Video: Determining the Travel Time for a Wave Pulse along a String

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

03:14

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.

Letโ€™s start our solution by highlighting some of the important information weโ€™ve been given. Weโ€™re told that the length of the string is 3.00 meters, which weโ€™ll call ๐ฟ. Weโ€™re also told the string has a total mass of 5.00 grams, which weโ€™ll call ๐‘š. The string is under a tension force of 500.00 newtons, which weโ€™ll Call ๐น sub ๐‘ก. We want to know the time, weโ€™ll call it ๐‘ก, for a pulse to travel the entire length of the string, and weโ€™ll 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 ๐‘ฃ is equal to the square root of the tension force on a string divided by its linear mass density, ๐œ‡. 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 ๐‘ฃ 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 ๐ฟ. So the square root of ๐น sub ๐‘ก over ๐œ‡ is equal to ๐ฟ divided by ๐‘ก. When we rearrange this equation to solve for ๐‘ก, we find that ๐‘ก, 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œ‡ over ๐น sub ๐‘ก.

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

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 ๐‘ก is equal, to three significant figures, to 5.48 milliseconds. Thatโ€™s how long it takes a pulse to travel down the full length of the string.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.