A sound wave of a frequency of 2.00 kilohertz is produced by a string oscillating in the 𝑛 equals six mode. The linear mass density of the string is 0.0065 kilograms per metre and the length of the string is 1.50 metres. What is the tension in the string?
Let’s start by highlighting some of the important information we’ve been given. We’re told that the frequency of the sound waves produced by the string and therefore the frequency of the string that oscillates is 2.00 kilohertz. We’ll call that 𝑓. We’re also told that the linear mass density of the string is 0.0065 kilograms per metre, which we’ll call 𝜇. And we know that the length of the string is 1.50 metres, which we’ll call 𝐿. We want to know the string’s tension, which we’ll call 𝐹 sub 𝑡.
To start, let’s draw a diagram of the string oscillating in the 𝑛 equals six mode. When the string is oscillating in the 𝑛 equals six mode, that means three complete wavelengths are on the string from one end to the other. To move ahead in solving for the tension in the string, let’s recall two wave speed equations for waves on a string.
First, the speed of a wave along the string is equal to the square root of the string tension 𝐹 sub 𝑡 divided by the string mass per unit length 𝜇. And second, wave speed in general is equal to frequency times wavelength 𝜆. If we combine these two equations for wave speed, we see that the square root of 𝐹 sub 𝑡 divided by 𝜇 equals wave frequency 𝑓 times wavelength 𝜆. Rearranging this equation to solve for 𝐹 sub 𝑡, the string tension, we find it’s equal to 𝜇 times 𝑓 times 𝜆, where 𝑓 and 𝜆 are both squared.
In our problem statement, we’ve been told both 𝜇 and 𝑓, the frequency. We can solve for 𝜆 by looking back at our diagram and the 𝑛 equals six mode. Three wave lengths are in the string length 𝐿 or three times 𝜆 equals 𝐿. So 𝜆 equals 𝐿 over three or 1.50 metres divided by three, which is 0.50 metres. That’s how long one wavelength is.
Now, we can plug in all three values to our equation for 𝐹 sub 𝑡, the string tension. When we plug in these three values, we make sure to convert the frequency to units of hertz: 2.00 kilohertz equals 2.00 times 10 to the third hertz. When we calculate the tension force 𝐹 sub 𝑡, we find that it’s equal to 6500 newtons. That’s the tension in this vibrating string.