Question

Two piano strings are supposed to be vibrating at 220 Hz, but a piano tuner hears three beats every 2.5 s when they are played together.
(a) If one is vibrating at 220.0 Hz, what must be the frequency of the other (is there only one answer)?
(b) By how much (in percent) must the tension be increased or decreased to bring them in tune?

Answer 1

Answer:

a) 218.8 Hz or 221.2 Hz

b) Increased by 1.087% or decreased by 0.11%

Explanation:

Number of beats = 3

Time = 2.5 s

Frequency of beat = $F_1=\frac{3}{2.5}=1.2\ Hz$

a) If one string is vibrating at 220 Hz, then

$F_1-F_2=220-1.2=218.8 Hz$

                                        or

$F_1+F_2=220+1.2=221.2 Hz$

So, the other string is vibrating at 218.8 Hz or221.2 Hz

b) If frequency is 218.8 Hz

$\frac{F_2}{F_1}=\sqrt{\frac{T_1}{T_2}}\\\Rightarrow \frac{F_2^2}{F_1^2}=\frac{T_1}{T_2}\\\Rightarrow \frac{218.8^2}{220^2}=\frac{T_1}{T_2}\\\Rightarrow \frac{T_1}{T_2}=0.9891$

T, tension must be increased by = 1-0.9891 = 0.0108 = 1.087 %

T, tension must be increased by 1.087%

If frequency is 221.2 Hz

$\frac{221.2^2}{220^2}=\frac{T_1}{T_2}\\\Rightarrow \frac{T_1}{T_2}=1.011$

T, tension must be increased by = 1-1.011 = -0.011 = -0.11 %

T, tension must be decreased by 0.11%