A string with a linear mass density of 0.0060 kilograms per meter is tied to the ceiling, and a mass of 20 kilograms is tied to the free end of the string. The string is plucked, sending a pulse down the string. Find the speed of the pulse down the string.
In this statement, we’re told that the linear mass density of the string is 0.0060 kilograms per meter; we’ll call that 𝜇. We’re told that a mass of 20 kilograms is tied to one end of the string; we’ll call that value 𝑚. We want to know when the string is plucked, what is the speed of the pulse that moves down the string, which we’ll call 𝑣?
To begin, let’s draw a diagram of the situation. We have a string with one end tied to a ceiling and the other end fixed to a mass of value 20 kilograms. This mass puts the string under tension. When the string is plucked, a pulse travels down the string with some speed that we’ve called 𝑣, and it’s that speed we want to solve for.
Let’s recall the relationship between speed, tension, and linear mass density, 𝜇. Wave speed, 𝑣, is equal to the square root of the tension force acting on a string along its length divided by its linear mass density, 𝜇. Applying this relationship to our scenario, wave speed 𝑣 is equal to the square root of the tension force which is created by the weight force of the mass.
Therefore, this force is the mass itself multiplied by the acceleration due to gravity, 𝑔. That tension force is divided by 𝜇. We assume here that 𝑔 is exactly 9.8 meters per second squared. We’re given values of 𝑚 and 𝜇 and can plug in for the three variables in this equation. When we compute this square root, we find that the wave speed along the string, to two significant figures, is equal to 180 meters per second. That’s how fast the pulse travels down the string.