The lowest musical note on a grand piano has a string frequency of 27.5 Hz. The entire string is 2.00 m long and has a mass of 400. g. The vibrating section of the string is 1.90 m long. What tension force is needed to tune this string properly?

The tension force needed to tune the string properly is 2184.05 N

Explanation:

Frequency is inversely proportional to the length l of the wire and directly proportional to the square root of the tension and it can be shown that the frequency of vibration is given by.

f = (1/2*l) (T/m) ^ (0.5)

Where, T = tension in the string

f = frequency = 27.5 Hz

l = length = 1.9 m

m = mass per unit length

Total string length = 2 m

Mass of string = 400 g

Unit mass = 400 g/2 m = 200 g / m

Therefore

27.5 Hz = 1 / (2*1.9) * (T/0.2) ^ (0.5)

Therefore, T = 2184.05 N

The required tension is 2184.05 N