A student uses a 2.00-m-long steel string with a diameter of 0.90 mm for a standing wave experiment. The tension on the string is tweaked so that the second harmonic of this string vibrates at 22.0Hz . (ρsteel=7.8⋅103 kg/m3)
If you wanted to increase the first harmonic frequency by 40% , what would be the tension in the string?
To find the tension in the string, we can use the equation for the frequency of a standing wave on a string:
f = (n/2L) * √(F/μ),
where f is the frequency, n is the harmonic number, L is the length of the string, F is the tension in the string, and μ is the linear mass density of the string.
Given that the second harmonic frequency is 22.0 Hz, we can substitute the values into the equation and solve for the tension F:
22.0 = (2/2*2.00) * √(F/7.8*10^3).
Simplifying the equation, we have:
22.0 = √(F/7.8*10^3).
To increase the first harmonic frequency by 40%, we need to find the new tension for the first harmonic. Let's call it F'.
The frequency for the first harmonic is given by:
f' = (1/2L) * √(F'/μ).
Since we want to increase the first harmonic frequency by 40%, the new frequency (f'') will be:
f'' = 1.4 * f',
where f' is the original first harmonic frequency.
Substituting the values into the equation, we have:
1.4 * f' = (1/2*2.00) * √(F'/7.8*10^3).
Simplifying the equation, we have:
1.4 * f' = √(F'/7.8*10^3).
Now we can solve for the new tension F':
F' = (1.4 * f')^2 * 7.8*10^3.
Finally, let's substitute the value of f' we obtained from the first equation (22.0 Hz) to find the new tension F':
F' = (1.4 * 22.0)^2 * 7.8*10^3.
Calculating the value of F', we can get the tension in the string when increasing the first harmonic frequency by 40%.