Problem 3 - AP Physics 1 with 2 Practice Test 2nd

Question

What is the wavelength of sound produced at a frequency of \(300 \mathrm{~Hz}\) when the air temperature is \(20^{\circ} \mathrm{C} ?\) (A) \(1.10 \mathrm{~m}\) (B) \(1.30 \mathrm{~m}\) (C) \(0.80 \mathrm{~m}\) (D) \(1.14 \mathrm{~m}\)

Step-by-Step Solution

Answer

The wavelength of sound at 300 Hz in air at 20°C is 1.14 m.

Step 1: Identify the given values

Given are the frequency of sound, which is 300 Hz, and the air temperature, which is 20°C.

Step 2: Determine the speed of sound in air at 20°C

The speed of sound in air can be calculated using the formula: \[\text{speed of sound} = 331.4 + 0.6 \times T\] where T is the temperature in Celsius. Substituting 20°C, we get: \[\text{speed of sound} = 331.4 + 0.6 \times 20 = 331.4 + 12 = 343.4 \text{ m/s} \]

Step 3: Use the formula to calculate the wavelength

The wavelength ( \(\text{λ}\) ) of the sound can be calculated using the formula: \[\text{λ} = \frac{\text{speed of sound}}{\text{frequency}}\] Thus, substituting the values, we get: \[\text{λ} = \frac{343.4 \text{ m/s}}{300 \text{ Hz}} \] \[\text{λ} ≈ 1.145 \text{ m}\]

Step 4: Choose the correct answer

The closest value to our calculated wavelength of 1.145 m is option (D) 1.14 m.

Key Concepts

frequency

In physics, frequency refers to the number of waves that pass a particular point per unit of time. It is denoted by the symbol \(f\). The common unit for frequency is Hertz (Hz), which signifies one cycle per second.
For sound waves, frequency determines the pitch of the sound. Higher frequencies correspond to higher pitches, while lower frequencies correspond to lower pitches. In everyday life, frequency is the reason why musical notes have different tones.
For instance, in our problem, the given frequency is \(300 \text{ Hz}\). This means that 300 sound waves pass through a point every second. This frequency corresponds to a medium-pitched sound.
Understanding frequency is essential to solving problems involving wave properties such as wavelength and speed.

speed of sound

The speed of sound refers to how quickly a sound wave travels through a medium. This speed varies depending on several factors such as the medium (air, water, steel) and the temperature of the medium.
At room temperature (around \(20^{\text{°C}}\)), the speed of sound in air is approximately \(343 \text{ m/s}\). The speed can be calculated using the formula: \[\text{speed of sound} = 331.4 + 0.6 \times T\], where \(T\) is the temperature in Celsius.
In the problem at hand, given the air temperature of \(20^\text{°C}\), the speed of sound is: \[\text{speed of sound} = 331.4 + 0.6 \times 20 = 343.4 \text{ m/s}\]
It's important to remember that as the temperature increases, the speed of sound in air also increases, and vice versa. This connection helps explain why sound travels faster on a warm day than on a cold day.

wavelength formula

The wavelength of a sound wave is the distance between successive crests (or troughs) of the wave. It's denoted by the symbol \( \text{λ} \).
To find the wavelength, we use the formula: \[ \text{λ} = \frac{\text{speed of sound}}{\text{frequency}} \]
In our exercise, we know the speed of sound (\(\text{343.4 m/s}\)) and the frequency (\(300 \text{Hz}\)). Using the formula, we calculate: \[ \text{λ} = \frac{343.4 \text{ m/s}}{300 \text{ Hz}} \text{λ} ≈ 1.145 \text{ m} \]
This calculation gives us the wavelength as approximately 1.145 meters. The closest match among the options is \(1.14 \text{ m}\), which confirms that the correct choice is option (D).
Knowing how to use the wavelength formula is crucial for solving problems in wave physics. It allows us to understand the relationship between speed, frequency, and wavelength thoroughly.