Waves on a particular string travel with a speed of 17 m/s. By what factor should the tension in this string be changed to produce waves with a speed of 102 m/s
That is an increase in wave speed by a factor of
102/17 = 6.00
Since wave speed is proportional to the square root of string tension, T must increase by a factor of
6^2 = 36.
To determine the factor by which the tension in the string should be changed to produce waves with a speed of 102 m/s, we can use the relationship between wave speed, tension, and linear density.
The wave speed on a string depends on two properties: the tension in the string (T) and the linear density of the string (μ), given by the formula:
v = sqrt(T/μ)
where v is the wave speed.
In this case, the initial wave speed is 17 m/s, and we need to find the factor by which the tension should be changed to reach a wave speed of 102 m/s.
Let's define the initial tension as T1, and the final tension as T2. The initial wave speed (v1) is given by 17 m/s, and the final wave speed (v2) is 102 m/s. We want to find the ratio (T2/T1).
Using the formula, we can equate the initial and final wave speeds:
sqrt(T1/μ) = 17 m/s
sqrt(T2/μ) = 102 m/s
Now, we can divide the two equations to eliminate μ:
(sqrt(T2/μ)) / (sqrt(T1/μ)) = (102 m/s) / (17 m/s)
Taking the square of both sides to isolate the tensions:
(T2/μ) / (T1/μ) = (102 m/s)^2 / (17 m/s)^2
This simplifies to:
(T2 / T1) = (102 m/s)^2 / (17 m/s)^2
Substituting the values:
(T2 / T1) = (102^2 / 17^2)
Calculating the ratio:
(T2 / T1) = (10404 / 289)
(T2 / T1) ≈ 36
Therefore, to produce waves with a speed of 102 m/s, the tension in the string should be changed by a factor of approximately 36.