A rope with length 2.50 m and mass 0.10 kg is stretched and pulled to create transverse waves of frequency 40.0 Hz and wavelength of 0.750 m. How much tension is exerted on the rope?
Question
A rope with length 2.50 m and mass 0.10 kg is stretched and pulled to create transverse waves of frequency 40.0 Hz and wavelength of 0.750 m. How much tension is exerted on the rope?
Solution
To solve this problem, we need to use the formula for the speed of a wave, which is given by:
v = f * λ
where: v is the speed of the wave, f is the frequency of the wave, and λ is the wavelength of the wave.
First, we can find the speed of the wave by substituting the given values into the formula:
v = 40.0 Hz * 0.750 m = 30.0 m/s
Next, we use the formula for the speed of a wave on a stretched string, which is given by:
v = sqrt(T/μ)
where: T is the tension in the string, and μ is the linear mass density of the string.
The linear mass density can be found by dividing the mass of the string by its length:
μ = 0.10 kg / 2.50 m = 0.04 kg/m
We can rearrange the formula for the speed of a wave on a stretched string to solve for the tension:
T = μ * v^2
Substituting the values we have:
T = 0.04 kg/m * (30.0 m/s)^2 = 36.0 N
So, the tension exerted on the rope is 36.0 N.
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