Question

Find the frequency (in hertz) and the angular velocity of a repeating waveform whose period is $$1.00 \mathrm{s}$$.

Short answer

The frequency is 1 Hz, and the angular velocity is \(2\pi\) radians per second.

Step-by-step solution

Determine the Frequency

The frequency (f) of a waveform is the reciprocal of the period (T). The period is given as 1.00 s. Therefore, calculate the frequency using the formula: \[ f = \frac{1}{T} \]

Calculate the Frequency

Plug the given period into the formula to find the frequency: \[ f = \frac{1}{1.00 \, \text{s}} = 1 \, \text{Hz} \]

Determine the Angular Velocity

The angular velocity (\(\omega\)) is related to the frequency by the formula: \[ \omega = 2\pi f \]

Calculate the Angular Velocity

Use the calculated frequency to find the angular velocity: \[ \omega = 2\pi \times 1 \, \text{Hz} = 2\pi \, \text{rad/s} \]

Key concepts

Waveform Frequency

The frequency of a waveform reflects how often the wave repeats itself over a unit of time, typically expressed in hertz (Hz). Understanding frequency is crucial in fields such as acoustics, radio communications, and electronics. To visualize this, imagine the regular swells in the ocean as waves; the frequency describes how many swells pass by a fixed point every second. In the provided exercise, the frequency can be calculated simply by taking the reciprocal of the period. A period of 1 second results in a frequency of 1 Hz, meaning the wave completes one full cycle every second.

When learning about frequency, it's important to remember that it indicates not only the rate of oscillation but also helps to determine the pitch of a sound in acoustics or colors in optics. A higher frequency corresponds to a higher pitch or a bluer color.

Angular Velocity Calculation

Angular velocity describes the rate of change of an object's rotational angle per unit of time and is a fundamental concept when dealing with rotating systems — anything from a record player's disc to planets orbiting the sun. It is typically denoted as \(\omega\) and measured in radians per second (rad/s).

The relation between angular velocity and waveform frequency is direct; given the frequency in hertz, one can find angular velocity using the formula \(\omega = 2\pi f\). The factor \(2\pi\) comes from the fact that there are \(2\pi\) radians in a full circle. So, if a waveform has a frequency of 1 Hz, its angular velocity is \(2\pi\) rad/s, suggesting that in every second, the wave's phase progresses by \(2\pi\) radians, which is equivalent to one full cycle. Students should be comfortable with converting between these units to understand rotational dynamics more thoroughly.

Period of Waveform

The period of a waveform, denoted \(T\), defines the duration of time it takes to complete one cycle of the wave. For repeating waveforms, this is a constant value representing the inverse of the frequency; this relationship is symbolized by the equation \(T = \frac{1}{f}\). In other words, if a waveform has a period of 1 second, it means that each wave cycle takes 1 second to complete before starting again.

To grasp this concept, picture the pendulum of a clock as it swings; each complete swing takes a certain amount of time to return to its starting position. That interval is the period of the pendulum's motion. In electronics or physics, understanding the period of a waveform is essential for timing circuits, analyzing signals, or synchronizing systems. This fundamental characteristic of a wave helps to define its behavior and interaction with different mediums.