from miles per hour to meters per second, which is useful because the SI units are more widely used in scientific formulas.
To facilitate conversion between units, one should have a basic understanding of the relationships between different units of measurement. For example, knowing that 1 mile equals 1.61 kilometers, or that 1 hour consists of 3600 seconds, is vital for converting miles per hour to meters per second. It's a matter of multiplying or dividing by conversion factors: the numerical values that relate one unit to another. These factors are used sequentially to transform the original unit into the desired one, like a chain of stepping stones across a stream. In our hummingbird example, we converted the bird's speed from miles per hour to meters per second, which is useful because the SI units are more widely used in scientific formulas.
Planck's Constant
Planck's constant, denoted by 'h', is a crucial quantity in quantum mechanics. It relates the energy of a photon to the frequency of its associated electromagnetic wave and is a key component in the equation defining the momentum of particles on the quantum scale. The precise value of Planck's constant is
\[\begin{equation}h = 6.62607015 \times 10^{-34} m^2 kg / s\end{equation}\]
This tiny number plays a gigantic role in the realm of the very small. For instance, it's central to the de Broglie hypothesis, which posits that particles like electrons behave like waves, with wavelengths inversely proportional to their momentum. In practical terms, such as in our hummingbird's case, Planck's constant enables us to calculate the de Broglie wavelength—essentially the 'wave nature'—of any moving object, no matter how large or small, by dividing the constant by the object's momentum. It's this linkage between wave phenomena and particles—where 'h' is the bridge—that cemented the strange and wonderful dual nature of reality as illuminated by quantum physics.
\[\begin{equation}h = 6.62607015 \times 10^{-34} m^2 kg / s\end{equation}\]
This tiny number plays a gigantic role in the realm of the very small. For instance, it's central to the de Broglie hypothesis, which posits that particles like electrons behave like waves, with wavelengths inversely proportional to their momentum. In practical terms, such as in our hummingbird's case, Planck's constant enables us to calculate the de Broglie wavelength—essentially the 'wave nature'—of any moving object, no matter how large or small, by dividing the constant by the object's momentum. It's this linkage between wave phenomena and particles—where 'h' is the bridge—that cemented the strange and wonderful dual nature of reality as illuminated by quantum physics.
