A string has a linear density of 6.2 x 10-3 kg/m and is under a tension of 250 N. The string is 1.2 m long, is fixed at both ends, and is vibrating in the standing wave pattern shown in the drawing. Determine the (a) speed, (b) wavelength, and (c) frequency of the traveling waves that make up the standing wave.

Question

A string has a linear density of 6.2 x 10-3 kg/m and is under a tension of 250 N. The string is 1.2 m long, is fixed at both ends, and is vibrating in the standing wave pattern shown in the drawing. Determine the (a) speed, (b) wavelength, and (c) frequency of the traveling waves that make up the standing wave.

Answer 1

I can't see your drawing but if there are no nods of standing wave between the ends

λ =2L
v = sqrt(T/m₀)
λ =v/f => f=v/ λ

Answer 2

To determine the speed of the traveling waves that make up the standing wave, we can use the formula:

v = √(T/μ)

Where:
- v is the speed of the waves
- T is the tension of the string
- μ is the linear density of the string

Given:
- T = 250 N (tension of the string)
- μ = 6.2 x 10^(-3) kg/m (linear density of the string)

Substituting the values into the formula, we have:

v = √(250 N / 6.2 x 10^(-3) kg/m)

Now, let's calculate the speed.

v = √(250 / 6.2 x 10^(-3))

To simplify the calculation, we can express the scientific notation in decimal form.

v = √(250 / 0.0062) [Dividing 6.2 x 10^(-3) by 10^(-3) gives 0.0062]

v = √40322.58

v ≈ 200.8 m/s

Therefore, the speed of the traveling waves making up the standing wave is approximately 200.8 m/s.

Moving on to calculate the wavelength of the standing wave.

In a standing wave, there are two times the length of a full wavelength present. Thus, the length of the standing wave is equal to twice the length of the string. Therefore, the wavelength (λ) can be calculated using:

λ = 2L

Given:
- L = 1.2 m (length of the string)

Substituting the value, we have:

λ = 2 x 1.2

λ = 2.4 m

Therefore, the wavelength of the standing wave is 2.4 m.

Finally, let's calculate the frequency of the standing wave using the following formula:

v = fλ

Where:
- v is the speed of the wave
- f is the frequency of the wave
- λ is the wavelength of the wave

Rearranging the formula, we have:

f = v / λ

Given:
- v = 200.8 m/s (speed of the waves)
- λ = 2.4 m (wavelength)

Substituting the values, we have:

f = 200.8 / 2.4

f ≈ 83.7 Hz

Therefore, the frequency of the standing wave is approximately 83.7 Hz.