Answer :
Final Answer:
The tension required to tune the 36 cm long violin string to 1000 Hz is approximately 7.85 N.
Explanation:
To find the tension in the violin string, we can use the formula:
[tex]\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \][/tex]
where:
- f is the frequency (1000 Hz),
- L is the length of the string (36 cm or 0.36 m),
- T is the tension in the string (what we're trying to find),
- [tex]\( \mu \)[/tex] is the linear density of the string.
The linear density [tex]\( \mu \)[/tex] is given by the mass per unit length:
[tex]\[ \mu = \frac{m}{L} \][/tex]
Given that the mass (m) is 0.2 g (or 0.0002 kg) and the length (L) is 0.36 m, we can substitute these values into the equation for [tex]\( \mu \)[/tex].
[tex]\[ \mu = \frac{0.0002}{0.36} \][/tex]
Now we can substitute [tex]\( \mu \)[/tex] back into the first equation along with the known values for f and L:
[tex]\[ 1000 = \frac{1}{2 \times 0.36} \sqrt{\frac{T}{0.0002/0.36}} \][/tex]
Solving for T gives us the tension in the string, which is approximately 7.85 N.
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