### Video Transcript

A string is 3.00 meters long with a mass of 5.00 grams. The string is held taut with a tension of 500.00 newtons applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 meters of the string? Give your answer in milliseconds.

Letโs start our solution by highlighting some of the important information weโve been given. Weโre told that the length of the string is 3.00 meters, which weโll call ๐ฟ. Weโre also told the string has a total mass of 5.00 grams, which weโll call ๐. The string is under a tension force of 500.00 newtons, which weโll Call ๐น sub ๐ก. We want to know the time, weโll call it ๐ก, for a pulse to travel the entire length of the string, and weโll give our answer in milliseconds.

We can move toward our solution by recalling an equation for wave speed that depends on string tension and string mass per unit length: wave speed ๐ฃ is equal to the square root of the tension force on a string divided by its linear mass density, ๐. When we apply that relationship to our situation, we can connect it with another relationship for speed. Recall that, in general, the speed of an object ๐ฃ is equal to the distance it travels divided by the time it takes to travel that distance.

In our case, the distance the pulse travels is the length of the string ๐ฟ. So the square root of ๐น sub ๐ก over ๐ is equal to ๐ฟ divided by ๐ก. When we rearrange this equation to solve for ๐ก, we find that ๐ก, the time it takes the pulse to move down the length of the string, is equal to that length times the square root of ๐ over ๐น sub ๐ก.

Now in the problem statement, weโve been given values for ๐ฟ and ๐น sub ๐ก, but what about ๐? Recall that ๐ has units of kilograms per meter, and it is called the linear mass density of the string. So from ๐, we can substitute the mass of the string divided by its length. Algebraically, this simplifies our expression for the time ๐ก, which we can now write as the square root of ๐, the mass, times ๐ฟ, the length of the string, divided by the tension force acting on it, ๐น sub ๐ก.

When we plug in the values given for these three variables, converting the mass in grams to a mass in kilograms and plugging these values into our calculator, we find that the time ๐ก is equal, to three significant figures, to 5.48 milliseconds. Thatโs how long it takes a pulse to travel down the full length of the string.