### Video Transcript

A string has a linear mass density π equals 0.00700 kilograms per meter, a length πΏ equals 0.700 meters, a tension of πΉ sub π equals 110 newtons and oscillates in a mode π equals three. What is the frequency of the oscillations?

Weβre told in this problem statement that π is 0.00700 kilograms per meter, that the string has a length of 0.700 meters, that itβs under a tension of 110 newtons, and that it oscillates in the mode π equals three. We want to solve for the oscillation frequency, which weβll call π.

To start off, letβs draw a diagram of the scenario. In this situation, we have a string fixed at both ends. The string is under tension πΉ sub π, and it oscillates in the π equals three mode, which means that from one end to the other they are one and a half wavelengths on the string.

To begin solving for the frequency, letβs recall two equations for the speed of a wave. First when a wave is on a string, the wave speed π is equal to the square root of the tension that the string is under, πΉ sub π, divided by the stringβs mass per unit length, π. And, second we recall that in general for a wave, the wave speed π is equal to the wave frequency π times wavelength π.

When we apply these two relationships to our situation, we see they can be combined to write that the square root of πΉ sub π over π equals π times π or π equals one over π times the square root of πΉ sub π over π. If we look again in our diagram, we can see that thereβs a relationship between the length of the string πΏ and the wavelength π.

Because one and a half wavelengths fit along the string at any given time in this mode, πΏ is equal to three halves π or π equals two πΏ divided by three. We can substitute this expression for π into our equation for frequency.

Since weβve been given πΏ, π, and πΉ sub π in the problem statement, we can plug in for those values now. When we enter these values into a calculator, we find that, to three significant figures, the frequency of oscillation is 269 hertz. Thatβs how many wavelengths of this wave pass a given point on the string every second.