Problem 6 - AP Physics 1 with 2 Practice Test 2nd

Question

What is the period of a wave that has a frequency of \(12,000 \mathrm{~Hz} ?\) (A) \(36.25 \mathrm{~s}\) (B) \(0.000083 \mathrm{~s}\) (C) \(0.0275 \mathrm{~s}\) (D) \(12,000 \mathrm{~s}\)

Step-by-Step Solution

Answer

(B) 0.000083 seconds

Step 1: Understand the relationship between period and frequency

The period (\tau) of a wave is the reciprocal of its frequency (f). The formula to find the period is \ \[ \tau = \frac{1}{f} \]. Since frequency is given in Hertz (Hz), which is equivalent to cycles per second, the period will be in seconds (s).

Step 2: Substitute the given frequency into the formula

The given frequency is \(12,000 \; \text{Hz}\). Substitute this value into the formula: \ \[ \tau = \frac{1}{12,000} \]

Step 3: Compute the period

Calculate the reciprocal of \(12,000\): \ \[ \tau = \frac{1}{12,000} \approx 0.000083 \; \text{s} \]

Step 4: Match the computed period with the correct option

From calculating, the period is approximately \(0.000083 \; \text{s}\). Therefore, the correct option is (B) \( 0.000083 \mathrm{~s} \).

Key Concepts

wave frequency

Wave frequency is a measure of how often the wave oscillates over a given period. It is typically expressed in Hertz (Hz), which translates to the number of cycles per second. In the context of the given problem, the wave has a frequency of 12,000 Hz, meaning it completes 12,000 cycles every second. When you understand frequency, you're essentially grasping how 'frequent' the wave repeats its pattern in one second. This concept is important for various applications such as sound, light, and radio waves. Higher frequency means more cycles per second, while lower frequency means fewer cycles per second. To find the period of a wave, we need to understand the reciprocal relationship with its frequency. As you proceed, remember this fundamental definition: frequency tells you how fast the wave cycles happen over time.

reciprocal relationship

The reciprocal relationship between frequency and period is crucial in wave calculations. The period (\tau) of a wave is the time it takes for one cycle to complete. The formula connecting period and frequency (f) is simple: \[ \tau = \frac{1}{f} \] This means that if you know the frequency of a wave, you can easily find its period by taking the reciprocal of that frequency. In our exercise, we have a frequency of 12,000 Hz. To find the period, we use the formula as follows: \[ \tau = \frac{1}{12,000} \] Doing the math, we get \[ \tau \approx 0.000083 \, \text{s} \] This relationship shows how inversely the period and frequency interact. As frequency increases, the period decreases, and vice versa. The reciprocal relationship makes it easier to transition between understanding how often a wave occurs to how long each wave cycle takes.

unit conversion

Understanding unit conversion is essential when dealing with measurements. In our problem, frequency is given in Hertz (Hz), which stands for cycles per second. When we calculate the period, the resulting unit is seconds (s) because we are figuring out the time for one complete cycle. No additional conversion is necessary in this example because we're working within compatible units. However, in other scenarios, you might need to convert between different units of time such as milliseconds (ms) or microseconds (µs). For instance, \[ 1 \, \text{s} = 1,000 \, \text{ms} \] In this case, \[ 0.000083 \, \text{s} \approx 83 \, \text{\mu s} \] Learning how to switch between units ensures accuracy in your calculations and helps you apply these concepts in various real-world contexts. Always double-check that your units match up at each step in your calculations.