Two strings on a violin are stretched between two points 20.0 cm apart. The mass per length of the strings are the same. If the ratio of the frequencies of the fundamental modes is 1.5 to 1.0, then what is the ratio of the tensions in the strings?
A) 1.22
B) 1.88
C) 1.00
D) 2.25
E) 1.50

Question

Quizplus · Accepted Answer

Let the tension in the string with frequency ratio 1.5 be T1, and the tension in the string with frequency ratio 1.0 be T2.

From the wave equation, we know that the frequency of a standing wave on a string is proportional to the tension in the string and inversely proportional to the length of the string:

f = (1/2L)√(T/μ)

where L is the length of the string, T is the tension, μ is the mass per unit length, and f is the frequency.

Since the strings have the same mass per length, we can write:

f1/f2 = √(T1/T2)

Given that f1/f2 = 1.5/1.0 = 1.5, we can solve for the ratio of tensions:

1.5 = √(T1/T2)

Squaring both sides, we get:

2.25 = T1/T2

Therefore, the ratio of tensions in the strings is 2.25. The answer is (D).

Two Strings on a Violin Are Stretched Between Two Points