A distance of 5.00 cm is measured between two adjacent nodes of a standing wave on a 20.0 cm long string.
A) In which harmonic number is the string vibrating?
B) Find the frequency of this harmonic if the string has a mass of 1.75 x 10-2 kg and a tension of 875 N
To find the answers to these questions, you need to understand the fundamental principles of standing waves and harmonics. Let's break down each question and go through the steps to solve them.
A) In which harmonic number is the string vibrating?
Step 1: Recall the formula relating wavelength, frequency, and speed:
v = f * λ
Where:
v is the speed of the wave
f is the frequency of the wave
λ is the wavelength of the wave
Step 2: The string has a length of 20.0 cm, and there is a distance of 5.00 cm between adjacent nodes. Since adjacent nodes are located at one-half of a wavelength, we can infer that λ = 2 * 5.00 cm = 10.0 cm.
Step 3: We need to convert the length of the string and the wavelength to meters, as the SI unit for the speed of the wave (v) is meters per second. Dividing both measurements by 100 will give us the values in meters:
Length of the string: 20.0 cm / 100 = 0.20 m
Wavelength: 10.0 cm / 100 = 0.10 m
Step 4: Now, substitute the values obtained into the formula:
v = f * λ
f = v / λ
Step 5: The speed of the wave can be determined using the tension (T) and the linear mass density (μ) of the string. Since the question doesn't provide the linear mass density, we need to use another formula:
v = √(T / μ)
Step 6: Rearrange the formula for speed to solve for μ:
μ = T / v²
Step 7: Substitute the values from the question:
T = 875 N
v = √(T / μ)
μ = T / v²
Step 8: Determine the frequency using the obtained values:
f = v / λ
B) Find the frequency of this harmonic if the string has a mass of 1.75 x 10-2 kg and a tension of 875 N
Step 1: Recall the formula for frequency:
f = v / λ
Step 2: The value of v we obtained in the previous question remains the same, so we can directly use it.
Step 3: Substitute the values obtained in the previous question:
f = v / λ
And that's how you break down the problem and solve for the harmonic number and frequency. Remember to substitute the appropriate values into the equations and convert units if necessary.