The lowest note on a grand piano has a frequency of 28.1 Hz. The entire string is 2.00 m long and has a mass of 440 g The vibrating section of the string is 1.90 m long. What tension is needed to tune this string properly?
f=1/2L*sqrt(T/lin density)
lin density=m/L
f=1/2L*sqrt(T/(m/L))
Now this is very similar to one I sketched out for you earlier. Try to do it.
ok, so I came out with 293.3N...what am I doing wrong???
To calculate the tension needed to tune the string properly, we can use the formula for the frequency of a vibrating string:
f = (1/2L) * sqrt(T/μ)
where:
f = frequency of the string (in Hz)
L = length of the vibrating section (in m)
T = tension in the string (in N)
μ = linear density of the string (in kg/m)
In this case, we are given:
f = 28.1 Hz
L = 1.90 m
μ = mass of the string / length of the string = 440 g / 2.00 m = 0.22 kg/m
We can rearrange the formula to solve for T:
T = (4L^2 * μ * f^2)
Substituting the given values:
T = (4 * (1.90 m)^2 * 0.22 kg/m * (28.1 Hz)^2)
T ≈ 181.05 N
Therefore, a tension of approximately 181.05 N is needed to tune this string properly.