Frequency and Period Relationship Explained

The question asks to identify the frequency that has the longest period. To solve this, we need to understand the fundamental relationship between frequency and period in wave physics.

• Frequency (f) is defined as the number of cycles or oscillations per unit of time. It is typically measured in Hertz (Hz), where 1 Hz means one cycle per second.
• Period (T) is the time taken for one complete cycle or oscillation. It is measured in seconds (s).

The relationship between frequency and period is inversely proportional, meaning that if one increases, the other decreases. The mathematical formula connecting them is:

\[ T = \frac{1}{f} \]

Where:

• \( T \) is the period in seconds.
• \( f \) is the frequency in Hertz.

From this formula, we can deduce that to have the longest period (\(T\)), the frequency (\(f\)) must be the smallest. Conversely, a high frequency would result in a short period.

Comparing Frequencies for Longest Period

Let's convert all the given frequency options to a common unit, Hertz (Hz), to easily compare them:

• Option 1: 10 KHz (Kilohertz)
• 1 KHz = 1000 Hz
• So, 10 KHz = \(10 \times 1000\) Hz = 10,000 Hz
• Option 2: 1 KHz (Kilohertz)
• 1 KHz = 1000 Hz
• Option 3: 10 Hz
• Option 4: 1 Hz

Now, let's list the frequencies in Hertz for a clear comparison:

Identifying the Smallest Frequency

From the table above, we can clearly see the magnitudes of the frequencies:

• 10,000 Hz
• 1,000 Hz
• 10 Hz
• 1 Hz

The smallest frequency among these options is 1 Hz.

Conclusion on Longest Period

Since the period is inversely proportional to the frequency, the smallest frequency will correspond to the longest period. Therefore, a frequency of 1 Hz will have the longest period among the given options.

Let's quickly verify the periods for each, just for understanding:

• For 10 KHz (10,000 Hz): \( T = \frac{1}{10000} = 0.0001 \) seconds
• For 1 KHz (1,000 Hz): \( T = \frac{1}{1000} = 0.001 \) seconds
• For 10 Hz: \( T = \frac{1}{10} = 0.1 \) seconds
• For 1 Hz: \( T = \frac{1}{1} = 1 \) second

As calculated, 1 Hz results in the longest period, which is 1 second.