Problem 1 - Applied Physics

Question

Find the period of a wave whose frequency is \(50 \overline{0} \mathrm{~Hz}\).

Step-by-Step Solution

Answer

The period of the wave is 0.02 seconds.

Step 1: Understanding the relationship

The period of a wave \(T\) is the reciprocal of its frequency \(f\). This means that the formula to calculate the period is given by \(T = \frac{1}{f}\). To proceed, we need to apply this relationship.

Step 2: Substitute the given frequency

The frequency given is \(50 \overline{0} \, \text{Hz}\), which is simply \(50 \, \text{Hz}\) when expressed as a regular number. We'll substitute \(f = 50\) into the formula for the period. So, \(T = \frac{1}{50}\).

Step 3: Perform the calculation

Now calculate the value of the period \(T\) by dividing 1 by 50. This gives \(T = \frac{1}{50} = 0.02 \, \text{seconds}\). This is the period of the wave in seconds.

Key Concepts

Frequency

Frequency is a fundamental concept in wave mechanics and describes how often a wave's crests pass a fixed point in a period of time, usually measured in hertz (Hz). One hertz means one cycle per second. When you encounter a wave with a frequency of 50 Hz, like in our original problem, it indicates that 50 wave cycles occur each second.

• The higher the frequency, the more cycles occur each second.
• The frequency is a vital parameter as it determines the energy and other characteristics of the wave.

Frequency is used in various applications, such as in radios where different stations are found at different frequencies, or in light where different colors correspond to different frequencies.

Reciprocal Relationship

The concept of reciprocal relationship is crucial when dealing with wave mechanics, particularly when calculating the period of a wave. The period (T) and frequency (f) of a wave are reciprocals of each other. This means that multiplying the frequency by the period equals 1: \[ T \times f = 1 \]Using this elegant mathematical property allows us to switch between understanding wave frequency and wave period with ease.

• If you know the frequency, you can easily find the period using the formula: \( T = \frac{1}{f} \).
• Conversely, if you have the period, you can find the frequency with: \( f = \frac{1}{T} \).

This relationship helps in many practical situations, like timing cycles in electronic circuits or understanding acoustic vibrations.

Wave Mechanics

Wave mechanics is a branch of physics that studies how waves behave and interact with the environment and is fundamental for understanding many technological and natural processes. Waves can be found in various forms like sound waves, light waves, and water waves. Each of these waves carries energy from one place to another.

• Waves have amplitude, wavelength, frequency, and velocity as their defining properties.
• The interplay of these properties determines the characteristics and behavior of the wave.

The principles of wave mechanics explain how different mediums and conditions affect wave propagation, such as how sound travels faster through solids than through air.

Calculation of Period

Calculating the period of a wave is a straightforward process once you understand its direct relationship with frequency. The period is the duration of time it takes for one complete cycle of a wave to pass a point. In the original problem, you are asked to find the period for a wave with a frequency of 50 Hz.

• First, recognize the reciprocal relationship: the period is the inverse of the frequency.
• Using the formula \(T = \frac{1}{f}\), substitute the given frequency to find \(T = \frac{1}{50} = 0.02\) seconds.
• This result shows that each wave cycle takes 0.02 seconds to complete.

Understanding this calculation helps in designing systems that rely on periodic signals, such as clocks and communication devices.