A string of mass 2.5 kg is under a tension of 200 N. The length of the stretched string is 20 m. If a transverse jerk is struck at one end of the string, the time taken for the disturbance to reach the other end is:
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You can combine the two formulas into a single step to minimize calculation steps:
Plugging values directly into this consolidated form:
Step 1: Understand the situation. A jerk at one end of a tight string makes a wave that travels to the other end. We need the time for that wave to cross the full length.
Step 2: Recall the speed of a wave on a string. The speed depends on the tension $T$ and the mass per unit length $\mu$. The formula is $v = \sqrt{\dfrac{T}{\mu}}$. A tighter or lighter string carries the wave faster.
Step 3: Find the mass per unit length. The string has mass $2.5$ kg spread over $20$ m. So \[ \mu = \frac{2.5}{20} = 0.125 \ \text{kg/m}. \]
Step 4: Work out the wave speed. Put the numbers in: \[ v = \sqrt{\frac{200}{0.125}} = \sqrt{1600} = 40 \ \text{m/s}. \]
Step 5: Use speed, distance and time. Time equals distance divided by speed. The distance is the length $20$ m. \[ t = \frac{20}{40}. \]
Step 6: State the answer. Dividing gives the travel time. \[ \boxed{0.5 \ \text{s}} \]