A string of mass 2.5kg is under a tension of 200 N. The length of the stretched string is 20m. If the transverse jerk is struck at one end of the string , how long does the disturbance take to reach the other end.

To determine how long it takes for a disturbance to travel along a stretched string, we first need to calculate the wave speed on the string. The speed of a wave on a string can be found using the formula:

Wave Speed Formula

The wave speed (v) is given by the equation:

v = √(T/μ)

where:

• T is the tension in the string (in Newtons)
• μ is the linear mass density of the string (mass per unit length, in kg/m)

Step 1: Calculate the Linear Mass Density

First, we need to find the linear mass density (μ) of the string. This is calculated by dividing the mass of the string by its length:

μ = mass/length

Given that the mass of the string is 2.5 kg and its length is 20 m, we can substitute these values:

μ = 2.5 kg / 20 m = 0.125 kg/m

Step 2: Calculate the Wave Speed

Now that we have μ, we can substitute the values of tension (T = 200 N) and linear mass density (μ = 0.125 kg/m) into the wave speed formula:

v = √(200 N / 0.125 kg/m)

Calculating this gives:

v = √(1600 m²/s²) = 40 m/s

Step 3: Calculate the Time for the Disturbance to Travel

Now that we have the wave speed, we can find the time (t) it takes for the disturbance to travel the length of the string. The time can be calculated using the formula:

t = distance / speed

In this case, the distance is the length of the string (20 m) and the speed is 40 m/s:

t = 20 m / 40 m/s = 0.5 seconds

Final Result

Therefore, the disturbance takes 0.5 seconds to reach the other end of the string. This calculation illustrates how the properties of the string, such as tension and mass, directly influence the speed of wave propagation along it.